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Eindhoven University of Technology

MASTER

Evolutions and construction of wavelet transform on the similitude group

Sharma, U.

Award date:2012

Link to publication

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https://research.tue.nl/en/studentthesis/evolutions-and-construction-of-wavelet-transform-on-the-similitude-group(6a0397eb-46d6-47fd-90d8-ba0f705298da).html

Technische Universiteit Eindhoven

Department of Mathematics and Computer Science

Evolutions and Construction of Wavelet

Transform on the Similitude Group

Upanshu Sharma

Supervisor : dr. ir. R. Duits

Eindhoven: July 2012

Abstract

Enhancement of structures in noisy image data is relevant for many biomedical applicationsbut most commonly used enhancement techniques fail at crossings in an image.Utilizing the inherent symmetry in images a generic invertible wavelet transform on the 2DSimilitude group (SIM(2)) is constructed. This transforms an image into it’s scale orientationscore (scale-OS) which allows to differentiate between different scales and orientations in an im-age. Unlike some wavelet transform based techniques this method allows stable reconstructionof an image from the corresponding wavelet transform. Appropriate contextual-enhancementoperators on an image can be constructed by using the group structure of the scale-OS. This isdone via left-invariant diffusions in the wavelet domain (unlike the usual wavelet thresholding).It is shown that left-invariant operators on the scale-OS are Euclidean and scaling invariant,i.e. the result of these operators does not change under the translation, rotation and scaling ofthe image. Hence a framework of well-posed operators on an image is provided.Sharp Gaussian estimates for Green’s function of linear left-invariant diffusion on the SIM(2)group are presented by using subcoercive operators and it is illustrated that even linear diffu-sion in combination with greyvalue transforms successfully handles crossing curves.By using the Cartan connection and a left-invariant metric on SIM(2), general (non)lineardiffusion equation in covariant form is presented and it is shown that diffusion in this casetakes place only along exponential curves.Finally, nonlinear adaptive diffusion defined on the Euclidean motion group (SE(2)) known asCED-OS, is extended to multiple orientation scores and the improvements on (medical) imagesare exhibited.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 PDEs in Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Curvelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Invertible Orientation Score . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 In the context of related work . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Content and Structure of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Generalized Wavelet Transform on the Similitude Group 11

2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Unitary maps from a Hilbert space H to a functional Hilbert space CIK . . . . 13

2.3 Functional Hilbert spaces on Groups . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Wavelet Transformations constructed by Unitary Irreducible representa-tions of Locally Compact Groups . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Generalized Wavelet Transformation on Locally Compact Groups . . . . . . . . 16

2.5 The Discrete Analogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Stable reconstruction of image f from its orientation score Uf . . . . . . 20

2.5.2 Quantification of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.6 Design of proper wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The Similitude Group 29

iii

3.1 Geometry of the SIM(2) group . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.1 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 SIM(2) Group Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.3 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.4 Left invariant Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1.5 Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 SIM(2)-Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 SIM(2)-convolution vs SIM(2)-Correlation . . . . . . . . . . . . . . . . 39

4 Image Enhancement via Left Invariant Operations on Scale-OS 41

4.1 Legal Operators on the Scale-OS . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Left Invariant Linear Bounded Operations are Group Convolutions . . . . . . . 43

4.3 Basic Left Invariant Operations on Scale-OS . . . . . . . . . . . . . . . . . . . . 44

4.4 Introduction to Diffusions on SIM(2) group . . . . . . . . . . . . . . . . . . . . 45

4.5 Approximate Contour Enhancement Kernels on Scale-OS . . . . . . . . . . . . 47

4.5.1 Approximation of SIM(2) by a Nilpotent Group via Contraction . . . . 47

4.5.2 Gaussian Estimates for the Heat-kernels on SIM(2) . . . . . . . . . . . 49

4.6 Nonlinear Left Invariant Diffusions on SIM(2) . . . . . . . . . . . . . . . . . . 51

4.6.1 Design of the metric on SIM(2) . . . . . . . . . . . . . . . . . . . . . . 53

4.6.2 Extraction of spatial curvature from scale-OS . . . . . . . . . . . . . . . 59

4.6.3 Adaptive diffusion on Scale-OS . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Adaptive SE(2) Diffusion on Scale-OS . . . . . . . . . . . . . . . . . . . . . . . 62

4.7.1 CED-OS Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.7.2 Advantage of scale adaptive diffusion . . . . . . . . . . . . . . . . . . . . 65

4.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 Summary and Future Research 75

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Acknowledgements 76

Appendix A 79

A.1 Results from Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.2 Derivation for basis of Lie Algebra . . . . . . . . . . . . . . . . . . . . . . . . . 81

A.3 Left-Invariant Differential Operators on SE(2)- An intuitive explanation . . . . 81

A.4 Exponential map on SIM(2) is surjective . . . . . . . . . . . . . . . . . . . . . 82

A.5 Implementation of G-convolution, G = R2 n SO(2) . . . . . . . . . . . . . . . . 84

A.5.1 Implementation of SE(2)-convolution via Discretization . . . . . . . . . 84

A.5.2 Implementation of SE(2)-convolution via Steerability . . . . . . . . . . 85

Appendix B 87

B.1 Regularized Derivatives and Local Features . . . . . . . . . . . . . . . . . . . . 87

B.1.1 General Regularized Derivatives in Scale-OS . . . . . . . . . . . . . . . . 88

B.1.2 Gaussian Derivatives in Scale-OS . . . . . . . . . . . . . . . . . . . . . . 88

B.2 Horizontal Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Bibliography 92

List of Figures

1.1 Medical images with crossing elongated curves . . . . . . . . . . . . . . . . . . 2

1.2 Comparison of enhancement via different non linear diffusion techniques . . . . 3

1.3 Illustration of curvelet construction . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Activation patterns in the primary visual cortex of a tree shrew . . . . . . . . . 6

1.5 Intuitive illustration of a orientation score . . . . . . . . . . . . . . . . . . . . . 7

2.1 B-splines in the log-scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 Cake-kernel description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Mψ for the cake-kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Invertible nature of wavelet transform . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Reconstruction via summing over scales . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Scale-OS of Retinal image for fixed orientation . . . . . . . . . . . . . . . . . . 27

2.7 Scale-OS of the Retinal image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Difference between cartesian and left-invariant derivatives . . . . . . . . . . . . 34

3.2 Schematic view of a SE(2)-convolution and an R3-convolution . . . . . . . . . 39

3.3 Difference between a kernel and it’s ”mirrored” version . . . . . . . . . . . . . . 40

4.1 Schematic view of image processing via invertible scale-OS . . . . . . . . . . . . 42

4.2 Image processing on scale-OS via normalization . . . . . . . . . . . . . . . . . . 46

4.3 Image processing on scale-OS via Monotonic Grey-value transformations . . . . 46

4.4 Enhancement kernel via Gaussian estimates at different scales and orientations 51

4.5 Comparison of enhancement via Gaussian estimates for heat kernels . . . . . . 52

vii

4.6 Enhancement via Gaussian estimates . . . . . . . . . . . . . . . . . . . . . . . . 52

4.7 Numerical schemes used for CED-OS . . . . . . . . . . . . . . . . . . . . . . . . 64

4.8 Scale invariance of left-invariant operators on scale-OS . . . . . . . . . . . . . . 65

4.9 Disadvantage of scale-invariance on medical images . . . . . . . . . . . . . . . . 66

4.10 Idea behind CED-OS-I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Idea behind CED-OS-II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.12 Comparison between linear and nonlinear diffusion on scale-OS (test-image) . . 68

4.13 Comparison between linear and nonlinear diffusion on scale-OS (Retinal image) 68

4.14 CED-OS on SE(2)-OS and scale-OS for test-image with same scale structure . 69

4.15 CED-OS on SE(2)-OS and scale-OS of a multi-scale test-image . . . . . . . . . 70

4.16 CED-OS on SE(2)-OS and scale-OS of a collagen tissue . . . . . . . . . . . . . 71

4.17 Comparision between CED-OS on SE(2)-OS and scale-OS of X-ray catheters . 72

4.18 Comparision between CED-OS on SE(2)-OS and scale-OS of bone-microscopyimage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.1 Intuitive explanation of left-invariant tangent vectors and differential operators 82

“...all actions in their entirety culminate in knowledge”.

- The Bhagavad Gita

Chapter 1

Introduction

1.1 Background

Elongated structures such as fibres and blood vessels are commonly found in the human bodyand often require analysis for diagnostic purposes. To this end many medical imaging tech-niques exist to acquire an image of these structures. Often used techniques include magneticresonance imaging (MRI), microscopy, X-ray flouroscopy, fundus imaging etc.

Many (bio)medical questions related to such images require the detection and tracking of theelongated structures. Usually the medical images acquired are noisy and therefore enhancementis an important preprocessing step for subsequent feature detection. Enhancement of noisyimages has been a topic of successful research for several years. However most methods fail toconsistently handle elongated structures that cross each other in an image, since it is usuallyassumed that at any position in the image there is either no elongated structure or a singleelongated structure. The recent framework of ”Invertible Orientation Scores” (OS), on whichthis study is based, developed by Duits [12] in collaboration with Almsick [39] and used forenhancement of elongated structures by Franken [21] successfully addresses this issue.

The OS framework does not include the notion of scales in an image. In this study we explicitlyinclude the notion of scaling in the construction of OS and develop mathematical methods forenhancement based on this modified OS. Images in Figure 1.1 illustrate some typical medi-cal images containing elongated crossing structures, suggesting that appropriate handling ofcrossings is relevant.

1.2 Literature Survey

In this section we briefly review a few recent and relevant techniques in image analysis, adaptivediffusion techniques by Perona and Malik [30] and Weickert [41], curvelets by Donoho andCandes [9, 10] and the invertible orientation score framework by Duits [15, 17]. We concludethis section by explaining how our work relates to this existing literature.

1

2 Introduction

Figure 1.1: Medical images with crossing elongated curves. Top left: section of a retinal fundusimage; top right: microscopy image of bone tissue; bottom left: X-ray catheters; bottom right:collagen tissue of a tissue engineered heart valve

1.2.1 PDEs in Image Denoising

Partial differential equations (PDEs) have led to an entire new field in image processing andcomputer vision as they offer several advantages:

• Deep mathematical results with respect to well-posedness are available, such that stablealgorithms can be found.

• They allow a reinterpretation of several classical methods under a novel unifying frame-work. This includes many well-known techniques such as Gaussian convolution, medianfiltering, dilation or erosion.

• The PDE formulation is genuinely continuous. Thus, their approximations aim to beindependent of the underlying grid and may reveal good rotational invariance, an oftenimportant requirement.

PDE-based image processing techniques are mainly used for smoothing and restoration pur-poses. Many evolution equations for restoring images can be derived as gradient descentmethods for minimizing a suitable energy functional, and the restored image is given by thesteady-state of this process. Typical PDE techniques for image smoothing regard the originalimage as initial state of a parabolic (diffusion-like) process, and extract filtered versions fromits temporal evolution. This whole evolution is called a scale-space.

Literature Survey 3

Figure 1.2: From left to right: input image f of the famous Van Gogh self portrait; computedon comparable slices uf (·, s) in a linear scale representation C = 1; Perona and Malik nonlinearscale space representation [30]; Weickert’s coherence-enhancing diffusion [41]

Mathematically, a scale space representation uf : Rd × R+ → R of an image f : Rd → R isobtained by solving an evolution equation on the additive group (Rd,+). The most commonevolution equation, in image analysis, is the diffusion equation

∂suf (x, s) = ∇x · (C(uf (·, s))(x)∇xuf )(x, s)

uf (x, 0) = f(x),

where C : L2(R2) ∩ C2(R2) → C2(R2) is a function which takes care of adaptive conductiv-ity; that is C(uf (·, s))(x) models the conductivity depending on the differential structure at(x, s, uf (x, s)). If C = 1 the solution is given by convolution uf (x, s) = (Gs ∗ f)(x) with a

Gaussian kernel Gs(x) = 1(4πs)d/2

e−‖x‖4s with scale s = 1

2σ2 > 0.

A special class of parabolic equations called nonlinear diffusion filters bridge the gap betweenscale-space and restoration ideas. Methods of this type have been proposed for the first time byPerona and Malik [30]. In order to smooth an image and to simultaneously enhance importantfeatures such as edges, they apply a diffusion process whose diffusivity is steered by derivativesof the evolving image. They pointed out that nonlinear adaptive isotropic diffusion is achievedby replacing C = 1 by C(uf (·, s))(x) = c(‖∇xuf (x, s)‖), where c : R+ → R+ is some smoothstrictly decaying positive function vanishing at infinity. The idea behind this choice is thatdiffusion should not occur when the (local) gradient value is large. By restricting to c > 0 oneensures that the diffusion is always forward and thus ill-posed backward diffusion is avoided.The common choices are,

c(t) = e− c

(λ/t)2p , c(t) =1

(t/λ)2p + 1, (1.1)

involving parameters p > 1/2, λ > 0. A further improvement of the Perona and Malik schemeis the ”coherence-enhancing diffusion” (CED) proposed by Weickert [41], which also uses thedirection of the gradient ∇xuf . The diffusion constant c is now replaced by a diffusion matrix:

S(uf (·, s))(x) = (Gσ ∗ ∇uf (·, s)(∇uf (·, s))T )(x), (1.2)

S(uf (·, s))(x) = αI + (1− α)e− c

(λ1(S(uf (·,s))(x))−(λ2(S(uf (·,s))(x)))2 e2(S(uf (·, s))(x))eT2 (S(uf (·, s))(x)),

where α ∈ (0, 1), c > 0, σ > 0 are parameters and S with eigenvalues λi(S(uf (·, s))(x)i=1,2

is used to get a measure of local anisotropy e− c

(λ1(S(uf (·,s))(x))−(λ2(S(uf (·,s))(x)))2 together with theorientation estimate e2(S(uf (·, s))(x)), which is the eigenvector of S corresponding to smallesteigenvalue. Figure(1.2) illustrates the visually appealing results due to nonlinear diffusions.However this method fails in image analysis applications with crossing curves as the directionof gradient at crossing locations is ill-defined, [16].

4 Introduction

Figure 1.3: Illustration of curvelet transform construction in Fourier domain. From left toright: basic wavelet at a particular scale a, γa00; support of basic wavelet; tiling of the frequencydomain into wedges for curvelet construction. Figures extracted from [28].

1.2.2 Curvelets

The discrete wavelet transform is a widely used tool for Mathematical analysis and signalprocessing, but has the disadvantage of poor directionality, which undermines it’s usage inimage denoising applications. In 1999, an anisotropic geometric wavelet transform, namedridgelet transform (also called first-generation curvelet transform), was proposed by Candesand Donoho [8]. Later, a considerably simpler second-generation curvelet transform based ona frequency partition technique was proposed by the same authors, see [9, 10]. The second-generation curvelet transform has been successfully used for many different applications, see[28] for more details. Here we briefly summarize the continuous curvelet transform (CCT) inR2 as presented in [9].

Let x and ξ denote the spatial and frequency variables respectively and (ρ, ϕ) the polar coordi-nates in the frequency domain. Consider a pair of windows W (ρ) and V (ϕ) (radial and angularwindows respectively) which are smooth, non-negative and real valued, with W taking positivereal arguments and supported on ρ ∈ (−1/2, 2) and V taking real arguments and supportedon ϕ ∈ [−1, 1]. These windows obey the admissibility conditions

∞∫0

W (aρ2)da

a= 1 ∀ρ > 0 ,

1∫−1

(V (u))2du = 1. (1.3)

The windows are used in the frequency domain to construct a family of analyzing elementswith three parameters: scale a > 0, location b ∈ R2 and orientation θ ∈ [0, 2π). At scale a,the family is generated by translation and rotation of a basic element γa00:

γabθ(x) = γa00(Rθ(x− b)) = γ100(Rθ(a−1(x− b)))

where Rθ is the planar rotation matrix effecting rotation by θ radians. The generating elementat scale a is defined in terms of Fourier coordinates (ρ, ϕ) by setting,

γa00(ρ, ϕ) = W (a.ρ).V (ϕ/√a).a3/4, 0 < a < a0. (1.4)

Therefore the support of each γa00 is a polar wedge defined by the support of W and V appliedwith scale-dependant window widths in each direction. Note that these curvelets are highlyoriented. Figure1.3 illustrates the idea of this construction graphically.

Equipped with this family of curvelets a continuous curvelet transform Γf , a function onscale/location/direction space, is defined by

Γf (a,b, θ) = 〈γa,b,θ, f〉, a < a0,b ∈ R2, θ ∈ [0, 2π). (1.5)

Literature Survey 5

Note that a0 is a fixed number - the coarsest scale in the problem. It is fixed once and forall and must obey a0 < π2 for the above construction to work properly. Below we give animportant result which illustrates the invertible nature of the curvelet transform.

Theorem. Let f ∈ L2(R2) have a Fourier transform vanishing for ξ < 2/a0. Let the V and Wobey the admissibility condition (1.3). For such high-frequency functions we have a reproducingformula,

f(x) =

∫Γf (a,b, θ)γa,b,θ(x)µ(da, db, dθ),

and a Parseval formula

‖f‖L2=

∫|Γf (a,b, θ)γa,b,θ|2µ(da, db, dθ),

where in both cases, µ denotes the reference measure dµ = 1a3 dadbdθ.

This transform can be extended to functions containing low frequencies using a single coarse-scale isotropic wavelet.

Note that the Curvelet transform samples g 7→ (Ugψ, f) with ψ = γ100 and Ugψ(x) =ψ( 1

aRθ(x − b)) where g = (x, a, θ) and so the operators on the curvelet transform shouldbe left-invariant. However due to the discrete construction it is not straightforward to guar-antee left-invariance of the operators on the curvelet transform. Thereby it is not possible toensure that the effective operator in the image domain are Euclidean invariant, whereas in ourcase this is ensured, see Section 4.1.

1.2.3 Invertible Orientation Score

The idea of orientations score (OS) is inspired by the early visual system of mammals, inwhich receptive fields exist that are tuned to various locations and orientations. Thereby asimple cell receptive field can be parametrized by its position and orientations. Figure 1.4illustrates this phenomenon. In essence the primary visual cortex ”transforms” the perceivedimage, which can be seen as a map of luminance values for each spatial positions (x, y), to athree dimensional representations, i.e. a map of orientation confidence for each position andorientation (x, y, θ). This redundant representation of the image simplifies subsequent analysisin higher visual cortex regions. We briefly explain the concept of invertible OS developed in[12, 15]. The Euclidean motion group (i.e. the group of planar rotations and translations)SE(2) is defined as SE(2) = R2 oSO(2) where SO(2) is the group of planar rotations. An OSUf : SE(2) → C of an image f : R2 → R is obtained by means of an anisotropic convolutionkernel ψ : R2 → C via

Uf (g) =

∫R2

ψ(R−1θ (y − x))f(y)dy, g = (x, θ) ∈ SE(2),

where ψ(−x) = ψ(x) and Rθ ∈ SO(2) is the 2D counter-clockwise rotation matrix. Assumeψ ∈ L2(R2), then the transform Wψ which maps images f ∈ L2(R2) can be rewritten as

Uf (g) = (Wψf)(g) = (Ugψ, f)L2(R2),

where g 7→ Ug is a unitary (group-)representation of the Euclidean motion group SE(2) intoL2(R2) given by Ugf(y) = f(R−1

θ (y − x)) for all g = (x, θ) ∈ SE(2) and for all f ∈ L2(R2).The general wavelet reconstruction results from [1, 23] are not applicable to the transform

6 Introduction

Figure 1.4: Activation patterns in the primary visual cortex of a tree shrew, shown for twodifferent orientation stimuli that were presented to the tree shrew. The orientation sensitivityof the different cells is color-coded. Figures extracted from [7].

f 7→ Uf , since the representation U is reducible. To accommodate reducible representations,in [11] a more general wavelet transform is proposed. With this wavelet transform the OS,Uf : SE(2) → C is constructed by means of an admissible vector ψ ∈ L(R2) such that the

transform Wψ is unitary onto the unique reproducing kernel Hilbert space CSE(2)K of functions

on SE(2) with reproducing kernel K(g, h) = (Ugψ,Uhψ), which is a closed vector subspace ofL2(SE(2)). For the abstract construction of the unique reproducing kernel space CI

K on a setI see [5]. Note that,

(Wψf)(x, θ) = (Ux,θψ, f)L2(R2) = (FTxRθψ,Ff)L2(R2) = F−1(RθFψ · Ff)

where the rotation and translation operators on L2(R2) are defined by Rθf(y) = f(R−1θ y)

and Txf(y) = f(y − x). This leads to the essential Plancherel formula,

‖Wψf‖2CSE(2)K

=

∫R2

2π∫0

|(FWψf)(ω, θ)|2 1

Mψ(ω)dωdθ

=

∫R2

2π∫0

|(Ff)(ω)|2|Fψ(RTθ ω)|2 1

Mψ(ω)dωdθ

=

∫R2

|(Ff)(ω)|2dω = ‖f‖L2(R2),

where Mψ ∈ C(R2,R) is given by Mψ(ω) =2π∫0

|Fψ(RTθ ω)|2dθ. If ψ is chosen such that Mψ = 1

then we gain L2 norm preservation. But this is not possible as ψ ∈ L2(R2) ∩ L1(R2) impliesthat Mψ is a continuous function vanishing at infinity. In practice, however, because of finitegrid sampling, U is restricted to the space of disc limited images,

L%2(R2) = f ∈ L2(R2)| supp(Ff) ⊂ B0,%.

Since the wavelet transform Wψ maps the space of images L2(R2) unitarily onto the space

Literature Survey 7

Figure 1.5: Illustration of the orientation score construction of an image containing a singlecircle. Since the orientation changes linearly as one traverses the circle, the resulting responsein a 3D orientation score is a spiral. Note that the orientation dimension (displayed vertically)is 2π periodic.

of orientation scores CSE(2)K (provided Mψ > 0) the original image f : R2 → R can be re-

constructed from it’s orientation score Uf : SE(2) → C by using the adjoint of the wavelettransform,

f =W∗ψWψ[f ] = F−1

ω 7→ 2π∫0

F [Uf (·, θ)](ω)M−1ψ (ω)dθ

.Remarks:

• With restriction to L%2(R2), Mψ = 1 implies equality of CSE(2)K and L2(SE(2)) norms i.e.

‖U‖CSE(2)K

= ‖U‖L2(SE(2)) for all U ∈ CSE(2)K .

• With restriction to L%2(R2), Mψ = 1 means that ψ has equal length over each irreduciblesubspace of L%2(R2) so that each irreducible subspace of L%2(R2) is mapped unitarily toeach irreducible subspace of L%2(SE(2)) = u ∈ L%2(SE(2))| u(·, θ) ∈ L%2(R2),∀θ ∈ [0, 2π)and the unitarity result is due to Plancherel formula on SE(2). See [17, App.A] for moredetails.

For examples of wavelets ψ for which Mψ = 1|B0,% and details on fast approximative reconstruc-tion by integration over angles see [12]. For details on image processing (particularly enhance-ment and completion of crossing elongated structures) via orientation scores, see [13, 16, 17, 18].An intuitive illustration of the relation between an elongated structure in a 2D image and thecorresponding elongated structure in an orientation score is given in Figure 1.5.

1.2.4 In the context of related work

PDE techniques [30, 41], usually fail at crossings, [16], which are commonly encountered inmedical images. The orientation score framework solves this problem by constructing evolu-tions on a Lie group (SE(2)) rather than the usual space of images, R2 [16, 17, 18]. Most ofthe noise in (medical) images exists at lower scales and this suggests the need for scale basedenhancement. Further the utility of curvelets (which is a scale based technique), [9, 10, 28],for enhancement also leads to a similar conclusion. In this study we introduce the notion ofscaling to the SE(2)-OS framework via continuous wavelet transform on the 2D Similitude

8 Introduction

group (SIM(2)) and arrive at a scale-OS. Wavelet transform on SIM(2) group was first usedby Antoine et al. for feature detection by construction of appropriate directional wavelets,[2, 3, 4]. Though similar in the basic idea we make use of a more general wavelet transformproposed in [11] for our study, where we do not rely on the celebrated result by Grossman et al.[23] and where the requirements for irreducible representations and the usual reconstruction viaL2-adjoint are dropped. The explicit construction of our transform is similar to the discrete-WT based construction utilized by Donoho et al. in [9, 10]. However it is important to notethat our construction has a group structure which allows us to define Euclidean and scalinginvariant operators on the scale-OS which is an improvement over all existing techniques. Weextend the ideas of PDE based techniques by constructing evolutions on the SIM(2) groupwhich like in the case of SE(2)-OS allows handling of crossing structures in an image.

1.3 Content and Structure of this Thesis

In Chapter 2, the concept of an abstract wavelet transform in a group theoretic setting isintroduced. This is followed by a generalized version of the wavelet transform. We then applythis transform to our case of interest, the 2D Similitude group (SIM(2)) and conclude thechapter with the design of appropriate wavelets and graphically illustrate the practical utilityof this design.

Chapter 3 deals with relevant differential-geometric properties of the SIM(2) group followed byan explicit formulation of the exponential and logarithmic curves in this case. The theoreticaland numerical issues pertaining to SIM(2)-convolutions are also discussed.

In Chapter 4, we begin by showing that all “legal” operators on orientation scores are leftinvariant. We then present sharp Gaussian estimates for Green’s function of linear diffusion onSIM(2) group. Adaptive nonlinear evolutions on SIM(2) in a differential geometric settingis presented by making use of the Cartan connection and a method for curvature estimationis proposed. This is followed by an extension of adaptive diffusion on the SE(2) group tomultiple orientation scores.

Finally in Chapter 5, we summarize this thesis and discuss possible improvements and futurework.

The main results of this thesis are:

1. Utilizing the inherent symmetry in images a generic invertible wavelet transform on theSimilitude group (SIM(2)) is presented, which transforms an image into it’s scale orien-tation score (scale-OS). This allows differentiating between different scales and angles inan image. This method allows stable reconstruction of the image from the correspondingwavelet transform.

2. We explicitly derive the exponential and logarithmic curves in the SIM(2)-group usingleft-invariant vector fields.

3. Using the wavelet transform mentioned above, we show that the left-invariant operatorson scale-OS are invariant under the operations of translation, rotation and scaling of theoriginal image. Thus we provide a framework for well-posed operators on scale-OS.

4. Sharp Gaussian estimates for the green’s function of the linear (non-adaptive) diffu-sion equation on the SIM(2)-group are presented by using subcoercive operators andwe illustrate that in scale-OS framework even linear diffusion combined with greyvaluetransformations leads to successful enhancement of crossing curves.

Content and Structure of this Thesis 9

5. By choosing an appropriate metric on SIM(2) and making use of the Cartan connectiona covariant form of nonlinear left-invariant diffusion equation on SIM(2) is presentedand it is shown that the diffusion only takes place along the exponential curves. Basedon these results we propose a method for extraction of spatial curvature from scale-OS.

6. Finally we extend the SE(2) based nonlinear adaptive diffusion (CED-OS) to multipleorientation scores and show improvements on medical images.

Chapter 2

Generalized Wavelet Transformon the Similitude Group

We begin with a few preliminaries followed by a review of the construction of unitary mapson functional Hilbert spaces given in [11], which leads to the famous result by Grossman etal. [23] in a group-theoretic setting. Though the results in [23] are applicable to our context,we make use of a more general theory presented in [11, 12], which allows the construction of(admissible) wavelets suited to our application area of signal/image processing. Finally wepresent a practical design for a wavelet transformation. We emphasise the importance of agroup structure as it will allow us to design appropriate (enhancement/contextual) operationson the wavelet transform of an image, see (2.14) and Section 4.1.

2.1 Preliminaries

• Lp(Ω, µ): The quotient space consisting of functions with finite Lp norm (0 < p < ∞),

i.e.( ∫

Ω

‖f |pdµ)1/p

<∞ on Ω with respect to the nilspace of the L2-norm, which consists

of all functions f on Ω with zero measure support, i.e. f = 0 almost everywhere.

• Images are assumed to be within L2(R2), unless explicitly stated otherwise.

• An image f ∈ L2(R2) is called disc-limited if the support of it’s Fourier transform isbounded, by say a sphere of radius %. The space of these images is given by

L%2(R2) = f ∈ L2(R2)| supp(Ff) ⊂ B0,%, % > 0.

• A Hilbert space is a vector space equipped with an inner product such that every Cauchysequence converges with respect to the norm induced by the inner product.

• B(X,Y ): The vector space consisting of continuous (i.e. bounded) linear operators fromX onto Y . In case, X = Y , we write B(X).

• The range of a linear operator A will be denoted byR(A) and the nilspace will be denotedby N (A).

• The Fourier transform F : L2(Rd)→ L2(Rd), is almost everywhere defined by

F(f)(ω) = f(ω) =1

(2π)d/2

∫Rd

f(x)e−iω·xdx.

11

12 Generalized Wavelet Transform on the Similitude Group

By Plancherel and Convolution theorem we have ‖F(f)‖2 = ‖f‖2 for all f ∈ L2(Rd) andthat F [f ∗ g] = (2π)d/2F [f ]F [g], for all f, g ∈ L2(Rd).

• We will use the short notation for the following groups:

– Aut(Rd)=A : Rd → Rd| A linear and A−1 exists– dilation group D(d)=A ∈ Aut(Rd)| A = aI, a > 0– orthogonal group O(d)=X ∈ Aut(Rd)| XT = X−1 – rotation group SO(d)=R ∈ O(d)| det(R) = 1– circle group T=z ∈ C|,|z| = 1, z = eiθ, θ =argz with group homomorphismτ : T→ SO(2) ⊂Aut(R2):

τ(z) = Rθ =

(cos θ − sin θsin θ cos θ

). (2.1)

• Let T and S be locally compact groups and let τ : T → Aut(S) be a group homomor-phism. The semi-direct product Sd oτ T is defined to be the group (which is againlocally compact) with underlying group (s, t)|s ∈ S, t ∈ T and group operation

(s, t)(s′, t′) = (sτ(t)s′, tt′). (2.2)

In this study we consider the following two groups:

– The group R2 oT, where τ is given by (2.1). The group product (2.2) is now givenby

gg′ = (b, eiθ)(b′, eiθ′) = (b +Rθb

′, ei(θ+θ′)), g = (b, eiθ), g′ = (b′, eiθ

′) ∈ R2 o T.

This group is the Euclidean motion group, SE(2).

– The group R2 o (T×D(1)), where τ is given by (2.1). The group product (2.2) isnow given by

gg′ = (b, a, eiθ)(b′, a′, eiθ′) = (b + aRθb

′, aa′, ei(θ+θ′)),

g = (b, eiθ), g′ = (b′, eiθ′) ∈ R2 o (T×D(1)).

This group is the Similitude group, SIM(2).

• A representation R of a group G into a Hilbert space H is a homomorphism R betweenG and B(H), the space of bounded linear operators on H. It satisfies Rgh = RgRh forall g, h ∈ G and Re = I. A representation R is irreducible if the only invariant closedsubspaces of H are 0 and H, otherwise reducible. We consider unitary representations,i.e, ‖Ugψ‖H = ‖ψ‖H for all g ∈ G and ψ ∈ H), which will be denoted by U rather thanR. Within the class of unitary representations we consider the representations of RdoTin L2(Rd) which are given by

(Ugψ)(x) =1√

det(τ(t))ψ((τ−1(t))(x− b)), with g = (b, τ(t)) ∈ Rd o T. (2.3)

These representations are called left-regular actions of G = Rd o T in L2(Rd).

• Let b ∈ Rd, a > 0 and g ∈ G with corresponding τ(g) ∈ Aut(Rd). Then the unitaryoperators f 7→ f , Tb,Da and Rg on L2(Rd) are defined by

f(x) = f(−x) Tbψ(x) = ψ(x− b)

Daψ(x =1

ad2

ψ(x

a) Rgψ(x) =

1√detτ(g)

ψ((τ(g))−1x). (2.4)

The mappings b 7→ Tb, g 7→ Rg, a 7→ Da are respectively left regular actions of Rd, D(d),G into L2Rd.

Unitary maps from a Hilbert space H to a functional Hilbert space CIK 13

• A functional Hilbert space1 is a Hilbert space consisting of complex valued functions onan index set I on which the point evaluation δa, is a continuous/bounded linear functionalfor all a ∈ I. Consequently, it has a Riesz representant Ka ∈ H

f(a) = 〈δa, f〉 = (Ka, f)H .

The function K : I × I → C given by K(a,b)H = (Ka,Kb)H = Kb(a) is called repro-ducing kernel. Note that the spaces L2(Rd) are not functional Hilbert spaces.

• The d-dimensional Gaussian kernel Gs at scale s is given by

Gs(x) =1

(4πs)d/2e−‖x‖24s . (2.5)

Occasionally, the Gaussian kernel will be parametrized by standard deviation σ and we

write Gσ = Gs, where we notice that s = σ2

2 .

2.2 Unitary maps from a Hilbert space H to a functionalHilbert space CI

K

In this section we present an abstract wavelet transform given in [11]. Recall that a functionalHilbert space is a Hilbert space such that point evaluation is continuous, and therefore by Reiszrepresentation theorem there exists a set Km|m ∈ I with

(Km, f)H = f(m), for all m ∈ I and f ∈ H.

Clearly the span of the set Km|m ∈ I is dense in CIK , for if f ∈ H is orthogonal to all Km

then f = 0 on I.

Define K(m,m′) = Km′(m) = (Km,Km′)H for all m,m′ ∈ I. K is called reproducing kerneland it is a function of positive type on I, i.e.

n∑i=1

n∑j=1

K(mi,mj)cicj ≥ 0, for all n ∈ N, c1, . . . , cn ∈ C, m1, . . . ,mn ∈ I.

Therefore to every functional Hilbert space there belongs a reproducing kernel, which is afunction of positive type. Conversely as mentioned in [5], a function K of positive type on setI, induces uniquely a functional Hilbert space consisting of functions on I with reproducingkernel K.

Let V = φm|m ∈ I be a subset of H. Define function K : I×I→ C by K(m,m′) = (φm, φm′).From this function of positive type the space CI

K can be constructed. From [11] we have thefollowing important result:

Theorem 2.1 (Abstract Wavelet Theorem). Define W : 〈V 〉 → CIK by

(Wf)(m) = (φm, f)H . (2.6)

Then W is a unitary mapping.

In our study we are interested in the case 〈V 〉 = H. In this case, Theorem 2.1 becomes thefollowing.

Theorem 2.2. If the span of V = φm|m ∈ I is dense in H, then the transform W : H → CIK

defined by(W[f ])(m) = (φm, f)H (2.7)

is a unitary mapping, i.e. ‖W[f ]‖CIK

= ‖f‖H .

1Also known as reproducing kernel Hilbert space.


2.3 Functional Hilbert spaces on Groups

From now on we will assume I to be a group G. Further assume the group to have a represen-tation on H, i.e. a map R : G → B(H), which satisfies RgRh = Rgh and Re = I, ∀g, h ∈ G,where e is the identity element in G.

Given ψ ∈ H (the wavelet), let g 7→ Rg be a representation of G onto H such that

Vψ = Rgψ|g ∈ G (2.8)

is dense in H and K(g, g′) = (Rgψ,Rg′ψ). Starting from Vψ we can construct the functionalHilbert subspace CGK . Then as a consequence of Theorem 2.2 we have the following result.

Theorem 2.3 (Wavelet Theorem for group representations). Let R be a representation of agroup G in a Hilbert space H. Define the function K : G×G→ C of positive type by

K(g, g′) = (Rgψ,Rg′ψ)H . (2.9)

Define the set Vψ = Rgψ|g ∈ G. Then the wavelet transformation Wψ : 〈V 〉 → CGK defined

by(Wψf)(g) = (Rgψ, f)H , (2.10)

is a unitary mapping.

In case 〈V 〉 = H, the wavelet transformation Wψ : H → CGK is unitary.

2.3.1 Wavelet Transformations constructed by Unitary Irreduciblerepresentations of Locally Compact Groups

Recall that a representation R of a group G in a Hilbert space H is irreducible if the onlyclosed subspaces of H which are invariant under all Rg for all g ∈ G are H and 0. Clearly

for every nonzero ψ ∈ H the subspace 〈Vψ〉, where Vψ = Rgψ|g ∈ G, is invariant under all

Rg with g ∈ G and since it is non-empty, 〈Vψ〉 = H. Hence the wavelet transformation is aunitary mapping for all ψ ∈ H.

The representation R is called square integrable if there exists ψ ∈ H with ψ 6= 0 and

Cψ :=

∫G

|Ugψ,ψ|2

(ψ,ψ)HdµG(g) <∞. (2.11)

ψ ∈ H with ψ 6= 0 such that Cψ < ∞ is known as the admissible wavelet2.If the grouprepresentation is unitary, irreducible and square integrable , then the functional Hilbert spacewill always be a closed subspace of L2(G), whenever the wavelet satisfies (2.11). This was firstshown by Grossmann et al.[23]. We now state a important theorem from [23].

Theorem 2.4 (The Wavelet Reconstruction Theorem). Let U be an irreducible, unitary andsquare integrable representation of a locally compact group G on a Hilbert space H. Let ψ ∈ Hsuch that (2.11) holds. Then the wavelet transform W : H → CGK defined by

(W[f ])(m) = (Ugψ, f)H

2In literature, see [3], we also encounter an alternative but equivalent definition of Cψ given by Cψ =

(2π)2∫R2

|Fψ(ω)|2‖ω‖2 dω. See [27] for more details.

Functional Hilbert spaces on Groups 15

is a linear isometry (up to a constant) from the Hilbert space H onto a closed subspace CGKψof L2(G, dµ):

‖Wψf‖2L2(G) = Cψ‖f‖2H . (2.12)

Here the space CGK is the functional Hilbert space with reproducing kernel

Kψ(g, g′) =1

Cψ(Ugψ,Ug′ψ).

The corresponding orthogonal projection Pψ : L2(G, dµ)→ CGKψ is given by

(Pψφ)(g) =

∫G

Kψ(g, g′)φ(g′)dµG(g′)/where/φ ∈ L2(G, dµ). (2.13)

Furthermore, Wψ intertwines U and the left regular representation L (i.e. Lg is given byLg(φ) = (h 7→ φ(g−1h))) on L2(G)

WψUg = LgWψ. (2.14)

Consequences of Theorem 2.4

• The inverse wavelet transform is the adjoint of the wavelet transform upto a constant:W−1ψ = 1

Cψ(Wψ)∗, which is given by,

W∗ψ(φ) =

∫G

(Ugψ)φ(g)dµG(g). (2.15)

This follows from a straightforward evaluation given in the appendix.

• Applying the Polarization identity to (2.12) it follows that (Wψ[f1],Wψ[f2])L2(G) =Cψ(f1, f2)H for all f1, f2 ∈ H.

• In the case where G = Roτ T and U a left regular action of G in L2(Rd), given by (2.3),which is unitary and if it is also irreducible one has the reconstruction formula:

f(x) =1

Cψ

∫τ(T )

∫Rd

1√det(τ(t))

Wψ[f ](t,b)ψ((τ−1(t))(x− b))dbdt (2.16)

for almost every x ∈ Rd.

• Operators on (multiple scale) wavelet transforms must commute with Lg, ∀g ∈ SIM(2),to ensure that the effective operator W∗ψ Φ Wψ on images commutes with Ug, ∀g ∈SIM(2).

Theorem 2.4 thus allows perfectly well-posed reconstruction of the image when the left regularaction U of the group G in Hilbert space H, is a irreducible representation. The followinglemma shows that we are allowed to use Theorem 2.4 for the SIM(2) case.

Lemma 2.5. The left regular action U of the SIM(2) group in L2(R2), given by

(Ub,eiθ,aψ)(x) =1

aψ

(R−1θ (x− b)

a

), a > 0, θ ∈ [0, 2π], b ∈ R2, (2.17)

is an irreducible unitary representation.


For proof, see [27, Pg. 51]. Often, this irreducibility condition is very strong and even simple(and often encountered) group representations do not satisfy it. The left regular action V ofthe Euclidean Motion group SE(2) in L2(R2), given by

Vb,eiθψ(x) = ψ(R−1θ (x− b)), θ ∈ [0, 2π], b ∈ R2,

is a reducible representation.

2.4 Generalized Wavelet Transformation on Locally Com-pact Groups

As mentioned above irreducibility is a strong condition. In [11], a generalized wavelet transformis introduced which does not have the condition of irreducibility. In this section we give acompilation of results from [11] which explicitly characterize CGK , when G = Rd o T , T is alocally compact group and U is the left regular action of G into L2(Rd) and then apply themto our case. At this point we mention that all locally compact groups have a (left/right) Haarmeasure defined on them. See appendix for the Haar measure for SIM(2). For more detailson locally compact groups see [20, Chap. 2].

ψ ∈ L2(Rd) is called an admissible wavelet if

0 < Mψ := (2π)d/2∫T

∣∣∣∣ F [Rtψ]√detτ(t)

∣∣∣∣2dµT (t) <∞ a.e on Rd, (2.18)

where Rt is defined in (2.4) and dµT (t) is the left-invariant Haar measure of T .

We define ψ almost everywhere on Rd by

ψ(x) =

∫T

1

detτ(t)(Rtψ ∗ Rtψ)(x)dµT (t), (2.19)

where ψ is defined in (2.4). Note that ψ is the inverse Fourier transform of Mψ.

The proof of the following lemma is very informative and is presented in the appendix.

Lemma 2.6. Let ψ be an admissible wavelet. Then the span of Vψ = Rgψ|g ∈ G, is dense

in L2(Rd), i.e. 〈Vψ〉 = L2(Rd).

Corollary 2.7. If the wavelet ψ is admissible, then the corresponding wavelet transform Wψ :L2(Rd)→ CGK is unitary.

Proof. Follows from Theorem 2.2 and Lemma 2.6.

Define the linear operator TMψon CGK by

[TMψ[φ]](b, t) = F−1

[ω 7→ (2π)−d/4M

−1/2ψ (ω)F [φ(·, t)](ω)

](b).

Operator TMψis well defined since Mψ > 0 a.e. on Rd and M

−1/2ψ F [φ(·, t)] ∈ L2(Rd) for all

t ∈ T .

Generalized Wavelet Transformation on Locally Compact Groups 17

Theorem 2.8. Let G = RdoT . Let ψ be an admissible wavelet. Then TMψφ ∈ L2(G, dµG(g))

for all φ ∈ CGK . Therefore (·, ·)TMψ : CGK × CGK → C defined by

(Φ,Ψ)Mψ= (TMψ

[Φ], TMψ[Ψ])L2(G), (2.20)

is an explicit characterization of the inner product on CGK , which is the unique functionalHilbert space with reproducing kernel K : G×G→ C given by

K(g, h) = (Ugψ,Uhψ)L2(Rd) = (Uh−1gψ,ψ)L2(Rd), g, h ∈ G. (2.21)

The wavelet transformation Wψ defined by

Wψ[f ](b, t) = (Ugψ, f)L2(Rd), f ∈ L2(Rd), g = (b, t) ∈ Rd o T, (2.22)

is a unitary mapping from L2(Rd) to CGK . The space CGK is a closed subspace of the Hilbert

space Hψ ⊗ L2(T ; dµT (t)det(τ(t)) ), where Hψ = f ∈ L2(Rd)|M−

12

ψ F [f ] ∈ L2(Rd) is equipped with

the inner product

(f, g) = (M− 1

2

ψ F [f ],M− 1

2

ψ F [g])L2(Rd;(2π)−d/2dx).

The orthogonal projection Pψ of Hψ ⊗ L2(T ; dµT (t)det(τ(t)) ) onto CGK is given by (Pψ[Φ])(g) =

(K(·, g,Φ))Mφ.

Proof. See [11, Pg. 27-30].

Remarks:

1. Since Wψ : L2(Rd)→ CGK is unitary, the inverse equals the adjoint and thus the image fcan be reconstructed from it’s orientation score Wψ[f ] by

f = W∗ψ[Wψ[f ]]

= F−1

[ω 7→ 1

(2π)d/2

∫T

F [Wψ[f ](·, t)](ω)F [Rtψ](ω)dµT

|det(τ(t))|M−1ψ

]. (2.23)

First we prove the following lemma.

Lemma 2.9. Let ψ ∈ L2(Rd) ∩ L1(Rd). Then (Wψf)(·, t)) ∈ L2(Rd) for all f ∈ L2(Rd)and t ∈ T .

Proof. Let f ∈ L2(Rd) and t ∈ T . Then

(TxRtψ, f)L2(Rd) =

∫Rd

(Rtψ)(x′ − x)f(x′)dx′ = Rtψ ∗R2 f, (2.24)

for all x′, where recall the definition of ψ from (2.4). Thus we arrive at a convolution ofa L1(R2) and a L2(R2) function, which is again a L2(R2) function.

This means that for all elements Φ ∈ CGK the function Φ(·, t) belongs to L2(R2) for fixedt ∈ T . Hence the Fourier transform of Φ(·, t) is well defined.

We prove the remark based on the following description of the adjoint map,

(Φ,Wψf)CGK = (W∗ψΦ, f)L2(Rd). (2.25)


The wavelet transform can be written as,

(Wψf)(x, t) = (TxRtψ, f)L2(R2) =

∫R2

Rtψ(x− x′)f(x′)dx′

= (Rtψ ∗R2 f) = F−1(FRtψFf). (2.26)

Thus using the Mψ inner product defined earlier and (2.26) we have,

(Φ,Wψf)CGK =

∫Rd

∫T

F [Φ(·, t)](ω)F [Wψf(·, t)](ω)(2π)−d/2M−1ψ (ω)

dµT (t)

|det(τ(t))|dω

=

∫Rd

∫T

F [Φ(·, t)](ω)F [Rtψ](ω)Ff(ω)(2π)−d/2M−1ψ (ω)

dµT (t)

|det(τ(t))|dω. (2.27)

Further expanding the R.H.S of (2.25) we have,

(W∗ψΦ, f)L2(Rd) = (F [W∗ψΦ],Ff)L2(Rd) =

∫Rd

F [W∗ψΦ](ω)Ff(ω)dω. (2.28)

Thus comparing (2.27) and (2.28), we arrive at the exact reconstruction formula,

f = W∗ψ[Wψ[f ]]

= F−1

[ω 7→ 1

(2π)d/2

∫T

F [Wψ[f ](·, t)](ω)F [Rtψ](ω)dµT

|det(τ(t))|M−1ψ (ω)

].

2. If Mψ = 1 on R2, then CGK is a closed subspace of L2(G).

Definition 2.10. The inner product on Hψ ⊗ L2(T ; dµT (t)det(τ(t)) ) induces a norm ‖.‖Mψ

: Hψ ⊗L2(T ; dµT (t)

det(τ(t)) )→ R+, which is given by

‖Φ‖Mψ=√

(Φ,Φ)Mψ=

∫R2

∫T

|F [Φ(·, t)](ω)|2(2π)−d/2M−1ψ (ω)

dµT (t)

det(τ(t))dω,

which is called the Mψ-norm.

In our specific case of G = SIM(2) and H = L2(R2), where note that R2 is a locally compactabelian group. Also let T := SO(2) × R+, which is a locally compact group. Recall that,SIM(2) = R2 oτ (SO(2)×R+)3 where the semi-direct product is defined to be the group withunderlying set R2 × (SO(2)× R+) and group operation,

(x, a, θ)(x′, a′, θ′) = (x + τ(a, θ)x′, aa′, θ + θ′), ∀(x, a, θ), (x′, a′, θ′) ∈ SIM(2), (2.29)

where τ(a, θ) = aRθ4.

Since both R2 and SO(2)×R+ are locally compact groups, SIM(2) is a locally compact groupas well and thus has a left-invariant Haar measure defined on it. As mentioned in Lemma 2.5there exists a unitary representation of SIM(2) in L2(R2) given by,

Ug=(b,a,θ)ψ(x) =1

aψ

(R−1θ (x− b)

a

), a > 0, θ ∈ [0, 2π], b ∈ R2, (2.30)

3We drop the subscript τ from the semi-direct product hereon for the ease of notation.4Rθ is the standard counter-clockwise rotation matrix.

The Discrete Analogue 19

We denote U : (x, t) = (x, a, θ) 7→ U(x,t) as

Ux,tf = TxRtf, t = (a, θ) (2.31)

where (Txf)(x′) = f(x′ − x) and (Rtf)(x′) = 1af( 1

aRθx′).

Given below is the interpretation of Theorem 2.8 in our context.

Corollary 2.11. The space of orientation scores is a reproducing kernel Hilbert space CSIM(2))K

which is a closed subspace of Hψ ⊗ L2(SO(2) × R+; dµT (t)det(τ(t)) ) which is a vector subspace5 of

L2(G). The inner product on CR2o(SO(2)×R+)K is given by (2.20) and is explicitly characterized

by means of the function Mψ given by,

Mψ(ω) = (2π)

∫T=SO(2)×R+

∣∣∣∣ (F [Rtψ])(ω)√detτ(t)

∣∣∣∣2dµT (t). (2.32)

The wavelet transform which maps an image f ∈ L2(R2) onto its orientation score Uf ∈CSIM(2)K is a unitary mapping: ‖f‖2L2(R2) = ‖Uf‖2Mψ

. Thus the image f can be reconstructed

from its orientation score Uf =Wψ[f ] by means of the adjoint wavelet transformation W∗ψ:

f =W∗ψ[Wψ[f ]] = F−1

[ω 7→ 1

(2π)

∫R+

2π∫0

F [Uf (·, a, eiθ)](ω)F [Ra,eiθψ](ω)dθda

aM−1ψ (ω)

].

(2.33)

Explicitly Mψ as defined in (2.32) can be written as,

Mψ(ω) = (2π)

∫T

∣∣∣∣F [Rtψ](ω)√detτ(t)

∣∣∣∣2dµT (t) = (2π)

2π∫0

∫R+

∣∣∣∣F [R(a,θ)ψ](ω)

a

∣∣∣∣2 daa dθ= (2π)

2π∫0

∫R+

∣∣∣∣a2ψ(aR−1θ ω)

a2

∣∣∣∣2 daa dθ = 2π

2π∫0

∫R+

|ψ(aR−1θ ω)|2 da

adθ (2.34)

= 2π

2π∫0

∫R

|ψ(eτR−1θ ω)|2dτdθ (2.35)

for all t = (a, θ) ∈ SO(2) × R+. Note that in the last equality we have made use of thesubstitution τ = loge(a). Irrespective of the choice of wavelet ψ, Mψ|ω=0 = 0 or ∞, as thestabilizer of ω = 0 is not compact. But this is an issue (dealt with in the next section) asthe admissibility condition requires that Mψ have a non-zero bounded value for almost everyω ∈ R2.

2.5 The Discrete Analogue

Hereon in this chapter we deal with the practical aspects of the implementation of the contin-uous wavelet transform discussed in the previous sections.

5i.e. is a subspace as a vector space, but is equipped with a different norm


Recall that SIM(2) = R2o(SO(2)×R+) where T = (SO(2)×R+) is a locally compact group,but not a compact group. We can choose SO(2) to be finite rotation group, denoted by TN(equipped with discrete topology) which is locally compact6 i.e.

TN = eik∆T |k ∈ 0, 1, . . . , N − 1,∆T =2π

N, for N ∈ N. (2.36)

On the other hand the scaling group R+ cannot be written in terms of a finite scaling groupdue to the following well known result from group theory.

Lemma 2.12. Every finite subgroup of the multiplicative group of a field is a cyclic subgroup.

See [33] for proof. The only finite subgroups of the group R∗ = (R/0, ∗) is the trivialsubgroup 1 and the 2nd order group generated by −1, −1, 1. The scaling group R+ isa subgroup of R∗ and therefore it does not have any finite subgroups other than the trivialsubgroup. So it is important to note that, we loose the inherent group structure in the discreteversion, unlike in the case of orientation score over the SE(2) group, see [12, Sec 4.4].

Consider the scaling group R+; this group consists of all positive reals greater than zero. Inthe discrete case we need to have a lower and an upper bound on the choice of the scales. Weassume that a ∈ [a−, a+] where 0 < a− < a+. We have the following discretization for scales,

DM =

e(τ−+k∆D)|k ∈ 0, 1, . . . M − 1,∆D =

eτ+ − eτ−

M

, for M ∈ N, (2.37)

where τ− = log(a−) and τ+ = log(a+).

Using the notation, tkl = (al, θk), k ∈ 0, 1, . . . , N − 1, l ∈ 0, 1, . . . ,M − 1 where al =τ− + k∆D and θk = k∆T, we write the discrete version of (2.31)

UN,Mf (b, al, θk) = (TbR(al,θk)ψ, f)L2(R2), (2.38)

which is the discrete orientation score of an image f ∈ L2(R2). We emphasize that we do notconsider a discrete subgroup of SIM(2)-group due to Lemma 2.12. The discrete version of Mψ

is,

Mψ(ω) =1

N

1

M

N−1∑k=0

M−1∑l=0

∣∣∣∣F(R(al,θk)ψ)(ω)

al

∣∣∣∣2 . (2.39)

2.5.1 Stable reconstruction of image f from its orientation score Uf

We assume that the support of the Fourier transform of our images is bounded by a annulus.The space of these images is a Hilbert space given by,

L%−,%+

2 (R2) = f ∈ L2(R2)| supp(F [f ]) ⊂ B0,%+\B0,%−, %+ > %− > 0. (2.40)

The reason for the assumption of an upper bound (%+) on the support of the Fourier trans-form of the images is the Nyquist theorem, which states that every band-limited function isdetermined by its values on a discrete grid. For example if uB : R2 → C is band-limited on asquare: supp(F [uB ]) ⊂ [−l/2, l/2]× [−l/2, l/2], then

uB(x, y) =∑

(k1,k1)∈Z2

uB(2πk1

l,

2πk2

l)sinc(

lx

2− k1π)sinc(

lx

2− k2π),

where the cutoff frequency % = l/2 is called the Nyquist frequency.

6Topological spaces with a discrete topology are locally compact.

The Discrete Analogue 21

Remark. From a practical point of view it does not make sense to store frequencies of ordergreater than l/2 if the signal contains l samples, as Fourier transform of a discrete imagef : 1, . . . , n × 1, . . . , n → C is again of the form Ff : 1, . . . , n × 1, . . . , n → C. Thus afinite width puts an upper bound on the domain of the Fourier transform.

The assumption of the lower bound (%−) on the Fourier transform has a theoretical as wellas a practical underpinning. Theoretically it allows us to avoid the origin where any choice ofwavelet ψ fails to satisfy the admissibility condition. The value for %− directly relates to thecoarsest scale we wish to detect in the spatial domain. Therefore the removal of extremely lowfrequencies from the image essentially corresponds to background removal in the image whichis often an essential pre-processing step in medical image processing, see [6, 42].

We wish to construct a wavelet transform

W%−,%+

ψ : L%−,%+

2 (R2)→ L2(SIM(2)) (2.41)

which requires that,

Ux,a,θψ ∈ L%−,%+

2 (R2), where a ∈ [a−, a+] and θ ∈ [0, 2π].

Therefore we choose

ψ ∈ L1(R2) ∩ L2(R2) with supp(F [ψ]) ⊂ B0,%+/a+\B0,%−/a− (2.42)

and therefore we satisfy supp(F [Ux,a,θψ]) ⊂ B0,%+\B0,%− , where a ∈ [a−, a+] and θ ∈ [0, 2π].

Note that in our current context, ψ ∈ L%−,%+

2 (R2) is called an admissible wavelet if

0 < Mψ = (2π)

2π∫0

a+∫a−

∣∣∣∣F [Ra,θψ]√detτ(t)

∣∣∣∣2 daa dθ <∞ on B0,%+\B0,%− , (2.43)

where Rt is given by Equation (2.4). Corresponding to (2.19) we can define ψ as,

ψ(x) =

2π∫0

a+∫a−

(Ra,θψ ∗ Ra,θψ)(x)da

adθ. (2.44)

By the compactness of SO(2) × [a−, a+] and since the convolution of two L1(R2) functionsis again in L1(R2), we have that ψ ∈ L1(R2), and so its Fourier transform is a continuousfunction on B0,%+\B0,%− .

Definition 2.13. Let ψ be an admissible wavelet in the sense of (2.43). Then the wavelet

transform W%−,%+

ψ : L%−,%+

2 (R2)→ L2(SIM(2)) is given by

(W%−,%+

ψ [f ])(g) = (Ugψ, f)L2(R2), f ∈ L%−,%+

2 (R2),

for almost every g = (x, a, θ) ∈ R2 o (SO(2)× [a−, a+]).

2.5.2 Quantification of Stability

The usual way to quantify well-posedness/stability of an invertible linear transformation A :V → W from a normed space (V, ‖ · ‖V ) to a normed space (W, ‖ · ‖W ) is by means of thecondition number

cond(A) = ‖A−1‖‖A‖ =

(supx∈V

‖x‖V‖Ax‖W

)(supx∈V

‖Ax‖W‖x‖V

)≥ 1. (2.45)


The closer it approximates 1, the more stable the operator and its inverse is. Note that thecondition number depends on the norms imposed on V and W . We wish to apply this generalconcept to the wavelet transformation which maps image f to its scale orientation score Uf .Recall from the previous section that the wavelet transform is a unitary mapping from the

space L2(R2) to the space CSIM(2)K respectively equipped with the L2-norm and the Mψ-norm.

Thus choosing these norms the condition number becomes 1. However from a practical pointof view it is more appropriate to impose the L2(SIM(2))-norm on the scale orientation scoresince it does not depend on the choice of the wavelet ψ and since we also employ a L2-normon the space of images.

Theorem 2.14. Let ψ be an admissible wavelet, with Mψ(ω) > 0 for all ω ∈ B0,%+\B0,%− .

Then the condition number cond(W%−,%+

ψ ) of the wavelet transformationW%−,%+

ψ : L%−,%+

2 (R2)→L2(G), (G = SIM(2))7 is defined by

cond(W%−,%+

ψ ) = ‖(W%−,%+

ψ )−1‖‖(W%−,%+

ψ )‖ =

(sup

f∈L%−,%+

2 (R2)

‖f‖L2(R2)

‖Uf‖L2(G)

)(sup

f∈L%−,%+

2 (R2)

‖Uf‖L2(G)

‖f‖L2(R2)

)

and satisfies

1 ≤ (cond(W%−,%+

ψ ))2 ≤(

sup%−≤‖ω‖≤%+

M−1ψ (ω)

)(sup

%−≤‖ω‖≤%+Mψ(ω)

).

Proof. Since Mψ > 0 and is continuous on the compact set B0,%+\B0,%− = ω ∈ R2|%− <

‖ω‖ < %+, sup%−<‖ω‖<%+

Mψ(ω) = max%−<‖ω‖<%+

Mψ(ω) do exist. The same holds for M−1ψ as well.

Further for all f ∈ L%−,%+

2 (R2), the restriction of the corresponding orientation scores, to fixed

orientations and scales also belong to the same space, i.e. Uf (·, a, eiθ) ∈ L%−,%+

2 (R2), where θ ∈[0, 2π] and a ∈ [a−, a+]. This follows from (2.26) which gives, F [Uf ](ω) = F [Rtψ](ω)F [f ](ω).Recall from Corollary 2.11, that ‖f‖2L2(R2) = ‖Uf‖2Mψ

and thus we have,

(cond(W%−,%+

ψ ))2 =

(sup

f∈L%−,%+

2 (R2)

‖Uf‖Mψ

‖Uf‖L2(G)

)(sup

f∈L%−,%+

2 (R2)

‖Uf‖L2(G)

‖Uf‖Mψ

)

≤(

sup%−≤‖ω‖≤%+

M−1ψ (ω)

)(sup

%−≤‖ω‖≤%+Mψ(ω)

).

Further we have, 1 = ‖(W%−,%+

ψ )−1(W%−,%+

ψ )‖ ≤ ‖(W%−,%+

ψ )−1‖‖(W%−,%+

ψ )‖.

Corollary 2.15. The stability of the (inverse) wavelet transformationW%−,%+

ψ : L%−,%+

2 (R2)→L2(SIM(2)) is optimal if M%

ψ(ω) = constant for all ω ∈ R, with %− ≤ ‖ω‖ ≤ %+.

Thus in general, the closer the functionMψ approximates the constant function, say 1B0,%+\B0,%−,

the better the Mψ norm on CSIM(2)K approximates the L2(SIM(2)) norm, the better the sta-

bility of reconstruction. In case of a good approximation, i.e. Mψ ≈ 1B0,%+\B0,%−, the recon-

struction formula in Corollary 2.11 can be simplified to,

f ≈ F−1

[ω 7→ 1

(2π)

a+∫a−

2π∫0

F [Uf (·, a, eiθ)](ω)F [Ra,eiθψ](ω)dθda

a

]. (2.46)

7Note that while the norm is L2(SIM(2))-norm we only consider an interval in the scaling group.

Design of proper wavelets 23

2.6 Design of proper wavelets

In this sequel a wavelet ψ ∈ L2(R2)∩L1(R2) with Mψ smoothly approximating 1B0,%+\B0,%−, is

called a proper wavelet. The entire class of proper wavelets allows for a lot of freedom in thechoice of ψ. In practice it is mostly sufficient to consider wavelets that are similar to the longelongated patch one would like to detect and orthogonal to structures of local patches whichshould not be detected, in other words employing the basic principle of template matching.We restrict the possible choices by listing below certain practical requirements to be fulfilledby our transform.

1. The wavelet transform should yield a finite number of orientations (N0) and scales (M0).This requirement is obvious from an implementation point of view.

2. The wavelet should be strongly directional, in order to obtain sharp responses on orientedstructures.

3. The transformation should handle lines, contours and oriented patters. Thus the waveletshould pick up edge, ridge and periodic profiles.

4. In order to pick up local structures, the wavelet should be localized in spatial domain.

To ensure that the wavelet is strongly directional, we require that the support of the waveletbe contained in a convex cone in the Fourier domain, [2].

The following lemma gives a simple but practical approach to obtain proper wavelets ψ, withMψ(ω) = 1B0,%+\B0,%−

. Note that we make use of polar coordinates (ρ, ϕ), ρ = ‖ω‖, ω =

(ρ cosϕ, ρ cosϕ).

Lemma 2.16. Let τ−, τ+ be chosen such that τ− = log(a−) and τ+ = log(a+), where 0 <a ∈ [a−, a+] is the finite interval of scaling. Let A : SO(2)→ C\R− and B : [τ−, τ+]→ C\R−such that

2π

2π∫0

|A(ϕ)|dϕ = 1,

τ+∫τ−

|B(ρ)|dρ = 1, (2.47)

then the wavelet ψ = F−1[ω →√A(ϕ)B(ρ)] has Mψ(ω) = 1 for all ω ∈ B0,%+\B0,%− .

Proof. From (2.35) and (2.43), for all ω ∈ B0,%+\B0,%− we have,

Mψ(ω) = 2π

2π∫0

%+∫%−

|ψ(eτR−1θ ω)|2dτdθ = 2π

2π∫0

%+∫%−

|√A(ϕ− θ)B(eτρ)|2dτdθ = 1,

for all ω ∈ B0,%+\B0,%− .

Lemma 2.16 can be translated into discrete framework R2 o (TN × DM ), recall (2.36) and(2.37), where condition (2.47) is replaced respectively by,

1

N

N−1∑k=0

|A(ϕ− θk)| = 1 and1

M

M−1∑l=0

|B(ealρ)| = 1. (2.48)

where we have made use of discrete notations introduced in (2.38).


If moreover 2π2π∫0

√|A(ϕ)|dϕ ≈ 1 and

τ+∫τ−|√B(ρ)|dρ ≈ 1, we have a fast and simple approxi-

mation for the reconstruction:

f(x) = 2π

2π∫0

τ+∫τ−

(Wψf)(x, τ, θ)dτdθ ≈ F−1[ω 7→ (√Mψ ∗ F [f ](ω))](x), for a.e. ω ∈ B0,%+\B0,%− .

Mψ ≈ 1B0,%+\B0,%−⇒ f ≈ f ∈ L%

−,%+

2 (R2) (2.49)

Note that we have made use of the description of (Wψf) given in (2.26). We need to fulfil therequirement Mψ(ω) ≈ 1 with an appropriate choice of kernel satisfying the condition givenabove, to achieve this simple reconstruction.

The idea is to “fill the cake by pieces of the cake” in the Fourier domain. In order to avoidhigh frequencies in the spatial domain, these pieces must be smooth and thereby must overlap.A choice of B-spline based functions in the angular and the radial direction is an appropriatechoice for such a wavelet kernel.

The kth order B-spline denoted by Bk is defined as,

Bk(x) = (Bk−1 ∗B0)(x), B0(x) =

1 if − 1/2 < x < +1/20 otherwise

(2.50)

B-splines have the property that they add up to 1. For more details see [38].

Based on the requirements and considerations above we propose the following kernel

ψ(x) = F−1R2 [ω →

√A(ϕ)B(ρ)](x)Gσs(x), (2.51)

where Gσs is a Gaussian window that enforces spatial locality cf. requirement 5.

A : SO(2)→ R+, is defined by,

A(ϕ) = Bk(

(ϕ mod2π)− π/2sϕ

), (2.52)

where sϕ = 2πNθ

(Nθ denotes the number of orientations chosen) and Bk denotes the kth orderB-spline.

B : [%−, %+]→ R+, is defined as,

B(%) =

Ns−1∑l=0

Bk(

log[ρ]

sρ− l), (2.53)

where sρ = (log[a+] − log[a−])/Ns , with Ns equals the number of chosen scales and a−, a+

are predefined scales, based on %−, %+ respectively. The motivation for log based splineconstruction (B(%)) for the scaling group will become clear in the next chapter. See Figure 2.1for a plot of these B-splines. It is easily checked that,

Ns−1∑l=0

Bk(

log[ρ]

sρ− l)

= 1 and

Ns−1∑l=0

√Bk(

log[ρ]

sρ− l)≈ 1.

For the second approximation, see Figure 2.1. Similar results also follow for the radial kernel.Therefore for this choice of kernels we can make use of the simple reconstruction given in (2.49)


10 20 30 40 50

0.10.20.30.40.50.60.7

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

10 20 30 40 50

0.2

0.4

0.6

0.8

10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

Figure 2.1: Plot for the B-splines given in (2.53). Values chosen: a− = %− = 10−8, a+ =

%+ = 50, Ns = 7. From Left to Right: Plot of each B-spline, Bk(

log[ρ]sρ− l)

. Ob-

serve that the B-splines are skewed which is due to non linear sampling in the scale di-

mension; Notice thatNs−1∑l=0

Bk(

log[ρ]sρ− l)

= 1; Plot of square root of the B-spline, i.e.√Bk(

log[ρ]sρ− l)

;Ns−1∑l=0

√Bk(

log[ρ]sρ− l)≈ 1.

i.e. the image can be reconstructed from its scale-OS by adding over all scales and orientations.Moreover the real valued part of the kernel is appropriate for line detection and the imaginarypart of the kernel is appropriate for edge detection. In the following pages we give variousplots and figures for our choice of wavelet as well as the resulting Scale-OS of an image.

The structure of our kernels (Figure 2.2) in the Fourier domain looks very similar to curveletsin the Fourier domain as shown in Figure 1.3. Note that unlike the cake-kernels used in thisstudy which are based on a group-theoretic approach, the curvelets are based on the discretewavelet transform and do not involve a group theoretical structure, which we will exploit indesigning enhancement operators later on, in Section 4.1.


Figure 2.2: Plots of the graph of ψ = F−1R2 [ω →

√A(ϕ)B(ρ)], with A(ϕ), B(ρ) given in (2.52)

and (2.53) respectively determined by the discrete Fourier transform of ω →√A(ϕ)B(ρ) sam-

pled on a 100× 100 equidistant grid. Values chosen: Ns = 6, Nθ = 12. From top to bottom:Row1- (Spatial domain) Plots of the real part of kernels at a fixed orientation, but differentscales.Row2- (Spatial domain) Plots of the real part of kernels at a fixed scale, but different orienta-tions.Row3- (Spatial domain) Plots of the imaginary part of kernels at a fixed scale, but differentorientations.Row4- (Frequency domain)Plots of the of kernels at a fixed orientation, but different scales.Row5- (Frequency domain)Plots of the of kernels at a fixed scale, but different orientations.Row6- (Frequency domain)Plots of sum over all orientations at a fixed scale.


Figure 2.3: From left to right: Mψ; ψ = F−1[Mψ]; Sum of all rotated and translated kernelsin Fourier domain; Sum of all rotated and translated kernels in Spatial domain.

Figure 2.4: Illustration of invertible nature of wavelet transform on a section of Retinal Fundusimage (200× 200 pixels). Left: Original image; Right: Result of approximative reconstructionby only adding over all scales (Ns = 6) and orientations (Nθ = 12) of scale-OS (2.49). Asexpected the fast approximative reconstruction is very close to exact reconstruction.

Figure 2.5: Reconstruction per scale by summation over all orientations at a fixed scale; fromleft to right the first scale (low detail) to the last scale (high detail).

Figure 2.6: Scale-OS corresponding to different scales at a fixed orientation, i.e. following thesame direction.


Figure 2.7: Scale-OS of the Retinal Fundus image. Rows: Fixed orientation; Columns: Fixedscale

Chapter 3

The Similitude Group

In this chapter we introduce the differential-geometric structure of the SIM(2) group which isessential for designing left-invariant enhancement-contextual operators on the scale-OS. This isfollowed by introducing SIM(2)-convolution which is required for linear operators on scale-OS.

3.1 Geometry of the SIM(2) group

To perform analysis on the local structure in scale orientation scores, for instance in detectingthe presence of an oriented structure locally, differential geometry is an essential tool. In thissection we establish the basics of differential geometry on SIM(2). We briefly introduce Liegroups and Lie Algebras followed by applying this general setting to the specific case of theSIM(2) group. Group properties as well as the concept of left-invariant vectors and vectorfields is defined and the notion of exponential and logarithmic mapping is introduced. For ananalogous study in the case of Euclidean motion group SE(2), see [12, 21].

3.1.1 Lie Groups and Lie Algebras

A Lie Group G is an infinite group, such that the group elements are parametrized by a finitedimensional differentiable manifold endowed with a metric, i.e. there is a notion of distancebetween group elements. The fact that group elements form a manifold makes it possible touse differential calculus on Lie groups.

Formally, a (real/complex) Lie group is a set G such that,

(a) G is a group.

(b) G is a finite dimensional (real/complex) smooth manifold.

(c) The group multiplication in (a) of the product manifold

µ : G×G→ G : (x, y)→ xy

and the group inversion operation in (a)

ι : G×G : x→ x−1

are analytic functions relative to the structure in (b).

29

30 The Similitude Group

By looking at the differential properties at the unity element e of G we obtain the correspondingLie Algebra denoted by Te(G), which is a vector space endowed with a binary operator calledthe Lie bracket or commutator, [·, ·] : Te(G)× Te(G)→ Te(G) with the following properties

(a) Bilinearity: [aX + bY, Z] = a[X,Z] + b[Y, Z] and [Z, aX + bY ] = a[Z,X] + b[Z, Y ] for allX,Y, Z ∈ Te(G).

(b) Anticommutativity: [X,Y ] = −[Y,X] for all X,Y ∈ Te(G).

(c) Jacobi Identity: [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0 for all X,Y, Z ∈ Te(G).

The Lie group and the Lie algebra relate to each other by an exponential map, exp : Te(G)→ G,which can be defined in following equivalent ways:

(a) If G is a (N -dimensional) matrix Lie Group1, then both g ∈ G and X ∈ Te(G) can beexpressed as N × N square matrices G ∈ CN×N and X ∈ CN×N respectively, and theexponential map coincides with the matrix exponential and is given by the ordinary seriesexpansion,

G = exp(X) = IN×N +

∞∑n=1

Xn

n!, (3.1)

where IN×N is the N -dimensional identity matrix.

(b) Let X ∈ Te(G). Then, exp(X) = γ(1), where

γ : R→ G (3.2)

is the unique one parameter subgroup of G whose tangent vector at the identity is equalto X. It follows from the chain rule that, exp(tX) = γ(t). The map γ can be constructedas the integral curve of the left invariant vector fields associated with X. Note that, theintegral curve exists for all real parameters follows from left translating the solution nearzero.

The matrix representations of Lie groups are often employed, as they simplify computations.For instance, the Lie bracket is found by,

[Xi,Xj ] = XiXj −XjXi. (3.3)

A set of Lie algebra elementsAi(or the corresponding matrix representation Xi), i ∈ 1, 2, . . . , N,only form a valid N -dimensional basis for a Lie algebra if they are closed under the Lie bracket,i.e. one can write

[Ai, Aj ] =

N∑k=1

ckijAk, ∀ j ∈ 1, 2, . . . , N, (3.4)

where ckij are the structure constants. The table of Lie brackets [Ai, Aj ], for all i, j ∈ 1, 2, . . . , Ncompletely describes the structure of the Lie algebra. If the exponential mapping is surjective,it also completely captures the structure of the Lie group. Two Lie algebras are isomorphic ifthe table of Lie brackets is the same.

For more details on Lie groups and Lie algebras see [32]. In the rest of this chapter we considerthe particular case of 2D similitude group, SIM(2) and clarify the concepts mentioned in thissection in the context of SIM(2).

1Ado’s theorem states that every finite dimensional Lie algebra is isomorphic to a matrix Lie algebra. Forevery finite dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as itsLie algebra. So every abstract finite dimensional Lie algebra is the Lie algebra of some (linear) Lie group.

Geometry of the SIM(2) group 31

3.1.2 SIM(2) Group Properties

The similitude group, SIM(2) = R2 o (R+×SO(2)), is the group of planar rotations (SO(2)),translations (R2) and scaling (R+). It can be parametrized by the group elements g = (x, a, θ)where x = (x, y) ∈ R2 are the two spatial variables that in the case of scale orientation scoreslabel the domain of the image f , a is the scale and θ is the orientation angle. Group elementswill be denoted by g or the explicit notation (x, a, θ)2. For g = (x, a, θ), g′ = (x′, a′, θ′) ∈SIM(2) the group product g ·SIM(2) g

′ is given by3,

g ·SIM(2) g′ = (x, a, θ) ·SIM(2) (x′, a′, θ′) = (x + aRθx

′, aa′, θ + θ′),

g−1 = (−a−1R−θx, a−1,−θ). (3.5)

The identity element is given by eSIM(2) = (0, 0, 1, 0)4. Note that the SIM(2) group is notcommutative, i.e. in general, gg′ 6= g′g. This is due to the presence of the semi-direct product,denoted by the symbol ”o”, between R2 and (R+ × SO(2)).

The Lie algebra of5 SIM(2) is spanned by

Te(SIM(2)) = spanA2, A3, A4, A1 ≡ spanex, ey, ea, eθ, (3.6)

where ex, ey, ea, and eθ are the unit vectors in the x, y, a and θ direction respectively. Thechoice of index in the notation Ai will become clear in the subsequent chapters. Te(G) isequipped with a Lie bracket given by

[A,B] = limt↓0

t−2(a(t)b(t)a(t)−1b(t)−1 − e), (3.7)

where a, b : R→ SIM(2) are any smooth curves in SIM(2) with a(0) = b(0) = e and a′(0) = Aand b′(0) = B. For example, to find the Lie brackets of SIM(2) we can use for A2 : a(t) =(t, 0, 1, 0), A3 : a(t) = (0, t, 1, 0), A4 : a(t) = (0, 0, t + 1, 0) and for A1 : a(t) = (0, 0, 1, t),yielding the following Lie brackets

[A2, A3] = 0, [A2, A4] = −A2, [A2, A1] = −A3,

[A3, A4] = −A3, [A3, A1] = A2, [A4, A1] = 0. (3.8)

Therefore, the structure constants ckij , satisfying [Ai, Aj ] =N∑k=1

ckijAk, are given by,

0 A3 −A2 0−A3 0 0 −A2

A2 0 0 −A3

0 A2 A3 0

,

where i enumerates vertically and j enumerates horizontally.

Next we express the Lie algebra Te(SIM(2)) by the isomorphism of the group SIM(2) tomatrix group SIM(2) of 3× 3 matrices,

SIM(2) 3 (x, a, θ)↔(aRθ x0T 1

)∈ SIM(2), where Rθ =

(cos θ − sin θsin θ cos θ

).

2Throughout this thesis these two alternative notations will be used interchangeably.3For the sake of simplicity of notation, hereon we drop ·G to denote group product of a group G.4Here on for the ease of notation, the identity element of any group is denoted by e and the group involved

can be understood from the context.5Note that the Lie algebra Te(G) of a Lie group G is also the tangent space at the identity element e of G.


Then the basis of the Lie algebra are given by,

X1 =

0 −1 01 0 00 0 0

, X2 =

0 0 10 0 00 0 0

,

X3 =

0 0 00 0 10 0 0

, X4 =

1 0 00 1 00 0 0

, (3.9)

where 1, 2, 3, 4 := θ, x, y, a. For a derivation of these basis elements see Appendix(SecA.2).

Consequently the commutators are given using (3.3),

[X2,X3] = 0, [X2,X4] = −X2, [X2,X1] = −X3,

[X3,X4] = −X3, [X3,X1] = X2, [X4,X1] = 0. (3.10)

Note that, as expected the Lie brackets for the matrix representations match the commutatorsgiven in (3.8).

3.1.3 Group Representations

On orientation scores U ∈ L2(SIM(2)) we have two different representations, defined by

(Lg U)(h) = (U L−1g )(h) = U(g−1h), g, h ∈ SIM(2), (3.11)

(Rg U)(h) = (U R−1g )(h) = U(hg), g, h ∈ SIM(2), (3.12)

where Lg andRg are the left and right regular representations respectively6. Lg, Rg : SIM(2)→SIM(2), are defined as,

left-multiplication : Lg(h) = gh, right-multiplication : Rg(h) = hg. (3.13)

A function Φ ∈ L2(SIM(2)) is called left invariant if it commutes with the left regular repre-sentation, i.e. Φ Lg = Lg Φ, for all g ∈ SIM(2).

3.1.4 Left invariant Vector Fields

Now we present the concept of left-invariant vector fields in the context of G = SIM(2). Avector field (now considered as a differential operator7) A on a group G is called left-invariantif it satisfies

Agφ = Ae(φ Lg) = Ae(h 7→ φ(gh)),

for all smooth functions φ ∈ C∞c (Ωg) where Ωg is an open set around g ∈ G and with the leftmultiplication Lg : G→ G as defined in the previous section. The linear space of left-invariantvector fields L(G) equipped with the Lie product [A,B] = AB−AB is isomorphic to Te(G) bymeans of the isomorphism,

Te(G) 3 A↔ A ∈ L(G)⇔ Ag(φ) = A(φ Lg) = A(h 7→ φ(gh)) = (Lg)∗A(φ)

for all smooth φ : G ⊃ Ωg → R.

6Note that, Lg ,Rg : SIM(2)→ L2(SIM(2))7Note that any tangent vector X ∈ T (G), where T (G) is the tangent space of lie group G can be considered

as a differential operator acting on a function U : G → R. So, for instance, if we are using Xe ∈ Te(G) in thecontext of differential operators, all occurrences of ei will be replaced by ∂i, which is the short-hand notationfor ∂

∂xi.


We define an operator dR : Te(G)→ L(G)8as,

(dR(A)φ)(g) = limt↓0

(Rexp(tA)φ)(g)− φ(g)

t, A ∈ Te(G), φ ∈ L2(G), g ∈ G (3.14)

and where R and exp are the right regular representation and the exponential map (definedpreviously) respectively. Using dR, we obtain the corresponding basis for left-invariant vectorfields on G:

A1,A2,A3,A4 := dR(A1), dR(A2), dR(A3), dR(A4) (3.15)

or explicitly in coordinates

A1,A2,A3,A4 = ∂θ, ∂ξ, ∂η, ∂β = ∂θ, a(cos θ∂x + sin θ∂y), a(− sin θ∂x + cos θ∂y), a∂a,(3.16)

where recall that as differential operators,

A1, A2, A3, A4 = ∂θ, ∂x, ∂y, ∂a.

For example, we have

A3 = (dR(A3)φ)(g) = limt↓0


t

= limt↓0

φ(g ·G exp(tA3))− φ(g)

t= lim

t↓0

φ(b + teξ, a, θ)− φ(b, a, θ)

t= ∂ξ,

where, g = (b, a, θ) ∈ SIM(2), ·G denotes the SIM(2) group product and eξ =

(−a sin θa cos θ

).

Left invariant vector fields constructed in such a way have a useful property that, if Xe ∈ Te(G)

is such that Xe =4∑j=1

ciAi, then for any g ∈ G, the corresponding tangent vector Xg, is given

by Xg =4∑j=1

ciAi.

To simplify the scale related left invariant differential operator, we introduce a new variable,τ = loge a, which leads to the following change in left invariant derivatives

A1,A2,A3,A4 = ∂θ, eτ (cos θ∂x + sin θ∂y), eτ (− sin θ∂x + cos θ∂y), ∂τ. (3.17)

Also note that, this change of variable, leads to an alternative description of the SIM(2) group,

g = (x, τ, θ)

gg′ = (x, τ, θ)(x′, τ ′, θ′) = (x + eτRθx′, τ + τ ′, θ + θ′),

e = (0, 0, 0, 0),

where x =

(xy

). Note that substituting τ = 0, leads to a = eτ = 1 which explains the

difference in identity element of the SIM(2) group in the log description.

Recall that in the construction of the wavelet kernel, a log-scale based definition was given.The underlying motivation for that construction is the aforementioned variable change, τ =loge(a)9.

8Proposed in [12, Sec 4.8.1].9For the sake of simplicity the base e will be dropped in loge.


Figure 3.1: The difference between cartesian derivatives and left-invariant derivatives, shownon a scale-OS (at a fixed scale) of an image with a single circle. From left to right, severalorientations are shown. Row 1: Scale-OS of a circle image at a fixed scale; Row 2: Cartesianderivative ∂y; Row 3 : Left-invariant derivative ∂ηComparing the derivatives ∂y and ∂η we observe that ∂η is invariant under rotation, i.e. theinterpretation of ∂η stays the same.

The set of differential operators A1,A2,A3,A4 = ∂θ, ∂ξ, ∂η, ∂τ is the appropriate set ofdifferential operators to be used in orientation scores instead of the set A1, A2, A3, A4 =∂θ, ∂x, ∂y, ∂a (which would seem to be the logical choice because of the sampling axes), sinceit has the following advantages,

• Since all 4 differential operators ∂θ, ∂ξ, ∂η, ∂τ are left-invariant, we can be sure thatall SIM(2)-coordinate independent linear and nonlinear combinations of these operatorsare left invariant. This advantage will become clear in the subsequent chapters when wedeal with evolutions on the SIM(2) group.

• At each scale, the derivatives ∂θ, ∂ξ, ∂η have a clear interpretation, since at each scale ∂ξis always the spatial derivative tangent to the orientation θ and ∂η is always orthogonalto this orientation. Figure 3.1 illustrates this for δη versus δy.

Note that when constructing higher order left-invariant derivatives, the order in which thederivatives are applied is important. This is due to the fact that unlike derivatives ∂x, ∂y, ∂a, ∂θwhich commute, the left-invariant derivatives ∂ξ, ∂η, ∂τ , ∂θ do not commute. The nonzerocommutators are given by

[∂τ , ∂ξ] = ∂ξ, [∂τ , ∂η] = ∂η, [∂θ, ∂ξ] = ∂η, [∂θ, ∂η] = −∂ξ, (3.18)

where recall that [Ai,Aj ] = AiAj −AjAi.

For a more intuitive explanation of left-invariant differential operators in the case of SE(2)-group (easily visualized as it excludes scale) see Appendix(Sec. A.3).


3.1.5 Exponential Map

In this section we explicitly evaluate the exponential (exp) map and the logarithm (log) map.Recall the following definition from Section 3.1.1:

Let X ∈ Te(SIM(2)). Then, exp(X) = γc(1), where

γc : R→ SIM(2) (3.19)

is the unique one parameter subgroup of SIM(2) whose tangent vector at the identity is equalto X. It follows from the chain rule that, exp(tX) = γc(t). This map γc, can be constructedas the integral curve of the left invariant vector fields associated with X.

An exponential curve is a curve γc : R → SIM(2) for which the components of the tan-gent vector expressed in the left invariant basis A1,A2,A3,A4 are constant over the entireparametrization (if necessary the reparametrization of t), i.e.

d

dtγc(t) =

4∑i=1

ciAi∣∣∣∣g=γc(t)

, for all t ∈ R. (3.20)

These lines are analogous to straight lines in Rn, which have a also have constant tangentvector relative to the cartesian basis ex, ey.

An exponential curve is obtained by using the exp mapping of the Lie algebra elements, (3.19).That is, an exponential curve passing through the identity element e ∈ SIM(2) at t = 0 canbe written as

γc(t) = exp

(t

4∑i=1

ciAi∣∣∣∣g=e

)= exp

(t

4∑i=1

ciAi

), (3.21)

and an exponential curve passing through g0 ∈ SIM(2) can be obtained by left multiplicationwith g0 = (x0, y0, e

τ0 , θ0), i.e. g0γc(t).

Below we present two different methods of arriving at these exponential curves. The firstmethod involves a matrix-algebra approach based on (3.1). In this method we calculate expo-nential curves is by using the matrix Lie algebra elements in (3.9) and the matrix exponentdefined in (3.1). Let m(g) ∈ SIM(2) represent the matrix representation of g = (x, y, a, θ) ∈SIM(2). Then the exponential curves can be determined by solving

g = g0 exp(t

4∑i=1

ciAi) (3.22)

which is equivalent to solving

m(g) = m(g0) m

(exp

(t

4∑i=1

ciAi

))

or m(g) = m(g0) exp

(t

4∑i=1

ciXi

). (3.23)

Recall that, this method works as exp map corresponds to the matrix exponential in case ofmatrix Lie groups.

Now we present a different method for evaluating exponential curves. This is a more elegantapproach based on the method of characteristics for PDEs. We begin by proving the followingresult.


Theorem 3.1. etdR(A) = RetA , ∀t > 0, ∀A ∈ Te(SIM(2)) and dR is defined in (3.14).

Proof. We need to prove that etdR(A)U = RetAU , where U ∈ D(dR(A))10, ∀A ∈ Te(G), ∀t > 0.Note that etdR(A)U is a solution for ∂W

∂t = dR(A)W with W (g, 0) = U . So to prove this

theorem, we need to show that,∂(RetAU)

∂t = dR(A)(RetAU), which follows from,

∂(RetAU)

∂t= limh→0

Re(t+h)AU − RetAUh

= limh→0

RehARetAU − RetAUh

= limh→0

[(RehA − I

h

)RetAU

]= dR(A)(RetAU). (3.24)

Note that the limit used above is defined on the space H = L2(SIM(2)). To justify the laststep of (3.24) we need to show that, U ∈ D(dR(A)) ⇒ RetAU ∈ D(dR(A)), which is provedin the next lemma.

Lemma 3.2. U ∈ D(dR(A))⇒ RetAU ∈ D(dR(A)) and generalized derivatives exist.

Proof. We first note that, U ∈ D(dR(A))⇒ (dR(A)U) ∈ H, where H = L2(SIM(2)). Then,RetAU ∈ D(dR(A)) follows as [dR(A)RetA ](U) = [RetAdR(A)](U), ∀U ∈ D(dR(A)).

[dR(A)RetA ](U)(g) =

[limh→0

RehA − Ih

](RetAU)(g)

= limh→0

RehARetA(U(g))−RetA(U(g))

h

= limh→0

RetARehA(U(g))−RetA(U(g))

h

= [dR(A)RetAdR](U)(g)

where we have used the fact that, Rh(U(g)) = U(gh) and etAehA = e(t+h)A = ehAetA.

Corollary 3.3. γc(t) = g0 exp

(t

4∑i=1

ciAi

)are the characteristics for the following PDE,

∂W∂t = −

4∑i=1

ciAiW

W (g, 0) = U.

(3.25)

Proof. Let dR(A) = A =4∑i=1

ciAi. Then the solution for (3.25) is, W (·, t) = e−tdR(A)U =

Re−tAU , where the second equality follows from Theorem 3.1. Thus we have,

W (g, t) = U(ge−tA) ∀g ∈ SIM(2), ∀t > 0⇒W (g0etA, t) = U(g0) ∀t > 0. (3.26)

Remark. The computation of the exponential curves in a fixed frame of reference follows fromthe chain rule,

∂w

∂t=

4∑i=1

f i∂w

∂vi, (3.27)

and that 〈dvi, γ〉 = f i. Thus solving the ODE system γi = f i, i ∈ 1, 2, 3, 4, we arrive at theexponential curves.

10D(A) denotes the domain of the operator A.

SIM(2)-Convolution 37

Thereby the explicit formulation of the exponential curves g0γc(t) = (x(t), y(t), τ(t), θ(t)) pass-ing through g0 = x0, y0, e

τ0 , θ0 at t = 0 is,

x(t) =1

c21 + c24[eτ0c1

((− sin [θ0] + etc4 sin [tc1 + θ0]

)c2 +

(− cos [θ0] + etc4 cos [tc1 + θ0]

)c3)

+ c21x0 +

c4(eτ0(−cos [θ0] + etc4 cos [tc1 + θ0]

)c2 + eτ0

(sin [θ0]− etc4 sin [tc1 + θ0]

)c3 + c4x0

)]

y(t) =1

c21 + c24[eτ0c1

((cos [θ0]− etc4 cos [tc1 + θ0]

)c2 +

(− sin [θ0] + etc4 sin [tc1 + θ0]

)c3)

+ c21y0 +

c4(eτ0(− sin [θ0] + etc4 sin [tc1 + θ0]

)c2 + eτ0

(− cos [θ0] + etc4 cos [tc1 + θ0]

)c3 + c4y0

)]

τ(t) = tc4 + τ0

θ(t) = tc1 + θ0. (3.28)

Now we present an important result for the exp map in the SIM(2) group.

Theorem 3.4. The exponential map, exp : Te(SIM(2)) → SIM(2) is surjective, i.e. for allg ∈ SIM(2), there exists a X ∈ Te(SIM(2)) such that, g = exp(X).

An elegant proof for this theorem is provided in Appendix(Sec. A.4).

From the above theorem exp map is surjective and thus we can define the log map, log =(exp)−1 : SIM(2)→ Te(SIM(2)). To explicitly determine the log map, we solve for c1, c2, c3, c4

from the equality, g = exp

(t

4∑i=1

ciAi

), where g ∈ SIM(2) and ci’s are as defined earlier. This

is achieved by substituting g0 = e, i.e. x0 = y0 = θ0 = τ0 = 0, in (3.28) yielding,

c1 = θ/t

c2 =yθ − xτ − eτ (yθ − xτ) cos θ + eτ (xθ + yτ) sin θ

t (1 + e2τ − 2eτ cos θ)

c3 =−xθ − yτ + eτ (xθ + yτ) cos θ + eτ (yθ − xτ) sin θ

t (1 + e2τ − 2eτ cos θ)

c4 = τ/t, (3.29)

where recall that g = (x, y, eτ , θ) ∈ SIM(2). The formulae in (3.29) can be written in asimplified form,

c1 = θ/t

c2 =(yθ − xτ) + (−θη + τξ)

t (1 + e2τ − 2eτ cos θ)

c3 =−(xθ + yτ) + (θξ + τη)

t (1 + e2τ − 2eτ cos θ)

c4 = τ/t, (3.30)

where we have made use of the definition of ξ and η,

ξ = eτ (x cos θ + y sin θ), η = eτ (−x sin θ + y cos θ).

This explicit form of the log map will be used to approximate the solution for evolutions onSIM(2) in the coming chapters.

3.2 SIM(2)-Convolution

It will be shown in the next chapter, Section 4.2, that all linear left invariant operators on thespace of scale-OS are a group convolution with an appropriate kernel. In this section we briefly


treat the mathematical properties of SIM(2)-convolution. The generalized group convolutionin the case of the SIM(2)-group can be written as,

(Ψ ∗SIM(2) U)(g) =

∫SIM(2)

Ψ(h−1g)U(h)dh,

where U ∈ L2(SIM(2)) (in our case a scale orientation score of an image) and Ψ ∈ L2(SIM(2))is the SIM(2) convolution kernel. Explicitly,

(Ψ ∗SIM(2) U)(x, a, θ) =

∫R2

2π∫0

∫R+

Ψ

(1

a′R−1θ′ (x− x′),

a

a′, θ − θ′

)U(x′, a′, θ′)

1

(a′)3dx′da′dθ′,

(3.31)where the measure dµSIM(2)(h) = dh = 1

(a′)3 dx′da′dθ′ is the left-invariant Haar measure

defined on SIM(2) and g = (x, a, θ), h = (x′, a′, θ′) ∈ SIM(2).

Below we summarize a few properties of the SIM(2)-convolution. Most of these propertieshave a fairly straightforward proof. Assume that Ψ,Ψ1,Ψ2, U, U1, U2 ∈ L2(SIM(2)). Then,

• Associativity:

(Ψ1 ∗SIM(2) Ψ2) ∗SIM(2) U = Ψ1 ∗SIM(2) (Ψ2 ∗SIM(2) U).

This also holds for scalar multiplication, i.e.

a(Ψ ∗SIM(2) U) = ((aΨ) ∗SIM(2) U) = (Ψ ∗SIM(2) (aU)),

for all a ∈ C.

• Distributivity:

(Ψ1 + Ψ2) ∗SIM(2) U = Ψ1 ∗SIM(2) U + Ψ1 ∗SIM(2) U

andΨ ∗SIM(2) (U1 + U2) = Ψ ∗SIM(2) U1 + Ψ ∗SIM(2) U2. (3.32)

• Identity: δ ∗SIM(2) U = U ∗SIM(2) δ = U , where δ : SIM(2) → C is defined byδ(g) = δ(x, y, a, θ) = δ(x)δ(y)δ(a)δ(θ), g = (x, y, a, θ) ∈ SIM(2)11. Note that in ourcase of scale-OS we can reproduce back the data using the kernel, δ : (x, y, a, θ) 7→δ(x)δ(y)δ(θ − θ0)δ(τ − τ0), where θ0, τ0 belong to the set of discrete orientations andscales(log sampling), used in the construction of scale-OS, respectively.

• Invariance: A SIM(2)-convolution is left invariant on it’s left-input, i.e.

Lg(Ψ ∗SIM(2) U) = Ψ ∗SIM(2) (LgU), for all g ∈ SIM(2). (3.33)

See Def. 4.1 for the definition of left invariant operator. This follows from the discussionin the previous subsection. Note that (3.33) is similar to the translation invarianceproperty of the usual convolution on Rd.

• Differentiation Rule: D(Ψ∗SIM(2)U) = ((DΨ)∗SIM(2)U), whereD denotes a left invariantderivative operator (see Section 3.1.4). This property follows from the following argument

D(Ψ ∗SIM(2) U)(g) = D∫

SIM(2)

(Lh Ψ)(g)U(h)dh =

∫SIM(2)

(D Lh Ψ)(g)U(h)dh

=

∫SIM(2)

(Lh (D Ψ))(g)U(h)dh = ((DΨ) ∗SIM(2) U)(g). (3.34)

Note that this property does not hold for general differential operators.

11Here the δ defined on a one dimensional space are the usual dirac delta functions.

SIM(2)-Convolution 39

Figure 3.2: Schematic view of a SE(2)-convolution (left) and an R3-convolution (right). Anyconvolution K ∗ f can be envisioned by the kernel K ”moving” over the entire domain of fwhere the result of the convolution is found by taking inner products at all positions. Thekernel is indicated as a box, and the 3 axes denote the orientation score domain and 3D imagedomain respectively. As indicated by the arrows, in both cases the kernel is moving over thex, y-plane. In the case of the SE(2)-convolution, there is an additional twist when moving inthe orientation dimension.

Below we explicitly state the SE(2)-convolution and relate it to SIM(2)-convolution. This isimportant as it leads to an easier implementation of the SIM(2)-convolution. Let Ψ′, U ′ ∈L2(SE(2)). Explicitly,

(Ψ′ ∗SE(2) U′)(x, θ) =

∫R2

2π∫0

Ψ′(R−1θ′ (x− x′), θ − θ′)U(x′, θ′)dθ′dx′, (3.35)

where g = (x, θ) ∈ SE(2). Figure 3.2 schematically explains how an SE(2)-convolution canbe envisioned.

Consider (Ψ ∗SIM(2) U), where Φ, U ∈ L2(SIM(2)) and g = (x, a, θ) ∈ SIM(2);

(Ψ ∗SIM(2) U)(g) =

∫R2

2π∫0

∫R+

Ψ

(1

a′R−1θ′ (x− x′),

a

a′, θ − θ′

)U(x′, a′, θ′)

1

(a′)3dx′da′dθ′

=

∫R+

[Ψττ ′ ∗SE(2) U(·, τ ′, ·)

](x, θ)dτ ′ (3.36)

where note that we have transformed the scale a as τ = log(a). For arbitrarily fixed values ofτ, τ ′, Ψτ

τ ′ : SE(2)→ C is defined as,

Ψττ ′(x, θ) =

1

(eτ ′)2Ψ( x

eτ ′,τ

τ ′, θ). (3.37)

Note that as Ψ ∈ L2(SIM(2)), Ψττ ′ ∈ L2(SE(2)) and hence the convolution is well defined.

We use this straightforward relation to implement SIM(2)-convolution in terms of SE(2)-convolution. In the Appendix(Sec A.5) we briefly describe the implementation of SE(2)-convolution. For a complete treatment of SE(2)-convolution see [21, Ch.3].

3.2.1 SIM(2)-convolution vs SIM(2)-Correlation

For the sake of completeness we present the concept of SIM(2)-correlation.


Figure 3.3: Example showing the difference between a kernel and its corresponding ”mirrored”kernel. Left: an arbitrary SE(2)-kernel Ψ. Right: Ψ according to, Ψ(x, θ) = Ψ(−R−1

θ x,−θ),is a mirrored version of Ψ with an additional twist over the orientation axis. Note that anSE(2)-convolution with the kernel on the left is the same as an SE(2)-correlation with thekernel on the right and vice versa.

Analogously to the R2-correlation, the SIM(2)-correlation is defined as

(Ψ ?SIM(2) U)(g) =

∫SIM(2)

Ψ(g−1h)U(h)dh, (3.38)

where Ψ, U ∈ L2(SIM(2)) and Ψ(g) = Ψ(g−1)for all g ∈ SIM(2). This shows that Ψ ∗SIM(2)

U = Ψ ?SIM(2) U . Explicitly this yeilds,

(Ψ ?SIM(2) U)(x, a, θ) =

∫R2

2π∫0

∫R+

Ψ

(1

aR−1θ (x′ − x),

a′

a, θ′ − θ

)U(x′, a′, θ′)

1

(a′)3dx′da′dθ′,

(3.39)where Ψ(x, a, θ) = Ψ(− 1

aR−1θ x, 1

a ,−θ). This structure of SIM(2) group indicates that trans-forming an SIM(2)-convolution kernel into an SIM(2)-correlation kernel is not only accom-plished by mirroring the x, y, θ axes but also involves a rotation as well as scaling, i.e. the cor-relation kernel would involve a twisting and scaling of the convolution kernel. Since visualizingthis kernel is difficult for the SIM(2) case due to the presence of four variables, (x, y, a, θ), weillustrate this difference between correlation and convolution graphically, using the a SE(2) ker-nel. Let Ψ ∈ L2(SE(2)), and the corresponding correlation kernel, Ψ(x, θ) = Ψ(−R−1

θ x,−θ).Note that this can be graphically visualized as ”twisting” of the kernel and not scaling, seeFigure 3.3.

Note that , it is preferable to use SIM(2)-convolution instead of SIM(2)-correlation asSIM(2)-correlation does not fulfill the properties of associativity, identity and the differen-tiation rule.

Chapter 4

Image Enhancement via LeftInvariant Operations on Scale-OS

We start by describing “legal” operators on scale-OS and then show that such legal linearoperators are group convolutions with an appropriate kernel. We then give an approximationof the contour enhancement heat kernels for the linear left-invariant diffusion process on theSIM(2) group. We then introduce adaptive diffusion on the SIM(2) group in a rigorousmanner using a differential-geometric approach. We conclude the chapter by applying SE(2)based adaptive diffusion (CED-OS) to the scale-OS.

4.1 Legal Operators on the Scale-OS

Let ψ be an admissible wavelet as described in Section 2.4 and let G = SIM(2), then thereexists exists a 1-to-1 correspondence between bounded operators Φ ∈ B(CGK) on orientationscores and bounded operators Υ ∈ B(L2(Rd)):

Υ[f ] = (Wψ)∗[Φ[Wψ[f ]]], f ∈ L2(Rd), (4.1)

which allows us to relate operations on orientation scores to operations on images in a robustmanner. To get a schematic view of the operations, see Figure 4.1.

Recall from Corollary 2.11 that CGK is the space of orientation scores as a closed linear subspaceof Hψ, which is a vector subspace of L2(G). For proper wavelets we have (approximative) L2-norm preservation, so then we have L2(G) ∼= Hψ. In this section we set L2(G) = Hψ, to avoidtechnical subtleties concerning approximations.

At this point we remark that if, Φ : L2(G) → L2(G) is a bounded operator on L2(G), thenthe range of restriction of this operator to the subspace CGK of orientation scores need not becontained in CGK , i.e. Φ(Uf ) need not be the orientation score of an image. Recall the adjointmapping of Wψ : L2(Rd)→ L2(G), given by

(Wψ)∗(V ) =

∫G

Ugψ V (g)dµG(g), V ∈ L2(G).

The operator Pψ = Wψ(Wψ)∗ is the orthogonal projection on the space of orientation scoresCGK . This projection can be used to decompose the manipulated orientation score:

Φ(Uf ) = Pψ(Φ(Uf )) + (I − Pψ)(Φ(Uf )).

41

42 Image Enhancement via Left Invariant Operations on Scale-OS

Figure 4.1: A brief schematic view on image processing via invertible scale orientation scores.The dotted arrow Υ represents the effective operation on the image. See text for details.

Notice that the orthogonal complement (CGK)⊥, which equals R(I − Pψ), is exactly the null-space of (Wψ)∗ as N ((Wψ)∗) = (R(Wψ))⊥ = (CKG )⊥ and so

[(Wψ)∗ Φ Wψ][f ] = [(Wψ)∗ Pψ Φ Wψ][f ], (4.2)

for all f ∈ L2(Rd) and all Φ ∈ B(L2(G)).

Definition 4.1. An operator Φ : L2(G)→ L2(G) is left invariant iff

Φ[Lhf ] = Lh[Φf ], for all h ∈ G, f ∈ L2(R2), (4.3)

where the left regular action Lg of g ∈ G onto L2(G) is given by

Lgψ(h) = ψ(g−1h) = ψ

(1

aR−1θ (b′ − b),

a′

a, θ′ − θ

), (4.4)

with g = (b, a, θ) ∈ G, h = (b′, a′, θ′) ∈ G.

Theorem 4.2. Let Φ be a bounded operator on CGK , Φ : CGK → L2(G). Then the uniquecorresponding operator Υ on L2(Rd), which is given by Υ[f ] = (Wψ)∗ Φ Wψ[f ] is Euclideaninvariant, i.e. UgΥ = ΥUg for all g ∈ G if and only if Pψ Φ is left invariant, i.e. Lg(Pψ Φ) =(Pψ Φ)Lg, for all g ∈ G.

Proof. As,

Wψ[Ug[f ]](h) = (Uhψ,Ugf)L2(Rd) = (Ug−1hψ, f)L2(Rd) = Lg[Wψ[f ]](h)

we conclude that,WψUg = LgWψ, for all g ∈ G. (4.5)

Moreover,

(WψUgf, U)L2(G) = (LgWψ, U)L2(G) ⇔ (f,Ug−1(Wψ)∗U)L2(G) = (f, (Wψ)∗Lg−1U)L2(G)

for all U ∈ L2(G), f ∈ L2(Rd), g ∈ G and therefore we have,

Ug(Wψ)∗ = Lg(Wψ)∗, for all g ∈ G. (4.6)

(Necessary condition) Assuming that (Pψ Φ) is left invariant it follows from (4.2), (4.5) and(4.6) that

Υ[Ugf ] = (Wψ)∗ Φ Wψ Ug[f ]

= (Wψ)∗ (Pψ Φ) Wψ Ug[f ]

= (Wψ)∗ (Pψ Φ) Lg W%ψ[f ] (4.7)

= (Wψ)∗ Lg (Pψ Φ) Wψ[f ]

= Ug (Wψ)∗ (Pψ Φ) Wψ[f ]

= Ug (Wψ)∗ Φ Wψ[f ] = Ug[Υ[f ]]

Left Invariant Linear Bounded Operations are Group Convolutions 43

for all f ∈ L2(R2) and g ∈ G. Thus we have ΥUg = UgΥ for all g ∈ G.

(Sufficient condition) Now suppose Υ is Euclidean invariant. Then again by (4.5) and (4.6) wehave that,

(Wψ)∗ Φ Lg Wψ[f ] = (Wψ)∗ Lg Φ Wψ[f ],

for all f ∈ L2(R2) and g ∈ G. Since the range of Lg∣∣CGK

and the range of (Pψ Φ) is contained

in CGK and since (Wψ)∗∣∣CGK

= (Wψ)−1, we have (Pψ Φ) Lg Wψ = Lg (Pψ Φ) Wψ. As

the range of Wψ equals CGK , we have, Lg (Pψ Φ) = (Pψ Φ) Lg for all g ∈ G.

Practical Consequence: Euclidean invariance of Υ is of great practical importance, since theresult should not be essentially different if the original image is rotated or translated. Inaddition in our construction scaling the image also does not affect the outcome of the operation.Note that it is not a problem when the mapping Φ : CGK → L2(G) maps an orientation scoreto an element in L2(G)\CGK as long as one is aware from (4.2) that PψΦ : CGK → CGK yields thesame result.

4.2 Left Invariant Linear Bounded Operations are GroupConvolutions

We start off by showing that all left invariant bounded linear operators on the space of orien-tation scores are group convolutions(defined appropriately). Note that this is possible as thegroup under consideration has a measure defined on itself (left/right invariant Haar measure).

Theorem 4.3 (Dunford-Pettis). Let X be a measurable space, with measure µ : X → R+. Let1 ≤ p < ∞. Let A be a bounded linear operator from Lp(X) into L∞(X), then there exists aK ∈ L1(X ×X) such that

supx

(∫X

|K(x, y)|qdµ(y)

) 1q

= ‖A‖,

with q > 0 such that 1p + 1

q = 1 and for all f ∈ Lp(X) we have that

(Af)(x) =

∫X

K(x, y)f(y)dµ(y)

for almost every x ∈ X.

For a proof see [40]. In our case of the SIM(2) group this translates as follows: All boundedlinear operators Φ : L2(SIM(2))→ L2(SIM(2)) are kernel operators, i.e. there exists a kernelK : SIM(2)× SIM(2)→ C such that,

(Φ(U))(g) =

∫SIM(2)

K(g, h)U(h)dh, g, h ∈ SIM(2) (4.8)

and the kernel satisfies,

supg

∫SIM(2)

|K(g, h)|2dh

12

<∞.


Note that there is a subtlety involved which allows the usage of Dunford-Pettis theoremin our case. We assume, Φ : L2(SIM(2)) → L2(SIM(2)) ∩ L∞(SIM(2)), such that, Φ :L2(SIM(2))→ L2(SIM(2)) and Φ : L2(SIM(2))→ L∞(SIM(2)) is a bounded operator1.

If Φ is left invariant (as required), i.e. Φ Lp U = Lp Φ U (recall Def. 4.1), using (4.8) weobtain,

(Φ Lp U)(g) =

∫SIM(2)

K(g, h)(Lp U)(h)dh

=

∫SIM(2)

K(g, h)U(p−1h)dh (4.9)

=

∫SIM(2)

K(g, ph)U(h)dh (substitute h← ph)

and

(Lp Φ U)(g) =

∫SIM(2)

K(p−1g, h)U(h)dh. (4.10)

Note that in the arguments made above the choice of p ∈ SIM(2) and U ∈ L2(SIM(2)) isarbitrary. For the right-hand sides of (4.9) and (4.10) to be equal, K(g, ph) = K(p−1g, h) musthold for all p, g, h ∈ SIM(2). Replacing g by pg we obtain K(pg, ph) = K(g, h), and thus wesee that the kernel K should be invariant under the left-multiplication of its arguments by anarbitrary element of SIM(2). So K(g, h) = K(e, g−1h), whence if we define a new functionΨ : SIM(2) → C such that, Ψ(g) = K(e, g−1) for g ∈ SIM(2), then using (4.8), we arrive atthe SIM(2) convolution

(Ψ ∗SIM(2) U)(g) =

∫SIM(2)

Ψ(h−1g)U(h)dh. (4.11)

(4.11) is a straightforward generalization of Rn-convolution to any other Lie-group, by replacingSIM(2) by any other group. For e.g. we obtain the Rn- convolution if we use the translationgroup Rn in (4.11), i.e.

(f ∗Rn g)(x) =

∫Rn

f(x− x′)g(x′)dx′, (4.12)

where f, g ∈ L1(Rd).

4.3 Basic Left Invariant Operations on Scale-OS

In image analysis, Euclidean invariant differential operators are used for corner/line/edge/blobdetection. Often, these differential invariants are expressed in a local coordinate system (gaugecoordinates). See [19] for construction of such detectors using gauge coordinates.

Below we present a few simple global left invariant operators used to enhance elongated struc-tures.

1This is a practical assumption as output of the operators acting on the orientation scores must be boundedin addition to being square integrable.

Introduction to Diffusions on SIM(2) group 45

• Normalization. See Figure (4.2).

[Φ(Uf )](b, a, θ) =

Uf (b, a, θ)/

(∫R2

|Uf (x, a, θ)|pdx

) 1p

, 1 < p <∞

Uf (b, a, θ)/maxx∈R2

|Uf (x, a, θ)| , p =∞.

• Grey-value Transformations.

Φ(Uf ) = (Uf −mingUf (g))p, p > 0.

This operation can be used to enhance the strongly oriented structures in the score andreduce the noise/weakly oriented structures in the score, see Figure (4.3). Note thatthis operation does not correspond to a simple grey-value transformation on the originalimage as (f)p 6= (Wψ)∗[(Wψ[f ])q].

4.4 Introduction to Diffusions on SIM(2) group

In this section we apply the general theory of diffusions on Lie groups, [14], to the SIM(2)group and consider the following left-invariant second-order evolution equations,

∂tW (g, t) = QD,a(A1,A2,A3,A4)W (g, t),

W (·, t = 0) =Wψf(·),(4.13)

where W : SIM(2)× R+ → C and QD,a is the following quadratic form on L(SIM(2)),

QD,a(A1,A2,A3,A4) =

4∑i=1

−aiAi +

4∑j=1

DijAiAj

, ai, Dij ∈ R, D := [Dij ] ≥ 0, DT = D.

(4.14)When Dij = Diiδij , i, j ∈ 1, 2, 3, 4, the quadratic form becomes,

QD,a(A1,A2,A3,A4) = [−a1∂θ−a2∂ξ−a3∂η−a4∂τ+D11(∂θ)2+D22(∂ξ)

2+D33(∂η)2+D44(∂τ )2].(4.15)

The first order part of (4.15) takes care of transport (convection) along the exponential curves2,deduced in Section 3.1.5. The second order part takes care of diffusion in the SIM(2) group.Note that the non-commutative nature of the SIM(2) group, recall (3.18), makes these evolu-tion equations complicated. Also note that these evolution equations are left-invariant as theyare constructed by linear combinations of left-invariant vector fields.

Hormander, [26], gave necessary and sufficient conditions on the convection and diffusion pa-rameters, respectively a = (a1, a2, a3, a4) and D = Dij , in order to get smooth Green’s func-tions of the left-invariant convection-diffusion equation (4.13) with generator (4.15). By theseconditions the non-commutative nature of SIM(2), in certain cases, takes care of missingdirections in the diffusion tensor. Below we formulate the Hormander’s theorem.

A differential operator L defined on a manifold M of dimension n ∈ N , n <∞, is called hypo-elliptic if for all distributions f defined on an open subset of M such that Lf is C∞ (smooth), fmust also be C∞. In his paper [26], Hormander presented a sufficient and essentially necessary

condition for an operator of the type L = c + X0 +r∑i=1

(Xi)2, r ≤ n, where Xi are vector

2Later in the chapter we prove that diffusion in our case take place only along exponential curves.


Figure 4.2: Image processing via Normalization of the SOS layers. Normalization of theorientation layers in the scale-OS reveals the most contrasting lines. Left: Original Image,Center: Normalization for p <∞, Right: Normalization for p =∞.

Figure 4.3: Image processing via Monotonic Grey-value transformation. Left: Noisy image,Right: Grey-value transformation for p=3.

Approximate Contour Enhancement Kernels on Scale-OS 47

fields on M , to be hypo-elliptic. This condition referred to as the Hormander’s condition, isthat among the set

Xj1, [Xj1, Xj2], [Xj1, [Xj2, Xj3]], . . . , [Xj1, [Xj2, [Xj3, . . . Xjk]]]| ji ∈ 0, 1, . . . r, (4.16)

there exist n elements which are linearly independent at any given point in M . In case M is aLie group and we restrict ourselves to left-invariant vector fields, then it is sufficient to checkwhether the vector fields span the tangent space at the unity element. So the necessary andsufficient conditions for smooth (resolvent) Green’s functions on SIM(2)\e on the diffusionand convection parameters (D, a) in the generator (4.15) of (4.13) for diagonal D are

1, 2, 4 ⊂ i| ai 6= 0 ∨Dii 6= 0 ∨ 1, 3, 4 ⊂ i| ai 6= 0 ∨Dii 6= 0. (4.17)

By results in Section 4.2, the solutions of (4.13) are given by a SIM(2)-convolution with the

corresponding Green’s function KD,at ,

W (g, t) = (KD,at ∗SIM(2) U)(g) =

∫SIM(2)

KD,at (h−1g)U(h)dµSIM(2)(h). (4.18)

In this study we are predominantly concerned with contour enhancement. So we consider aspecial case of (4.13) where

Dij = Diiδij , i, j ∈ 1, 2, 3, 4, D33 = 0 and a = 0, (4.19)

and arrive at Gaussian estimates of the Green’s function for this evolution equation. Notethat this particular choice of parameters satisfies the Hormander’s condition and therefore theGreen’s function of this diffusion process will be smooth. The choice a = 0, reflects that we donot allow any transport along the curves in the SIM(2) group while the choice of the diffusiontensor D is based on the fact that we wish to diffuse along the elongated structures and notorthogonal to it.

4.5 Approximate Contour Enhancement Kernels on Scale-OS

Compared to diffusion on Rn, where the Green’s function is simply a Gaussian, the Green’sfunction of diffusion on non-commutative groups is complicated. In [13, 17], the authors derivethe exact Green’s function for (4.13) with conditions similar to (4.19) for the SE(2) case.Explicit and exact formulas for heat kernels for the SIM(2) case do not exist in the literature.

However there exists a general theory, see [29, 35], which provides Gaussian estimates forGreen’s function of left-invariant diffusions on Lie groups, generated by subcoercive operators.In this section we employ this general framework to our case of interest. We first carry outthe method of contraction which serves as an essential pre-requisite for the Gaussian estimatesand approximation kernels later in this chapter.

4.5.1 Approximation of SIM(2) by a Nilpotent Group via Contrac-tion

We follow the general framework by ter Elst and Robinson [35], which involves semigroups onLie groups generated by subcoercive operators. In their work we consider a particular case


by setting the Hilbert space, H = L2(SIM(2)), the group G = SIM(2) and the right-regularrepresentation R. Furthermore we consider the algebraic basis A1 = ∂θ,A2 = ∂ξ,A4 = ∂τleading to the following filtration of the Lie algebra

g1 := spanA1,A2,A4,⊂ g2 = [g1, g1] = spanA1,A2,A3,A4 = L(SIM(2)). (4.20)

Based on this filtration we assign the following weighs to the generators:

w1 = w2 = w4 = 1 and w3 = 2. (4.21)

For e.g. w2 = 1 since A2 occurs first in g1, while w(3) = 2 since A3 occurs in g2 and not g1.Now based on these weights we define the following dilations on the Lie algebra Te(SIM(2))(recall Ai = Ai|e),

γq

(4∑i=1

ciAi

)=

4∑i=1

qwiciAi, ∀ci ∈ R

γq(x, y, τ, θ) =

(x

qw2,y

qw3,τ

qw4, ei

θqw1 ,

)with the weights wi defined in (4.21), and for 0 < q ≤ 1 we define the Lie product [A,B]q =γ−1q [γq(A), γq(B)]. Let (SIM(2))t be the simply connected Lie group generated by the Lie al-

gebra (Te(SIM(2)), [·, ·]q)3. The group products on these intermediate groups (SIM(2))q∈(0,1]

are given by,

(x, y, τ, θ)·q(x′, y′, τ ′, θ′) = (x+eτq[cos(qθ)x′−q sin(qθ)y′], y+eτq[sin(qθ)

qx′+cos(qθ)y′], τ+τ ′, θ+θ′).

(4.22)Note that the dilation on the Lie algebra coincides with the pushforward of the dilation on thegroup γq = (γq)∗ and thus the left-invariant vector fields on (SIM(2))t are given by

Aqi |g = (γ−1q Lg γq)∗Ai,

∀q ∈ (0, 1] which leads to,

Aqi |gφ = (γ−1q Lg)∗(γq)∗Aiφ = (γ−1

q Lg)∗(γq)(Ai)φ = twi(γ−1q Lg)∗(Ai)φ

= qwi(γ−1q )∗Ai|gφ = qwiAi|γ−1

q g(φ γq)

for all smooth complex valued functions φ defined on a open neighbourhood around g ∈SIM(2). So for all g = (x, y, τ, θ) ∈ SIM(3) we have,

At1|g = q(1

q∂θ) = ∂θ

Aq2|g = q

[eτq(

cos(qθ)

q∂x +

sin(qθ)

q2∂y

)]= eτq

(cos(qθ)∂x +

sin(qθ)

q∂y

)(4.23)

Aq3|g = q2

[eτq(− sin(qθ)

q∂x +

cos(qθ)

q2∂y

)]= eτq (−q sin(qθ)∂x + cos(qθ)∂y)

Aq4|g = ∂τ .

For example,

Aq3|g = (γ−1q Lg γq)∗A3 = (γ−1

q Lg)∗(γq)∗A3 = (γ−1q Lg)∗(q2∂y)

= q2(γ−1q )∗(∂η) = q2eτq(γ−1

q )∗(− sin(qθ)∂x + cos(qθ)∂y)

= q2eτq(− sin(qθ)γ−1q (∂x) + cos(qθ)γ−1

q (∂y)) = q2eτq(− sin(qθ)

q∂x +

cos(qθ)

q2∂y).

3Recall that (SIM(2))q = expq(Te(SIM(2))), Section 3.1.1.

Approximate Contour Enhancement Kernels on Scale-OS 49

Note that [Ai, Aj ]q = γ−1q [γq(Ai), γq(Aj)] = γ−1

q qwi+wj [Ai, Aj ] =4∑k=1

qwi+wj−wkckijAk and

therefore

[A1, A2]q = A3, [A1, A3]q = −q2A2

[A4, A2]q = qA2, [A4, A3]q = qA3. (4.24)

Analogously to the case q = 1, (SIM(2))q=1 = SIM(2) there exists an isomorphism of the Liealgebra at the unity element Te((SIM(2))q) and the left-invariant vector fields on the groupL((SIM(2))q):

(Ai ↔ Aqi and Aj ↔ Aqj)⇒ [Ai, Aj ]q ↔ [Aqi ,Aqj ]. (4.25)

It can be verified that the left invariant vector fields Aqi satisfy the same commutation relationsas (4.24).

Consider the case H ≡ limq↓0

(SIM(2))q. In this case the left-invariant vector fields are given by

A01 = ∂θ, A0

2 = ∂x + θ∂y, A03 = ∂y, A0

4 = ∂τ . (4.26)

So, the homogeneous nilpotent contraction Lie group equals

H3 = limq↓0

(SIM(2))q and SIM(2) = (SIM(2))q=1/(0 × 0 × 0 × 2πZ), (4.27)

with the Lie algebra L(H) = span∂θ, ∂x+θ∂y, ∂y, ∂τ and recall from Section 3.1.4, L(SIM(2)) =span∂θ, ∂ξ, ∂η, ∂τ.

4.5.2 Gaussian Estimates for the Heat-kernels on SIM(2)

According to the general theory in [35], the heat-kernels Kq,Dt : (SIM(2))q → R+ (i.e. kernels

for contour enhancement whose convolution yields diffusion on SIM(2))q) on the parametrizedclass of groups (SIM(2))q, q ∈ [0, 1] in between SIM(2) and its nilpotent Heisenberg approx-imation (SIM(2))0 satisfy the Gaussian estimates

|Kq,Dt (g)| ≤ Ct− 5

2 exp

(−b‖g‖2q

4t

), with C, b > 0 (constant), g ∈ (SIM(2))q , (4.28)

where the norm ‖ · ‖q : (SIM(2))q → R+ is given by ‖g‖q = |log(SIM(2))q (g)|q. log(SIM(2))q :(SIM(2))q → Te((SIM(2))q), is the logarithmic mapping on (SIM(2))q, which was computedexplicitly for the case of (SIM(2))q=1 = SIM(2) in Section 3.1.5, and where the weightedmodulus, see [35, Prop 6.1], in our special case of interest is given by,∣∣∣∣ 4∑

i=1

ciqAqi

∣∣∣∣q

=√|c1q|2/w1 + |c2q|2/w2 + |c3q|2/w3 + |c4q|2/w4

=√

((c1q)2 + (c2q)

2 + (c4q)2) + |c3q|, (4.29)

where recall the weightings from (4.21).

Remark. The constants b, C in (4.28) can be taken into account by the transformationst 7→ t

b , U 7→Cb2U respectively.

Lemma 4.4. A sharp estimate for the front factor constant C in (4.28) is given by

C =1

4πD11D22

1√D44

. (4.30)


Proof. According to general theory in [35] the choice of constant C is uniform for all groups(SIM(2))q, q ∈ [0, 1]. Thus to determine C we need to determine the front factor constant forthe green’s function G of the following resolvent equation(

(D11(A01)2 +D22(A0

2)2 +D44(A04)2)

)G(x, y, τ, θ) = +δe

where A0 denote the basis for L(H) (recall H = (SIM(2))q↓0) and e is the identity of H. SinceA0

i i=1,2 are independent of τ the Green’s function in our case is separable, i.e.

G(x, y, τ, θ) = GD11,D22

(SE(2))q=0(x, y, θ) ·GD44

R (τ)

where GD11,D22

(SE(2))q=0(x, y, θ) and GD44

R (τ) are Green’s function for linear diffusion on SE(2) and Rrespectively. The Green’s function for the Heisenberg case of SE(2) have been derived explicitlyin [17] and the front factor is given by 1

4πD11D22t2and the Green’s function of diffusion on R is

GD44

R (τ) = |τ |√tD44

. Thus the choice of sharp estimate follows from multiplication of the front

factor constants.

Using this theory we arrive at the Gaussian estimates for the case SIM(2) = (SIM(2))q=1.Using the definition of ci, i ∈ 1, 2, 3, 4 from (3.30) we arrive at,

|Kq=1,Dt (g)| ≤ 1

4πt52D11D22

√D44

exp

(−1

4t

(θ2

D11+

(c2)2

D22+

τ2

D44+

|c3|√D11D22D44

))(4.31)

where,

c2 =(yθ − xτ) + (−θη + τξ)

t (1 + e2τ − 2eτ cosθ), c3 =

−(xθ + yτ) + (θξ + τη)

t (1 + e2τ − 2eτ cosθ).

A problem with these estimates is that they might not be differentiable everywhere. Thisproblem can be solved by applying the estimate

|a|+ |b| ≥√a2 + b2 ≥ 1√

2(|a|+ |b|),

which holds for all a, b ∈ R, to the exponents of our Gaussian estimates. Thus we estimate theweighted modulus by the equivalent (∀q > 0) weighted modulus | · |q : Te((SIM(2))q) → R+

by

∣∣∣∣ 4∑i=1

ciqAqi

∣∣∣∣q

:= 4

√((c1q)

2 + (c2q)2 + (c4q)

2)2 + |c3q|2, yielding the Gaussian estimate,

|Kq=1,Dt (g)| ≤ 1

4πt52D11D22

√D44

exp

(−1

4t

([θ2

D11+

(c2)2

D22+

τ2

D44

]2

+|c3|2

D11D22D44

)).

(4.32)

Figure 4.4 shows the typical structure of these enhancement kernels. In Figure 4.6 we providethe enhanced image for a variety of cases. The results clearly indicate that even with lineardiffusion we achieve very good results due to the natural construction of the transform andoperations on the SIM(2) group. As seen in the figure, we are able to handle straight andcurved lines for different scales. In addition this method can also enhance an image with singlescale repeating structure.

Figure 4.5 clearly indicates the advantage of scale-OS constructed using the SIM(2) groupover the OS constructed over SE(2) group, see [15, 17], in the handling of multi-scale images.Though there is definite improvement due to usage of SIM(2) diffusion on SE(2) OS as

Nonlinear Left Invariant Diffusions on SIM(2) 51

Figure 4.4: Plots of enhancement kernels generated on a 41 × 41 grid at different scales andorientations. Parameters chosen: D11 = 0.05, D22 = 1, D44 = 0.01, t = 0.7.

compared to SE(2) diffusion4 on SE(2) OS, there are added artefacts such as unnecessaryincrease in luminescence. One obvious reason for this is that we convolve each SIM(2) kernelwith the entire SE(2) OS thus leading to additional diffusion in multiple regions. Though thiscauses an increase in the brightness of thin lines, there is also increased brightness in the thicklines.

4.6 Nonlinear Left Invariant Diffusions on SIM(2)

From (4.13) we recall that the general left-invariant second order evolution equation under con-sideration in the context of scale-OS. In this section we consider nonlinear evolutions withoutconvection given by,

∂tU(g, t) = QD(U),a=0U(g, t), g ∈ SIM(2), t > 0,

U(·, t = 0) =Wψf(·), g ∈ SIM(2),(4.33)

where the conductivity D(U) depends on the local differential structure of (g, t) 7→ U(g, t).The advantage of this more elaborate nonlinear, adaptive diffusions on invertible scale-OS isthat the domain of the scale-OS is the similitude group, which has a much richer structurethan the domain of images R2, allowing us to deal with crossing curves (with different scales)in images where the commonly used nonlinear adaptive diffusion techniques usually fail, seeSection 1.2.1.

We begin by choosing an appropriate metric on SIM(2). We then use a Cartan connection towrite (4.33) in the covariant derivative form and prove that diffusions locally take place onlyalong exponential curves in SIM(2) group even in the nonlinear adaptive case. We finallypresent a method to extract spatial curvature from the scale-OS. The ideas used in this sectionare analogous to nonlinear adaptive diffusion on the SE(2) group by Duits and Franken [16, 18].

4See [17] for the Gaussian estimates of the Green’s function of linear diffusion on the SE(2) case


Figure 4.5: Comparison of enhancement via Gaussian estimates for heat kernels for differentcases. From left to right: original image; original image+noise; enhancement on SE(2)-OSvia Gaussian estimates for SE(2) heat kernels, see [17]; enhancement on SE(2)-OS via Gaus-sian estimates for SIM(2) heat kernels; enhancement on scale-OS via Gaussian estimates forSIM(2) heat kernelsParameters used: Nθ = 20, Ns = 5; SE(2) estimates: D11 = 0.00015, D22 = 1, t = 15;SIM(2) estimates: D11 = 0.05, D22 = 1, D44 = 0.01, t = 0.7

Figure 4.6: Plot of enhancement on a scale-OS via linear diffusion on SIM(2) group usingGaussian estimated provided in this section.From left to right: Original Image; Original Image+Noise; Grey-Scale Transformation (p =1.6); Contour enhancementRow 1:Image with curved contours; Row 2:Image with repeating single scale structure; Row3:Noisy microscopy image of bone tissue; Row 4:Noisy microscopy image of muscle cell


4.6.1 Design of the metric on SIM(2)

In order to generalize the coherence enhancing diffusion schemes (see Section 1.2.1) on imagesto scale-OS we have to replace the left invariant vector fields ∂x, ∂y on the additive group(R2,+), by the left-invariant vector fields on SIM(2) as in (4.13). In order to keep track oforthogonality and parallel transport in such diffusions we need an invariant first fundamentalform G on SIM(2), rather than the trivial, bi-invariant (i.e. left and right invariant), firstfundamental form on (R2, T (R2)), where each tangent space Tx(R2) is identified with T0(R2)by standard parallel transport on R2, i.e. GR2(x,y) = x · y = x1y1 + x2y2.

Recall Theorem 4.2 which essentially states that operators Φ on scale-OS should be left-invariant (i.e. Lg Φ = Φ Lg) and not right-invariant in order to ensure that the effectiveoperator Υψ on the image is a Euclidean invariant operator. This suggests that the firstfundamental form required for our diffusions on SIM(2) should be left-invariant. The follow-ing theorem characterizes the formulation of a left-invariant first fundamental form w.r.t theSIM(2) group.

Theorem 4.5. The only real valued left-invariant (symmetric, positive, semidefinite) firstfundamental form G : SIM(2)× T (SIM(2))× T (SIM(2)→ C on SIM(2) are given by,

G =

4∑i=1

4∑j=1

gij ωi ⊗ ωj , gij ∈ R. (4.34)

where the dual basis ω1, ω2, ω3, ω4 ⊂ (L(SIM(2)))∗ of the dual space (L(SIM(2)))∗ of thevector space L(SIM(2)) of left-invariant vector fields spanned by

A1,A2,A3,A4 = ∂θ, eτ (cos θ∂x + sin θ∂y), eτ (− sin θ∂x + cos θ∂y), ∂τ, (4.35)

obtained by applying the operator dR : Te(G)→ L(G), defined as

(dR(A)φ)(g) = limt↓0


t, A ∈ Te(G), φ ∈ L2(G), g ∈ G, (4.36)

to the standard basis in the Lie algebra

A1, A2, A3, A4 = ∂θ, ∂x, ∂y, ∂a ⊂ Te(SIM(2)), (4.37)

is given by

ω1, ω2, ω3, ω4 = dθ, 1

eτ(cos θdx+ sin θdy),

1

eτ(− sin θdx+ cos θdy), dτ. (4.38)

Proof. Recall that dR yields the fundamental isomorphism between the Lie algebra Te(SIM(2))and L(SIM(2)). The dual basis (4.38) satisfy 〈Ai,Aj〉 = δij . A first fundamental formG : SIM(2)× T (SIM(2))× T (SIM(2)→ C on SIM(2) by definition is

• left invariant if ∀h, g ∈ SIM(2) ∀X,Y ∈ T (SIM(2)) : Gh(Xh, Yh) = Ggh((Lg)∗Xh, (Lg)∗Yh).

• right invariant if ∀h, g ∈ SIM(2) ∀X,Y ∈ T (SIM(2)) : Gh(Xh, Yh) = Ggh((Rg)∗Xh, (Rg)∗Yh).

• inversion-invariant if ∀h, g ∈ SIM(2) ∀X,Y ∈ T (SIM(2)) : Gre(g)((re)∗Xg, (re)∗Yg) =Gg(Xg, Yg).

• Ad-invariant if ∀h, g ∈ SIM(2) ∀X,Y ∈ T (SIM(2)) : Ghgh−1(Ad(h)Xg, Ad(h) ∗ Yg) =Gg(Xg, Yg).


• inversion-invariant if ∀h, g ∈ SIM(2) ∀X,Y ∈ T (SIM(2)) : Grh(g)((rh)∗Xg, (rh)∗Yg) =Gg(Xg, Yg).

The dual tangent space (Tg(SIM(2))∗, g ∈ SIM(2) is spanned by ω1|g, ω2|g, ω3|g, ω4|g.Thus for all g ∈ SIM(2) there exist numbers gij ∈ R, i, j ∈ 1, 2, 3, 4 such that

Gg =

4∑i=1

4∑j=1

gij(g) ωi∣∣g ⊗ ωj

∣∣g. (4.39)

G is left-invariant iff Gg(Ai|g,Aj |g) = Ge((Lg−1)∗Ai|g, (Lg−1)∗Aj |g) = Ge(Ai, Aj), ∀i, j ∈1, 2, 3, 4 ∀g ∈ SIM(2) which implies that ∀i, j ∈ 1, 2, 3, 4, gij(g) = gij .

We consider the Maurer-Cartan form on SIM(2) (discussed in the remainder of the subsection)and impose the following left-invariant, first fundamental form Gβ : SIM(2) × T (SIM(2)) ×T (SIM(2))→ C on SIM(2),

Gβ,δ =

4∑i=1

4∑j=1

gij ωi ⊗ ωj = dθ ⊗ dθ + β2dξ ⊗ dξ + β2dη ⊗ dη + δ2dτ ⊗ dτ. (4.40)

Here the parameters β (physical dimensions equal 1/[Length]) and δ (dimensionless) shouldbe considered as a fundamental parameters which relate the notion of distances in the sets(x, τ, θ)| x ∈ R2, (x, τ, θ)| τ ∈ R and (x, τ, θ)| θ ∈ [0, 2π).

In order to understand the meaning of the next two theorems we need some definitions fromdifferential geometry. See [34] for more details on these concepts.

Definition 4.6. Let M be a smooth manifold and G be a Lie group. A principle fiber bundlePG := (P,M, π,R) above a manifold M with structure group G is a tuple (P,M, π,R) suchthat P is a smooth manifold (called the total space of the principle bundle), π : P → M is asmooth projection map with π(P ) = M and π(u · a) = π(a), ∀u ∈ P, a ∈ G, R a smooth rightaction Rgp = p · g, p ∈ P, g, h ∈ G. Finally it should satisfy the ”local triviality” condition:For each p ∈M there is a neighbourhood U of p and a diffeomorphism t : π−1(U)→ U ×G ofthe form t(u) = (π(u), φ(u)) where φ satisfies φ(u ·a) = φ(u)a where the latter product is in G.

Definition 4.7. A principle fiber bundle PG := (P,M, π,R) is commonly equipped with aCartan-Ehresmann connection form ω. This by definition is a Lie algebra Te(G)-valued 1-formω : P × T (P )→ Te(G) on P such that

ω(dR(A)) = A for all A ∈ Te(G)

ω((Rh)∗A) = Ad(h−1)ω(A) for all vector fields A in G. (4.41)

It is also common practice to relate principle fiber bundles to vector bundles. Here one usesan external representation ρ : G→ F into a finite-dimensional vector space F of the structuregroup to put an appropriate vector space structure on the fibers π−1(m)| m ∈ M in theprinciple fiber bundles.

Definition 4.8. Let P be a principle fiber bundle with finite dimensional structure group G.Let ρ : G→ F be a representation in a finite-dimensional vector space F . Then the associatedvector bundle is denoted by P ×ρ F and equals the orbit space under the right action

(P × F )×G→ P × F given by ((u,X), g) 7→ (ug, ρ(g)X),

for all g ∈ G, X ∈ F and u ∈ P .


Theorem 4.9. The Maurer-Cartan form w on SIM(2) can be formulated as

wg(Xg) =

4∑i=1

〈ωi|g, Xg〉Ai, Xg ∈ Tg(SIM(2)), (4.42)

where ωi4i=1 is given by (4.38) and Ai = Ai|e; recall (4.35). It is a Cartan-Ehresmannconnection form on the principle fiber bundle P = (SIM(2), e, SIM(2),L(SIM(2))), whereπ(g) = e, Rgu = ug, u, g ∈ SIM(2). Let Ad denote the adjoint action of SIM(2) on itsown Lie algebra Te(SIM(2)), i.e. Ad(g) = (Rg−1Lg)∗, i.e. the push-forward of conjugation.Then the adjoint representation of SIM(2) on the vector space L(SIM(2)) of left-invariantvector-fields is given by

Ad(g) = dR Ad(g) w. (4.43)

The adjoint representation gives rise to the associated vector bundle SIM(2)×AdL(SIM(2)).

The corresponding connecting form on this vector bundle is given by

w = A2 ⊗ ω3 ∧ ω1 +A3 ⊗ ω1 ∧ ω2 +A2 ⊗ ω4 ∧ ω2 +A3 ⊗ ω4 ∧ ω3. (4.44)

Then w yields the following 4× 4 matrix-valued 1-form:

wkj (·) := −w(ωk, ·,Aj), k, j ∈ 1, 2, 3, 4 (4.45)

on the frame bundle, where the sections are moving frames5. Let µk4k=1 denote the sections inthe tangent bundle E := (SIM(2), T (SIM(2))) which coincides with the left-invariant vectorfields Ak4k=1. Then the matrix-valued 1-form (4.35) yields the Cartan connection 6 D on thetangent bundle (SIM(2), T (SIM(2))) given by the covariant derivatives

DX|γ(t)(µ(γ(t))) := D(µ(γ(t)))(X|γ(t))

=

4∑k=1

akµk(γ(t)) +

4∑k=1

ak(γ(t))

4∑k=1

wjk(X|γ(t))µj(γ(t)) (4.46)

=

4∑k=1

akµk(γ(t)) +

4∑k=1

˙γi(t)ak(γ(t))Γjikµj(γ(t)),

with ak = γi(t)(Ai|γ(t)ak), for all tangent vectors X|γ(t)= γi(t)Ai|γ(t) along a curve t 7→ γ(t) ∈

SIM(2) and all sections µ(γ(t)) =4∑k=1

ak(γ(t))µk(γ(t)). The Christoffel symbols in (4.46) are

constant Γjik = −cjik, with cjik the structure constants of the Lie algebra Te(SIM(2)).

Proof. As in Theorem 4.5, we set A1, A2, A3, A4 = ∂θ, ∂x, ∂y, ∂a as a basis for the Lie alge-bra Te(SIM(2)) and A1,A2,A3,A4 = dR(A1), dR(A2), dR(A3), dR(A4) = ∂θ, ∂ξ, ∂η, ∂τas the basis for the space L(SIM(2)) of left-invariant vector fields with corresponding dual ba-sis ω1, ω2, ω3, ω4 ⊂ (L(SIM(2)))∗. The Maurer-Cartan form ω : (SIM(2), T (SIM(2))) →Te(SIM(2)) is defined as

wg(Yg) = (Lg−1)∗Yg, (4.47)

where (Lg−1)∗ denotes the push- forward of the inverse left multiplication h 7→ Lgh = g−1hi.e. wg(Yg)φ = Yg(φ Lg−1) for all φ : Ωe → R smooth and defined on some open local set Ωe

5See [34, Ch.8] for more details on frame bundles and moving frames.6Following the definitions in [34], formally this is not a Cartan connection but a Koszul connection, see

[34, Ch.6] , corresponding to a Cartan connection, i.e. a the associated differential operator corresponding toa Cartan connection. We avoid these technicalities and use ”Cartan connection” (a Koszul connection in [34])and ”Cartan-Ehresmann connection form” (Ehresmann connection in [34]).


around the unity e. Recall that the left-invariant vector fields Ai4i=1 satisfy Ai|g= (Lg)∗Aiand therefore the dual elements (the corresponding co-vector fields) are obtained by the pull-back from Te(SIM(2)), i.e. ωi|g= (Lg−1)∗ω

i|e since 〈(L−1g )∗ω

i|e, (Lg)∗Aj〉 = 〈ωi|e, Aj〉 = δij .Now for any Xg ∈ Tg(SIM(2)), direct computation yields (4.42),

wg(Xg) = (Lg−1)∗Xg =

4∑i=1

〈ωi|e, (Lg−1)∗Xg〉Ai

=

4∑i=1

〈(Lg)∗ωi|e, Xg〉Ai =

4∑i=1

〈ωi, Xg〉Ai.

Now we show that the Maurer-Cartan form indeed forms a Cartan-Ehresmann connection form,recall Def. 4.7, on the principle fiber bundle, P = (SIM(2), e, SIM(2),L(SIM(2)). Now recallfrom Theorem 4.5 that the left invariant vector fields are obtained from the operator dR, i.e.Ai = dR(Ai). The Maurer-Cartan form does the reverse in the sense that it connects eachtangent space Tg(SIM(2)) to Te(SIM(2)). To this end note that

limh↓0

φ(g · ehAi)− φ(g)

h=: (dR(Ai))gφ = (Ai)gφ = (Lg)∗Aiφ = Ai(φ Lg) ∈ R

for all g ∈ SIM(2) and all smooth φ : Ωg → R. Therefore we have

∀i ∈ 1, 2, 3, 4 : w dR(Ai) = w(Ai) = Ai ⇔ w dR = I.

The second requirement in Def. 4.7 follows from the following computation,

wgh((Rh)∗Yg) = (Lh−1 Lg−1)∗((Rh)∗Yg) = (Lh−1)∗ (Lg−1)∗ (Rh)∗Yg = Ad(h−1)wgYg.

To show that equality (4.43) holds we note that the left multiplication Lg and the right mul-tiplication Rg commute and this leads to

(Rg−1Lg)∗ = (LgRg−1)∗ = (Lg)∗(Rg−1)∗ = (Lg)∗(Rg−1Lg)∗(L−1g )∗,

which implies that Ad(g) = dR Ad(g) w. The adjoint representation Ad : SIM(2) →GL(Te(SIM(2))) coincides with the derivative of the conjugate automorphism h 7→ conj(g)(h) =ghg−1 evaluated at e, i.e. Ad(g) = Deconj(g) = (Rg−1Lg)∗.

Remark. HereGL(Te(SIM(2))) is the collection of linear operators on the Lie algebra Te(SIM(2)).Note that each linear operator Q ∈ GL(Te(SIM(2))) on Te(SIM(2)) is 1-to-1 related to bi-linear form Q on (Te(SIM(2)))∗ × Te(SIM(2)) by means of

〈B,QA〉 = Q(B,A), for all B ∈ (Te(SIM(2)))∗, A ∈ Te(SIM(2))and

Q =

4∑i=1

Q(ωi|e, ·)Ai.

So a basis for GL(Te(SIM(2))) is given by ωi|e⊗Aj | i, j ∈ 1, 2, 3, 4. For the simplicity ofnotation we omit the overline and write ωi|e⊗Aj as it is clear from context whether we meanthe bilinear form or the linear mapping.

Recall Def. 4.8 of an associated vector bundle and set

P = SIM(2), M = e, G = SIM(2), F = L(SIM(2)), ρ = Ad, π(g) = e, Rgu = ug, (4.48)

where L(SIM(2)) denotes the Lie algebra of left invariant vector fields on SIM(2) and Ad theadjoint representation of SIM(2) into GL(L(SIM(2)) given by

Ad(g)X = (R−1g Lg)∗X, X ∈ L(SIM(2)), g ∈ SIM(2).


A connection w on a principle fiber bundle P is 1-to-1 related to a connection w on the vectorbundle P ×ρ F by means of

w =∑j

Aj ⊗ dxj ↔ w =∑j

ρ∗(Aj)⊗ dxj ,

where dxj are dual forms on F , see [31] for more details on this bijection. In our case wehave F = L(SIM(2)) and the corresponding dual forms ωj4j=1. Note that we applied the

convention mentioned in the remark. So in our case (4.48) the push-forward ρ∗ of ρ = Adequals

(Ad)∗ = (dR Ad∗)(Aj) = (dR ad w)(Aj)

= ad(Aj) = [·,Aj ]L(SIM(2)) =

4∑i,j,k=1

ckijAk ⊗ ωi.

Thereby the connection form on the vector bundle SIM(2)×AdL(SIM(2)) is given by

w =

4∑j=1

ad(Aj)⊗ ωj =

4∑i,j,k=1

ckijAk ⊗ ωi ⊗ ωj (4.49)

where ckij are the structure constants of the Lie group SIM(2), so ad(Aj)(Ai) = [Ai,Aj ] =∑4i,j,k=1 c

kijAk. Using the nonzero structure constants c312 = −c321 = −c213 = c231 = −c334 =

c343 = c242 = −c224 = 1 and by using the definition anti-symmetric products da∧ db = da⊗ db−db⊗ da in (4.49) we arrive at (4.44).

From w defined in (4.49) we can construct the 16-connection 1-form wkj (·)4k,j=1 via (4.45)which together form a 4× 4 matrix-valued 1-form on the frame bundle where the sections aremoving frames.

Let µk4k=1 denote the sections in (SIM(2), T (SIM(2))) which coincide respectively withthe left-invariant vector fields Ak4k=1. Then the Cartan connection D on (the vector bundleSIM(2)×

AdL(SIM(2)) is isomorphic to) the tangent bundle (SIM(2), T (SIM(2))) equals

D := d+ w with w(µk)(·) :=

4∑j=1

wjk(·)µj , (4.50)

or more precisely the covariant derivatives are given by

DX|γ(t)(µ(γ(t))) := (Dµ(γ(t)))(X|γ(t))

=

4∑k=1

akµk(γ(t)) +

4∑k=1

ak(γ(t))

4∑k=1

wjk(X|γ(t))µj(γ(t))

=

4∑k=1

akµk(γ(t)) +

4∑k=1

˙γi(t)ak(γ(t))Γjikµj(γ(t)),

with ak = γi(t)(Ai|γ(t)ak)(γ(t)), for all curves γ : R → SIM(2) and tangent vectors X|γ(t)=∑4

i=1 γi(t)Ai|γ(t)∈ Tγ(t)(SIM(2)) and all sections

µ(γ(t)) =

4∑k=1

ak(γ(t))µk(γ(t))

and where the Christoffel symbols Γkij are constant

Γkij = wkj (Ai) = −w(ωk,Ai,Aj) = −ckij = ckji.


As seen in the theorem above, we define the notion of covariant derivatives independent ofthe metric G on SIM(2), which is the underlying principle behind the Cartan connection.Although in principle these two entities need not be related, in Lemma 4.11 we show that theconnection induced above is metric compatible.

Definition 4.10. Let (M,G) be a Riemannian manifold (or pseudo-Riemannian manifold)where M and G denote the manifold and the metric defined on it respectively. Let ∇ denote aconnection on (M,G). Then ∇ is called metric compatible with respect to G if

∇ZG(X,Y ) = G(∇ZX,Y ) + G(X,∇ZY ), (4.51)

for all X, Y, Z ∈ T (M).

Lemma 4.11. The Cartan connection D on (SIM(2), T (SIM(2))) is metric compatible withrespect to Gβ,δ : SIM(2)× T (SIM(2))× T (SIM(2))→ C on SIM(2) defined in (4.40).

Proof. We first note that DAiGβ,δ(Aj ,Ak) = 0, i, j, k ∈ 1, 2, 3, 4 as the covariant derivativeof a scalar field is the same as partial derivative. Here Ai4i=1 denote the basis of the leftinvariant vector fields L(SIM(2)). The following brief computation

Gβ,δ(DAiAj ,Ak) + Gβ,δ(Aj , DAiAk) = Gβ,δ(clijAl,Ak) + Gβ,δ(Aj , cmikAm) = −ckij − cjik,

(4.52)

where we have used the fact that Γkij = ckji, along with non-zero structure constants c312 =

−c321 = −c213 = c231 = −c334 = c343 = c242 = −c224 = 1, leads to the result.

The next theorem relates the previous results on Cartan connections and covariant derivativesto the nonlinear diffusion schemes on SIM(2).

Theorem 4.12. The covariant derivative of a co-vector field a on the manifold (SIM(2),Gβ,δ)is a (0, 2)-tensor field with components ∇jai = Ajai − Γkijak, whereas the covariant derivative

of a vector field v on SIM(2) is a (1, 1)-tensor field with components ∇j′vi = Aj′vi+Γij′k′vk′ ,

where we have made use of the notation ∇j := DAj . The Christoffel symbols equal minus thestructure constants of the Lie algebra L(SIM(2)), i.e. Γkij = −ckij. The Christoffel symbols areanti-symmetric as the underlying Cartan connection D has constant torsion. The left-invariantequations (4.33) can be rewritten in covariant derivatives ∂sW (g, s) =

4∑i,j=1

Ai((Dij(W ))(g, s)AjW )(g, s) =4∑

i,j=1

∇i((Dij(W ))(g, s)∇jW )(g, s),

W (g, 0) =Wψf(g), for all g ∈ SIM(2), s > 0.

(4.53)

Both convection and diffusion in the left-invariant evolution equations (4.33) take place alongthe exponential curves in SIM(2) which are covariantly constant curves with respect to theCartan connection. These curves t 7→ (x(t), y(t), τ(t), θ(t)) in R2 × R+ × SO(2) are given by,

x(t) =1

c21 + c24[eτ0c1

((− sin [θ0] + etc4 sin [tc1 + θ0]

)c2 +

(− cos [θ0] + etc4 cos [tc1 + θ0]

)c3)+ c21x0

+ c4(eτ0

(−cos [θ0] + etc4 cos [tc1 + θ0]

)c2 + eτ0

(sin [θ0]− etc4 sin [tc1 + θ0]

)c3 + c4x0

)]

y(t) =1

c21 + c24[eτ0c1

((cos [θ0]− etc4 cos [tc1 + θ0]

)c2 +

(− sin [θ0] + etc4 sin [tc1 + θ0]

)c3)+ c21y0

+ c4(eτ0

(− sin [θ0] + etc4 sin [tc1 + θ0]

)c2 + eτ0

(− cos [θ0] + etc4 cos [tc1 + θ0]

)c3 + c4y0

)]

τ(t) = tc4 + τ0

θ(t) = tc1 + θ0, (4.54)

where c21 + c24 6= 0.


Proof. The first part of the proof follows from Theorem 4.9. The torsion tensor T (X,Y ) =DXY −DYX − [X,Y ] is constant follows from a simple computation

T (Ai,Aj) = DAiAj −DAjAi − [Ai,Aj ] =

4∑k=1

(ΓkijAk − ΓkjiAk − ckijAk) = −3

4∑k=1

ckijAk.

The covariant constant curves7 γ are by definition given by Dγ γ = 0 on the tangent bundle(SIM(2), T (SIM(2)))

Dγ γ = DγiAi|γ(t) γiAi|γ(t)= γiAi|γ(t)+γ

iγkΓjikAj = γiAi|γ(t)= 0, (4.55)

where we have made use of the Einstein’s summation convention and applied automatic sum-mation over double indices and where Γkij = −Γkji = ckji = −ckij . Note that the tangent vec-tors to these auto-parallel curves have constant coefficients with respect to A1,A2,A3,A4as ∀t > 0, γi(t) = 〈ωi|γ(t), γ(t)〉 = |γ(0), γ(0)〉 = ci ∈ R, i ∈ 1, 2, 3, 4. For smoothU : SIM(2)→ C one has

d

dtU(γ(t)) = lim

h→0

U(γ(t+ h)− U(γ(t)))

h= (dR(

4∑i=1

ciAi)U)(γ(t))

=

4∑i=1

ci(dR(Ai)U)(γ(t)) =

4∑i=1

ciAiU |γ(t), (4.56)

where γ(t) = g0et∑4i=1 c

iAi . Therefore these curves γ(t) coincide with the exponential curves inSIM(2), see (3.28). As mentioned earlier the connection D is a Koszul connection which hasthe property that ∇i(UAj)φ = ∇i(U∇j)φ = U∇i∇jφ + (∇iU)∇jφ for all U ∈ C1(SIM(2))and all smooth φ ∈ C∞(SIM(2)). Set U = Dij(W )(·, s) and φ = W (·, s) for all s > 0, useDij = Dji and Γkij = −γkji, take the sum over both indices i, j and (4.53) follows.

Remark. Though the connection D is Gβ,δ compatible (Lemma 4.11) , we have shown in theprevious theorem that our connection is not torsion free, i.e. the torsion tensor T (X,Y ) 6= 0for all X,Y ∈ T (SIM(2)). Therefore this connection does not coincide with the Levi-Civitaconnection, which is the unique metric compatible torsion free connection on a Riemannianmanifold. Consequently the auto-parallels which are the exponential curves, see (4.54), do notminimize the (sub-)Riemannian distance on (SIM(2),Gβ,δ).

4.6.2 Extraction of spatial curvature from scale-OS

Let U : SIM(2) → R+ be a smooth function on SIM(2). For instance this could be theabsolute value U = |Wψf | =

√(R(Wψf))2 + (I(Wψf))2 of an orientation score of an image.

In this subsection we provide algorithms for curvature estimation at each g0 ∈ SIM(2) in thedomain of U , by finding the exponential curves through g0 that fits U locally in an optimalway. To achieve this we first present a few preliminaries.

The inner product between two left-invariant vector fields is given by applying the first funda-mental form (4.40) on T (SIM(2))× T (SIM(2)) and is denoted by (·, ·)ν ,

(ciAi, cjAj)ν =

4∑i,j=1

gijci1cj2 = cθ1c

θ2 + β2cξ1c

ξ2 + β2cη1c

η2 + δ2cτ1c

τ2

7Also called ”auto-parallel” curves.


where we use the convention c1k = cθk, c2k = cξk, c

3k = cηk, c

4k = cτk, k = 1, 2. The norm of a

left-invariant vector field ciAi is given by

|ciAi|ν =√

(ciAi, ciAi)ν =√

(cθ)2 + (βcθ)2 + (βcη)2 + (βcτ )2 =: ‖c‖ν , (4.57)

with c = (c1, c2, c3, c4) ∈ R4. Note that the norm | · |ν : L(SIM(2)) → R+ is defined on thespace L(SIM(2)) of left-invariant vector fields on SIM(2), whereas the norm ‖ · ‖ν : R4 → R+

is defined on R4. The gradient dU of U : SIM(2)→ R+ which is a co-vector field is given by

dU =∂U

∂θω1 +

∂U

∂ξω2 +

∂U

∂ηω3 +

∂U

∂τω4.

The corresponding vector field is,

G−1β,δdU =

∂U

∂θ∂θ + β−2 ∂U

∂ξ∂ξ + β−2 ∂U

∂η∂η + δ−2 ∂U

∂τ∂τ ,

where G−1β,δdU : (T (SIM(2)))∗ → T (SIM(2)) is the inverse of the fundamental bijection be-

tween the tangent space and its dual and is defined as G−1β,δω

k = gkiAi, with gijgjl = δil . Thenorm of a co-vector field (such as dU) is given by

|aiωi|2ν = gijaiaj = (aθ)2 + β−2(aξ)

2 + β−2(aη)2 + δ−2(aτ )2 = ‖a‖ν−1 , with a = (a1, a2, a3, a4).(4.58)

If we differentiate a smooth function U : SIM(2) → R+ along an exponential curve γ(t) =g0 exp(t

∑ciAi) passing through g0 we get (by application of chain rule)

d

dtU(γ(t)) = 〈dU, γ′(t)〉 =

4∑i=1

ciAiU |γ(t). = c1Uθ(γ(t)) + c2Uξ(γ(t)) + c3Uη(γ(t)) + c4Uτ (γ(t)).

(4.59)

Now we return to our goal of finding the optimal exponential curve at position g0 ∈ SIM(2)given U : SIM(2)→ R+.

Definition 4.13. Consider the solution of the following minimization problem:

c∗ = arg minci4i=1

∣∣∣∣ ddtdU(γ(t))

∣∣∣∣t=0

∣∣∣∣2ν

| γ(t) = g0 exp(t

4∑i=1

ciAi) ; ‖c‖ν = 1

. (4.60)

Then we call the covariantly constant curve t 7→ g0 exp(t4∑i=1

ci∗Ai) the optimal exponential curve

at g0 ∈ SIM(2) given U : SIM(2)→ R+.

By means of (4.59) and the chain rule the energy in (4.60) can be rewritten as∣∣∣∣ ddtdU(γ(t))

∣∣∣∣t=0

∣∣∣∣2ν

= ‖∇(∇U)T (γ(0)) · γ′(0)‖2ν−1

=

∥∥∥∥∥∥∥∥∂θ(∂θU) ∂ξ(∂θU) ∂η(∂θU) ∂τ (∂θU)∂θ(∂ξU) ∂ξ(∂ξU) ∂η(∂ξU) ∂τ (∂ξU)∂θ(∂ηU) ∂ξ(∂ηU) ∂η(∂ηU) ∂τ (∂ηU)∂θ(∂τU) ∂ξ(∂τU) ∂η(∂τU) ∂τ (∂τU)

∣∣∣∣∣∣∣∣g0

c1

c2

c3

c4

∥∥∥∥∥∥∥∥

2

ν−1

=: ‖HU |g0c‖2ν−1 (4.61)

where ∇U := (∂θU, ∂ξU, ∂ηU, ∂τU). Note that this noncovariant Hessian HU is not the sameas the covariant Hessian form consisting of covariant derivatives of the Cartan connection


provided in (4.12), which equals

|∇i∇jU | = [∇iAjU ] = [AiAjU + ΓλijAλU ]

=

∂θ(∂θU) ∂θ(∂ξU) ∂θ(∂ηU) ∂θ(∂τU)∂ξ(∂θU) ∂ξ(∂ξU) ∂ξ(∂ηU) ∂ξ(∂τU)∂η(∂θU) ∂η(∂ξU) ∂η(∂ηU) ∂η(∂τU)∂τ (∂θU) ∂τ (∂ξU) ∂τ (∂ηU) ∂τ (∂τU)

. (4.62)

For example, in the 2nd row and 1st column we have ∇1∇2U = (∂θ∂ξ − c312∂η − c412∂τ )U =∂ξ(∂θU). The minimization problem in (4.60) can now be rewritten as

argminc

‖(HU)(g0) c‖2ν−1 | ‖c‖ν = 1

.

Set Mν := diag1, β−1, β−1, δ−1 ∈ GL(4,R) and HνU := MνHUMν . Then by the EulerLagrange theory the gradient of ‖(HU)c‖ = (c, (HU)TM2

ν (HU)c)1 at the optimum c∗ islinearly dependent on the gradient of the side condition, which can be written as (1−‖c‖2ν) =1− (c,M−2

ν c)1 = 0, i.e.

(HU(g0))TM2ν (HU(g0))c∗ = λM−2

ν c∗ ⇔ (HνU)T (HνU)c = λc,

for some Lagrange multiplier λ ∈ R, where c = M−1β c∗. Thus we have shown that the mini-

mization problem (4.53) requires eigensystem analysis of (HνU)THνU rather than eigensystemanalysis of the covariant Hessian given in (4.62). Making use of the discussion above we suggestthe following method for curvature estimation.

Compute the curvature of the projection x(s(t)) = PR2

(g0 exp(t

4∑i=1

c∗Ai)

)of the optimal

exponential curve in SIM(2) on the ground plane from an eigenvector c∗ = (cθ∗, cξ∗, c

η∗, c

τ∗).

This eigenvector of (Hν |U |)T (Hν |U |), with a 4× 4 Hessian

Hν |U | =

β2∂θ∂θ|U | β∂ξ∂θ|U | β∂η∂θ|U | β

δ ∂τ∂θ|U |∂θ∂ξ|U | ∂ξ∂ξ|U | ∂η∂ξ|U | ∂τ∂ξ|U |∂θ∂η|U | ∂ξ∂η|U | ∂η∂η|U | ∂τ∂η|U |∂θ∂τ |U | ∂ξ∂τ |U | ∂η∂τ |U | ∂τ∂τ |U |

, (4.63)

corresponds to the smallest eigenvalue. The curvature estimation is then given by the expres-sion κest = ‖x(s)‖sgn(x(s) · eη).

4.6.3 Adaptive diffusion on Scale-OS

In order to obtain adaptive diffusion on scale-OS we consider the following nonlinear left-invariant evolution equation on SIM(2):

∂tU(g, t) =

(β∂θ ∂ξ ∂ηβδ ∂τ )

(D11(U))(g, t) 0 0 0

0 (D22(U))(g, t) 0 0

0 0 (D33(U))(g, t) 0

0 0 0 (D44(U))(g, t)

β∂θ

∂ξ

∂ηβδ ∂τ

U(g, t),

for all g ∈ SIM(2), t > 0,

U(g, t = 0) =Wψ[f ](g) for all g ∈ SIM(2),

(4.64)


where the positive functions Dkk : L(SIM(2) × R+) ∩ C2(SIM(2) × R+) → C1(SIM(2) ×R+), k ∈ 1, 2, 3, 4 are given by

(g, t) 7→ (Dkk(U))(g, t) ≥ 0, U ∈ L(SIM(2)× R+).

These functions Dkk, k ∈ 1, 2, 3, 4 should be chosen dependent on the local Hessian HU(·, t)of U(·, t) such that, at strong orientations, D33 should be small so that we have anisotropicdiffusion in the spatial plane along the preferred direction ∂ξ, while at weak orientations D22

and D33 should be relatively comparable for isotropic diffusion.

In the previous subsection we discussed a method to obtain curvature estimates in scale-OS. This was done by finding the best exponential curve fit to the absolute value8 |Wψf | :SIM(2)→ R+. of the scale-OS Wψf : SIM(2)→ C. We can include curvature in the scheme(4.64) by replacing ∂ξ by ∂ξ + κ∂θ.

Remark. We note that ∂θ, ∂ξ + κ∂θ, ∂η, ∂τ are, in contrast to ∂θ, 1β∂ξ,

1β∂η,

1δ∂τ, not or-

thonormal with respect to the (·, ·)ν inner product (4.40). Therefore to make use of this differentset of coordinates, gauge coordinates aligned with the optimally fitting exponential curve willneed to be introduced. A similar approach is followed in the case of adaptive diffusion on SE(2),see [18, Sec 4.1].

4.7 Adaptive SE(2) Diffusion on Scale-OS

In this section we apply nonlinear adaptive diffusions on the 2D Euclidean motion group SE(2),called coherence enhancing diffusion on orientation scores (CED-OS), to each scale of our scale-OS9. We can apply CED-OS in our case as at each fixed scale the scale-OS is a function onthe SE(2) group. Though this is an essentially pragmatic approach, we will see later in thissection that even this approach handles crossing structures very well.

In this section we briefly present a few algorithmic details of CED-OS in this section. Theunderlying theoretical concepts are similar to those mentioned in Section4.6 and have beenavoided until absolutely necessary. See [18] for more details on the theoretical aspects ofCED-OS. We conclude this section by illustrating the results of enhancement via CED-OS onscale-OS.

Recall that for an image f ∈ L2(R2), the corresponding scale-OS Wψ(f) ∈ CSIM(2)K ⊂

L2(SIM(2)). For any a ∈ (0,∞), Wψ(f)(·, ·, a, ·) ∈ L2(SE(2)). Let Φ : L2(SIM(2)) →L2(SIM(2)) denote nonlinear adaptive diffusion (CED-OS) on the SE(2) group. In this sec-tion we propose the operator Λ on scale-OS defined as

Λ[(Wψf)(g)] =

m∑i=1

Φ[Wψf(·, ·, ai, ·)], (4.65)

where g ∈ R2 × [0, 2π) × [a−, a+] and aimi=1 is the discretization of [a−, a+]. Recall thedefintion of 0 < a− < a+ <∞ from Section 2.5.

4.7.1 CED-OS Implementation

Curvature estimation of a spatial curve using SE(2)-OS is similar to the scheme followed inSection 4.6.2 wherein we make use of the optimal exponential curve fit at each point. We

8The absolute value |Wψf | is phase invariant. This is important for local feature estimation. For e.g. thecurvature estimate at the border of a thick line should be the similar to the estimate on top of the line.

9Due to constraints of time, adaptive diffusions on the SIM(2) group have not been implemented.

Adaptive SE(2) Diffusion on Scale-OS 63

make use of the same notation as used in Section 4.6.2, the only difference being that in theSE(2) group setting there is no scaling group involved. Two methods suggested for curvatureestimation are as follows:

• Compute the curvature of the projection x(s(t)) = PR2(g0 exp(t∑3i=1 c

i∗Ai)) of the opti-

mal exponential curve in SE(2) on the ground plane from an eigenvector c∗ = (cθ∗, cξ∗, c

η∗).

This eigenvector of (Hβ |U |)T (Hβ |U |), with a 3× 3 Hessian

Hβ |U | =

β2∂θ∂θ|U | β∂ξ∂θ|U | β∂η∂θ|U |∂θ∂ξ|U | ∂ξ∂ξ|U | ∂η∂ξ|U |∂θ∂η|U | ∂ξ∂η|U | ∂η∂η|U |

, (4.66)

corresponds to the smallest eigenvalue. The curvature estimation is now given,

κest = ‖x(s)‖sgn(x(s) · eη) =cθ∗sign(cξ∗)√(cξ∗)2 + (cξ∗)2

.

Note that unlike the SIM(2) case curvature is a constant in this case.

• In this method we only the choice of optimal exponential curve to horizontal exponentialcurves. Horizontal curves are curves in the (SE(2), ω3) Sub-Riemannian manifold. Fora brief explanation of horizontal curves see App(Sec. B.2). We wish to diffuse alonghorizontal curves because typically the mass of a SE(2)-OS is concentrated around ahorizontal curve [16] and therefore this is a good and fast curvature estimation method.

Compute the curvature of the projection x(s(t)) = PR2(g0 exp(t∑3i=1 c

i∗Ai)) of the opti-

mal exponential curve in SE(2) on the ground plane from the eigenvector c∗ = (cθ∗, cη∗).

This eigenvector of (Hhorβ |U |)T (Hhor

β |U |), with a 3× 2 horizontal Hessian

Hhorβ |U | =

β2∂θ∂θ|U | β∂ξ∂θ|U |∂θ∂ξ|U | ∂ξ∂ξ|U |∂θ∂η|U | ∂ξ∂η|U |

, (4.67)

corresponds to the smallest eigenvalue. The curvature estimation is now given,

κhorest = ‖x(s)‖sgn(x(s) · eη) =cθ∗

cξ∗.

For numerical experiments on these curvature estimates on orientation scores of noisyimages, see [16].

For experiments in this study we enforce horizontality and restrict ourselves to the secondmethod.

We are interested in the nonlinear diffusion equation given by∂tW (g, t) = ∇ · (D(W )(g, t)∇W (g, t)), g ∈ SE(2), t ≥ 0,

W (g, 0) = U(g)(4.68)

where U : SE(2)→ C. The diffusion tensor is given by an adaptive mixture of diffusion alongthe exponential curve and β-isotropic diffusion

D(W )(g) = (1−Da)µ2

‖c‖2βccT +Dadiag1, 1, β2, 0 < Da < 1.


Figure 4.7: Finite difference scheme of (4.68), where second-order B-spline interpolation, [38],is employed for sampling on the grid of the moving frame eθ, eξ = cos θex + sin θey, eη =− sin θex + cos θey.

where we have to substitute Da = Da(s(W (·, t))) and =c∗(W (·, t))(g). s(W (·, t)) is a measureof orientation confidence and is determined by taking the Laplacian in the plane orthogonal tothe line which is calculated at g ∈ SE(2) by

s(g) = −∆|U |(g) = −((e1(g))T Hβ |U |e1(g) + (e2(g))T Hβ |U |e2(g)),

where Hβ |U | is defined in (4.66) and e1 and e2 are vectors orthonormal to the tangent vectorc. Below we briefly explain β-isotropic diffusion.

There is no inherent notion of isotropy in SIM(2). We can however define an artificial notionof β-isotropic diffusion, where the diffusion tensor is defined as

Disoβ = diag1, 1, β2.

The diffusion equation is β-isotropic in the sense that ‖∂ξ‖β = ‖∂η‖β = β2‖∂θ‖β = β and sowith respect to the metric on SE(2) (same as the SIM(2) case (4.40) but without the dτ ⊗ dτterm) we apply the same amount of diffusion in all directions. The β-isotropic diffusion equationcan be written as

∂tW = ((∂2ξ + ∂2

η) + µ2∂2θ )W = ((∂2

x + ∂2y) + µ2∂2

θ )W,

where W : SE(2)→ C.

Da in CED-OS is given by

Da(W (·, t)) = ζ

s(W (·, t))a maxg∈SE(2)

s(W (g, t)).

where ζ is defined as

ζ =

1 s ≤ 0

1− exp(−( cs )n) n > 0 and s > 0

exp(−( cs )n) n < 0 and s > 0,

with n ∈ Z\0. The explicit inclusion of adaptive curvature is achieved via a Gauge frameinterpretation of anisotropic diffusion in SE(2). For more details on the motivation for thechoice of Da and ζ and construction of Gauge frames see [21, Chapter 6].


Figure 4.8: Illustration of scale invariance of left-invariant operators on scale-OS. From leftto right: input image; enhancement via linear diffusion on input image sampled on 200× 200grid; enhancement via linear diffusion on input image sampled on 300× 300 grid.

Figure 4.7 illustrates the numerical schemes used for the discretization of (4.68). These schemesmake use of second-order B-spline interpolation, [38], for sampling on the grid of movingframes eθ, eξ = cos θex + sin θey, eη = − sin θex + cos θey. For the stability analysis of thesenumerical schemes see [16].

4.7.2 Advantage of scale adaptive diffusion

Figure 4.8 illustrates the idea of scale invariance. This is a useful mathematical propertyand intuitively it means that large particles in an image diffuse more than small particles.However in most medical imaging applications this is not desirable as the interest usually liesin a particular interval of scales. Figure 4.9 illustrates this idea. The advantage of scale basedapplication of CED-OS lies in the adaptability of diffusion parameters per scale, for e.g. byadapting the end time of diffusion per scale. This is a clear advantage of CED on scale-OS incomparison to CED on SE(2)-OS.

4.7.3 Results

In this section present the results of applying CED-OS at each layer of the scale-OS. Theparameters used in the construction of scale-OS are the same for all the images : k = 3 (orderof B-spline for construction), Ns = 4 (number of scales) and Nθ = 32 (number of orientations).k = 3, Nθ = 32 is the parameter used in the construction of SE(2)-OS.

Figure 4.10 and Figure 4.11 illustrate the main idea of this method. Figure 4.10 shows theresult of enhancement via applying CED-OS to each scale of the scale-OS and Figure 4.11 showsthe enhancement achieved at each scale. Parameters used for CED-OS: τ = 0.1, β = 0.058,endt = 0.5, 0.5, 0.5, 0.5.

In Figure 4.12 we compare enhancement via Gaussian estimates for linear diffusion on scale-OS(recall Section 4.5.2) with nonlinear diffusion via CED-OS per scale on scale-OS for a test-imagewith crossing lines. Parameters used for the linear diffusion scheme: D11 = 0.05, D22 = 1,D44 = 0.01, endt = 0.7. Parameters used for CED-OS on scale-OS : τ = 0.1, endt = 1, 1, 1, 1,β = 0.058.

In Figure 4.13 we compare enhancement via Gaussian estimates for linear diffusion on scale-OS(recall Section 4.5.2) with nonlinear diffusion via CED-OS per scale on scale-OS for a Retinal


Figure 4.9: Illustration of disadvantage of scale-invariance of diffusion (CED-OS on scale-OS)on retinal fundus image. Left: Fundus image; right: Enhancement via CED-OS per scale.Scale-invariant diffusion causes additional diffusion of the optic disc in the retina which isundesirable when the goal is blood-vessel extraction.

fundus image. Parameters used for the linear diffusion scheme: D11 = 0.05, D22 = 1, D44 =0.01, endt = 0.7. Parameters used for CED-OS on scale-OS : τ = 0.1, endt = 0.5, 0.5, 1.2, 1.5,β = 0.058. Figure 4.12 and Figure 4.13 clearly indicate that enhancement via nonlineardiffusion give better results than linear diffusion.

In Figure 4.14 we perform CED-OS on both the SE(2)-OS and the scale-OS. Parameterschosen for CED-OS on SE(2)-OS : τ = 0.1, endt = 3, β = 0.058. Parameters used for CED-OS on scale-OS : τ = 0.1, endt = 0.5, 0.5, 0.5, 0.5 β = 0.058. As seen in the results, for animage with same scale structures, CED-OS on scale-OS and CED-OS on SE(2)-OS give verysimilar results. Note that our schemes handle crossings in an image in a much better way thanstandard adaptive diffusion techniques suggested in [30] and [41]. For a comparison (in thecontext of crossing curves in a medical image) of CED technique suggested in [41] and CED-OSsee [16].

Figures 4.14, 4.15, 4.16, 4.17 and 4.18 compare enhancement via CED-OS on SE(2)-OS andscale-OS of different images. For all these images the parameters for CED-OS are: τ =0.1, and β = 0.058.

Figure 4.14 illustrates that for an image with uniform scale structure CED-OS on SE(2)-OS(endt = 10) and scale-OS (endt = 0.5, 0.5, 0.5, 0.5) have similar results.

Figures 4.15, 4.16, 4.17, 4.18 clearly indicate that CED-OS on scale-OS leads to better en-hancement when multiple scale structures are present in the image. These experiments areperformed on different medical images.


Figure 4.10: CED-OS applied to the scale-OS of a noisy test-image with concentric circles.Left: Noisy input image; Right: Result of enhancement at each scale on scale-OS.

Figure 4.11: Enhancement per scale of the test image in Figure4.10. As clearly visible mostof the noise is concentrated at the lower scales and this suggests the need for scale basedenhancement operators.


Figure 4.12: Comparison between behaviour of linear diffusion on scale-OS and CED-OS perlayer on scale-OS for a test-image. From left to right: original image; original image + noise;enhancement via Gaussian estimates for heat kernels of linear diffusion on SIM(2) group (seeSection 4.5.2); enhancement via CED-OS per scale layer of the scale-OS. As expected thenonlinear adaptive diffusion scheme gives much better results than the linear scheme.

Figure 4.13: Comparison between behaviour of linear diffusion on scale-OS and CED-OS perlayer on scale-OS for a Retinal fundus image. From left to right: input image; enhancementvia Gaussian estimates for heat kernels of linear diffusion on SIM(2) group (see Section 4.5.2);enhancement via CED-OS per scale layer of the scale-OS.


Figure 4.14: CED-OS on SE-OS, [16], and scale-OS. Top- left: original image; right: originalimage + noise. Bottom- left: CED-OS on SE(2)-OS; right: CED-OS at each scale in scale-OSand then adding the layers.


Figure 4.15: CED-OS on SE-OS and scale-OS for a multi-scale test-image. Top to bottom:input image; CEDOS on SE(2)-OS; CED-OS on scale-OS


Figure 4.16: CED-OS on SE-OS and scale-OS for a collagen tissue image. Top to bottom:input image; CEDOS on SE(2)-OS; CED-OS on scale-OS


Figure 4.17: CED-OS on SE-OS and scale-OS for a X-ray catheter image. Top to bottom:input image; CEDOS on SE(2)-OS; CED-OS on scale-OS


Figure 4.18: CED-OS on SE-OS and scale-OS for a bone-microscopy image. Top to bottom:input image; CEDOS on SE(2)-OS; CED-OS on scale-OS

Chapter 5

Summary and Future Research

5.1 Evolutions and Construction of Wavelet Transformon the Similitude Group (Summary)

The enhancement of structures in noisy image data is relevant for many (bio)medical imagingapplications. Commonly used image processing methods cannot appropriately handle elon-gated structures that cross each other. In this thesis we propose to extend the orientationscore framework (successful in handling crossing structures) to a scale orientation score frame-work by introducing the notion of scaling. This is needed because in many imaging applicationselongated structures with different scales (diameters) appear.

The construction of scale orientation score (scale-OS) is introduced in Chapter 2. We beginby reviewing the construction of unitary maps on functional Hilbert spaces and arrive at thecelebrated result for wavelet transform in a group theoretic setting by Grossman, Morlet andPaul. Then a more generalized version of the wavelet transform is presented which avoidsthe requirement of irreducibility of group representations. From this generalization it followedthat our wavelet transformation is a unitary mapping between the space of images with finitesupport in the Fourier domain and a functional Hilbert space. We then explicitly quantify thestability of the inverse wavelet transform and then present criterion for the choice of admissiblewavelets that allow stable invertible wavelet transformation. Finally a suitable wavelet for thepurpose of our applications is proposed.

In Chapter 3 differential-geometric structure of the SIM(2) group is introduced which is essen-tial for designing left-invariant enhancement-contextual operators on the scale-OS. We presentthe idea of left-invariance in the context of the SIM(2) group where we deal with left-invariantvector fields and derivatives. This is followed by an explicit derivation of the exponential andlogarithmic curves for SIM(2) group. We conclude by presenting SIM(2)-convolutions whichare essential for implementation of linear operators on scale-OS.

Chapter 4 deals with operators on scale-OS. The chapter begins by showing that operators onscale-OS should be left invariant so that the net operator on the corresponding image is Eu-clidean and scaling invariant, i.e. translation, rotation and scaling of the image does not affectthe result of left-invariant operations on the corresponding scale-OS. Using the Dunford-Pettistheorem it is shown that all linear left-invariant operators on scale-OS are essentially groupconvolutions (defined appropriately). We then introduce the generic left-invariant convection-diffusion equation on the SIM(2) group and using the results by Hormander provide conditionsunder which this equation has a smooth Green’s function. Sharp Gaussian estimates for the

75

76 Summary and Future Research

Green’s function of linear left-invariant diffusion on SIM(2) group generated by subcoerciveoperators are then introduced. This is achieved by approximating the SIM(2) group by anilpotent group via contraction. Results of enhancement via linear diffusions are then pre-sented. By choosing an appropriate metric on SIM(2) and employing the Cartan connectionwe present the nonlinear convection-diffusion equation in a covariant form and show that dif-fusion takes place only along the exponential curves in SIM(2). Following this, a method forextraction of spatial curvature in this abstract setting of the convection-diffusion equation isproposed. Finally by applying nonlinear adaptive diffusion defined on the SE(2) group weenhance the scale-OS per scale and present experiments. These experiments show that CEDapplication on scale-OS is preferable over CED on SE(2)-OS. The typical advantages are

• Adaptability of diffusion can be scale dependant. This is often required in medicalimaging applications, wherein a particular scale is more important than others.

• In the SE(2)-OS framework if |Re (U(x, θ) | (or |Im (U(x, θ)) |) is large, where the realpart corresponds to a line (and imaginary part to an edge), we do not know which scaleis responsible for the high response. This is not a problem in the scale-OS framework.

5.2 Future Research

In this section we discuss potentially interesting topics for future research related to the workdescribed in this thesis.

We have presented sharp Gaussian estimates for Green’s function of linear left-invariant diffu-sions on SIM(2). It is an interesting mathematical problem to explicitly derive these Green’sfunctions both for the Heisenberg group H = (SIM(2))0 and the SIM(2) group.

Implementation of curvature estimation in the SIM(2) setting and the resulting diffusionequations proposed in this study would be an interesting numerical problem.

Due to noncommutative structure of the Lie algebra of SIM(2), applying diffusion along anelongated structure also leads to some diffusion in the direction orthogonal to the elongatedstructure. This leads to undesirable blurring which is especially noticeable in the case of linearleft-invariant diffusions via Gaussian estimates. The introduction of a “thinning” mechanismaccomplished by adaptive morphological operations on scale-OS via left-invariant Hamilton-Jacobi equations is a promising approach to obtain sharper results.

This thesis focusses on contour enhancement (via diffusions on scale-OS) in images. Contourcompletion by considering convection based evolutions on scale-OS would be a challengingtopic in the further development of scale-OS based image processing.

Acknowledgements

Joseph Addison famously wrote, “There is not a more pleasing exercise of the mind thangratitude. It is accompanied with such an inward satisfaction that the duty is sufficientlyrewarded by the performance”.

First and foremost I would like to express my gratitude towards my supervisor Dr. RemcoDuits. This work would not have been possible without his constant guidance and encourage-ment. Working under him has made me realize that Mathematics in it’s purest form is mostfun when coupled with applications.

I would also like to thank Erik Bekkers for not only bringing his keen intuition but also muchneeded Mathematica skills to the table and helping me understand the world of imaging fromthe new perspective of an Engineer.

No acknowledgement is complete without thanking one’s well wishers. I would like to thank myclassmates for showing interest in my work and constantly encouraging me. A special mentionshould be made of the group of SSSIHL alumni in Eindhoven who invited me for numerousdinners and gave me a chance to enjoy great food and unwind.

Speaking of well wishers, I would like to acknowledge the wonderful lady living in the apartmentabove mine. The constant inflow of people in her house and the ensuing loud banter during thenights, allowed me to spend a few more hours working towards my project under the influenceof wonderful music in my ears.

A loving “thank you” to my favourite people in the world, my parents, for being my anchor.

Finally my gratitude to Bhagwan Sri Sathya Sai Baba to whom I owe everything.

It seems that my academic sojourn is far from over. At this moment I cannot help but recallthe immortal quote by Dory in Finding Nemo, “Just keep swimming”.

Appendix A

A.1 Results from Chapter 2

Theorem A.1 (Haar’s theorem(see [25] for proof)). Let G be a locally compact topologicalgroup.There is, up to a positive multiplicative constant, a unique countably additive, nontrivialmeasure µ on the Borel subsets of G satisfying the following properties:

1. µ(gE) = µ(E) for any g in G and Borel set E (left-translation-invariance).

2. µ(K) is finite for every compact set K.

3. Every Borel set E is outer regular:

µ(E) = infµ(U) : E ⊆ U, Uopen

4. Every Borel set E is inner regular:

µ(E) = supµ(K) : K ⊆ E, Kcompact.

Such a measure on G is called a left Haar measure.

It can also be proved that there exists a unique (up to multiplication by a positive constant)right-translation-invariant Borel measure ν satisfying the above regularity conditions and be-ing finite on compact sets, but it need not coincide with the left-translation-invariant measureµ. The left and right Haar measures are the same only for so-called unimodular groups.

Expression for left invariant Haar measure for SIM(2) and SE(2) groups

• Similitude group has the structure of a semi-direct product, SIM(2) = R2o(SO(2)×R+),where R2 is the group of translations, SO(2) that of rotations and R+ of dilations/scaling.For this particular case we compute the left invariant Haar measure.

If (b0, θ0, a0) is a fixed element of the group and (b, θ, a) arbitrary then,

(b′, θ′, a′) = (b0, θ0, a0)(b, θ, a) = (b0 + a0Rθb, θ0 + θ, a0a).

Noting that det[Rθ0 ] = 1 we arrive at,

db′ = a20db, dθ

′ = dθ, da′ = a0da.

Thus the measure dµ(b, eiθ, a) = 1a3 dbdθda, is the left invariant Haar measure for the

SIM(2) group.

79

80

• Euclidean Motion group, SE(2) = R2 o SO(2), where R2 is the group of translations,SO(2) that of planar rotations. The expression for the left invariant Haar measure canbe arrived at as above. dµ(b, θ) = dbdθ

Evaluation of (2.15):

From Theorem 2.4, we know that,

(Wψ[f ],Wψ[f ])L2(G) = Cψ(f, f)H .

This implies W∗ψWψ = CψI, where I : H → H is the identity operator, and therefore W−1ψ =

1Cψ

(Wψ)∗. Now we arrive at (2.15). By the definition of adjoint of an operator,

(φ,Wψ[f ])L2(G,dµG) = (W∗(φ), f)L2(R2), where φ ∈ CGKψ . (A.1)

Note that we have used L2(G, dµG) norm for CGKψ because CGKψ is a closed subspace of

L2(G, dµG). Here on we drop the usage of dµG from the notation of Hilbert space. Alsonote that we have used L2(R2) as the domain of our functions, which is the space to which allimages belong. (A.1) can be written as,∫

G

φ(g)(Wψ[f ])(g)dµG(g) =

∫R2

(W∗ψ[f ])(x)f(x)dx. (A.2)

The LHS can be written as,∫G

φ(g)(Wψ[f ])(g)dµG(g) =

∫G

φ(g)

[ ∫R2

(Ugφ)(x)f(x)dx

]dµG(g)

=

∫R2

[ ∫G

φ(g)(Ugφ)(x)dµG(g)

]f(x)dx. (A.3)

The last equality follows from Fubini’s theorem. Combining (A.2) and (A.3) we have therequired result.

Proof of Lemma 2.6: Let f ∈ 〈Vψ〉⊥. Thus

∀b∈Rd∀t∈T (Ub,tψ, f)L2(Rd) = 0. (A.4)

We know that,

(Ub,tψ, f)L2(Rd) = (FRtTψ,Ff)L2(Rd) = (F−1[FRtψFf ])(b).

Thus (A.4) implies

∀b∈Rd∀t∈T(

(b, t) 7→ F−1[FRtψFf ])(b)

)= 0

⇒(

(ω, t) 7→ FRtψ(ω)Ff(ω)

)= 0 a.e. on Rd o T

⇒(

(ω, t) 7→ |FRtψ(ω)Ff(ω)|2)

= 0 a.e. on Rd o T.

Therefore,

Mψ(ω)|(Ff)(ω)| = (2π)d/2∫T

∣∣∣∣FRtψ(ω)Ff(ω)√detτ(t)

∣∣∣∣2dµT (t) = 0 a.e. on Rd.

Since Mψ 6= 0 a.e. as ψ is an admissible wavelet, this implies that |Ff |2 = 0 which in turnimplies that f = 0.

Derivation for basis of Lie Algebra 81

A.2 Derivation for basis of Lie Algebra

The four basic sets of operations of dilation, rotation and the two translations, each constituteone-parameter subgroups of SIM(2). More precisely, these subgroups are generated by groupelements of the type

(x, a, θ) = (0T , et, 0), t ∈ R,or (x, a, θ) = (0T , 1, t), t ∈ [0, 2π),

or (x, a, θ) = (eit, 1, 0), t ∈ R, i = 1, 2,

where,

0 =

(00

), e1 =

(10

), e2 =

(01

),

constitute one-parameter subgroups. A general element of SIM(2) can be written as a productof elements of these subgroups:

(x, a, θ) = (e1b1, 1, 0)(e2b2, 1, 0)(0T , 1, θ)(0T , a, 0), where b =

(b1b2

).

Generically, writing elements in any one of these subgroups as g(t) and computing the derivative

at the identity ddtg(t)

∣∣∣∣t=0

, we obtain the four 3× 3 matrices,

X1 =

0 −1 01 0 00 0 0

, X2 =

0 0 10 0 00 0 0

,

X3 =

0 0 00 0 10 0 0

, X4 =

1 0 00 1 00 0 0

,

where 1, 2, 3, 4 := θ, x, y, a.

A.3 Left-Invariant Differential Operators on SE(2)- Anintuitive explanation

In Figure A.1(a), the left side depicts a curve γ : R→ SE(2) passing through the the identityelement e, such that the tangent vector Xe = cξex + cηey + cθeθ ∈ Te(SE(2)). When thecurve γ is left multiplied by g = (x, θ) ∈ SE(2), it yields a new curve gγ, shown on the rightside. At position g, the curve gγ has a tangent vector Xg ∈ Tg(SE(2)) which corresponds toXe ∈ Te(SE(2)). This operation of transporting vectors from Te(SE(2)) to Tg(SE(2)) is calledthe push-forward operation and is denoted by Xg = (Lg)∗Xe. Intuitively the push-forwardoperator allows the movement of tangent vectors to tangent spaces at other group elements.Using the notion of push-forward operator the concept of left-invariant vector field is introducedas a tangent vector field SE(2) → Tg(SE(2)), that fulfils the property Xg = (Lg)∗Xe, for allg ∈ SE(2) and a fixed Xe ∈ Te(SE(2)).

The Lie Algebra Te(SE(2)) is spanned by the basis ex, ey, eθ as shown on the left side ofFigure A.1(a). Though one could use the same basis for Tg(SE(2)) for all g ∈ SE(2), it ismore natural to introduce a new set of basis vectors constructed by push-forward of ex, ey, eθ,

eξ(g), eη(g), eθ(g) = (Lg)∗ex, (Lg)∗ey, (Lg)∗eθ = cosθex+sinθey,−sinθex+cosθey, eθ.

82

Figure A.1: Illustration of the concept of left-invariance, from two different perspectives: (a)considered as tangent vectors to the curves, i.e. Xg = cξeξ(g) + cηeη(g) + cθeθ(g), for allg ∈ SE(2), (b) considered as differential operators on a locally defined smooth function U , i.e.Xg = cξ∂ξ(g) + cη∂η(g) + cθ∂θ(g), for all g ∈ SE(2). The push forward (Lg)∗ : Te(SE(2)) →Tg(SE(2)) connects the tangent space at the unity element Te(SE(2)) to all tangent spacesTg(SE(2)) in a ”left-invariant” way.

In Figure A.1(b) we utilize the alternate description of a tangent vectors (Xe, Xg) as a differ-ential operators, acting on a function U : SE(2)→ R, i.e. we replace all occurrences of ei by∂i. Xg can be viewed as an operator that calculated the derivative of U at g in the directionspecified by Xg,

Xg(U) = (cξ∂ξ|g +cη∂η|g +cθ∂θ|g)U = (cξ(cos θ∂x+sin θ∂y)+cη(− sin θ∂x+cos θ∂y)+cθ∂θ)U,

yielding a scalar on the R-axis on the bottom of the figure. Alternatively we could also arriveat the same result by first translating and rotating U over g, i.e. Lg(U) = U Lg, such thatthe original neighbourhood ωg is shifted back to ωe giving,

Xg(U) = Xe(U Lg) = (cξ∂ξ + cη∂η + cθ∂θ)(U Lg).

A.4 Exponential map on SIM(2) is surjective

In this section we give a proof for the surjectivity of the exp map on SIM(2) group, cf.Theorem3.4 . This proof is based on the results given in [3, Chap.4].

Theorem A.2. The exponential map, exp : Te(SIM(2)) → SIM(2) is surjective, i.e. for allg ∈ SIM(2), there exists a X ∈ Te(SIM(2)) such that, g = exp(X).

Exponential map on SIM(2) is surjective 83

Proof. Any general element X ∈ Te(SIM(2)) has the form,

X(α, β) =

4∑i=1

ciXi =

c4 −c1 c2

c1 c4 c3

0 0 0

,

where ci ∈ R for all i and α =

(c4

c1

)and β =

(c2

c4

).

Corresponding to a 2-vector v consider, a 2× 2 matrix s(v),

s(v) =

(v1 −v2

v2 v1

), v =

(v1

v2

).

For any two vectors v, w we note that, s(v)s(w) = s(w)s(v) and s(v)w = s(w)v.

Using s, X(α, β) can be rewritten as,

X(α, β) =

(s(α) β0T 1

).

Taking the exponential of the matrix X(α, β) yields g ∈ SIM(2)

g = exp(X(α, β)) =

(ec

4

Rθ F (s(α))β0T 1

), (A.5)

where 0 ≤ θ < 2π and F (s(α))β is a 2× 2 matrix defined as the sum of the following infiniteseries,

F (s(α)) = I +s(α)

2!+

[s(α)]2

3!+ · · ·

Define, sinch : R → R as Sinch(u) = sinh(u)u , sinch(0) = 1. This function is a positive,

infinitely differentiable function as is, F (u) = eu/2sinch(u2 ). Expanding F (u) using Taylorseries we get,

F (u) = eu/2sinch(u

2) = 1 +

u

2!+u2

3!+ · · · .

For u 6= 0, we can also write,

F (u) = eusinch(u

2) =

eu − 1

u.

We now define the 2× 2 matrix valued function,

eA/2sinch(A

2) = F (A) = I +

A

2!+

A2

3!+ · · · , F (N) = I,

for any 2 × 2 real matrix A and where N and I are the 2 × 2 null and identity matricesrespectively. Further, if det(A) 6= 0 then,

F (A) = eA/2sinch(A

2) = A−1[eA − I].

Further if det(eA − I) 6= 0 we may also write,

(F (A))−1 =e−A/2

sinch(A/2)= [eA − I]A.

Hence for |α| 6= 0 (recall α =

(c4

c1

)and det(s(α)) = |α|),

F (s(α)) = s(α)−1[es(α) − I]

=1

(c1)2 + (c4)2

(c4 c1

−c1 c4

)(ec

4

cos(c1)− 1 −ec4sin(c1)

ec4

sin(c1) ec4

cos(c1)− 1

)

84

and

[F (s(α))]−1 =1

2(cosh(c4)− cos(c1))

(cos(c1)− e−c4 sin(c1)

−sin(c1) cos(c1)− e−c4

)(c4 −c1c1 c4

)Thus, (A.5) can be written as

g = exp(X(α, β)) =

(ec

4

Rθ es(α)/2sinch( s(α)2 )β

0T 1

), (A.6)

and thus writing g = (b, a, θ) in matrix form and comparing, we get the relations

c4 = log(a), β =

(c2

c3

)= [F (s(α))]−1b, (A.7)

between group parameters and the lie algebra. This shows that any group element (b, a, θ)can be written as the exponential of some element X(α, β) in the lie algebra.

A.5 Implementation of G-convolution, G = R2 n SO(2)

Recall that the G-convolution is given by (3.35). There are two reasonable options for imple-menting this expression, either by means of discretization or by expressing everything in termsof steerable components. Below we briefly discuss the discretization of the orientation scorefollowed by a brief treatment of both these approaches.

In order to discretize (SE(2)) orientation scores equidistant samples in the SE(2) group usingthe standard (x, y, θ) coordinates are taken. In the spatial dimensions, the sampling distanceis set equal to the sampling distance of the corresponding image. Assume that the image issquare with Ns × Ns pixels. In the orientation dimension, the number of samples taken isdenoted by No, and therefore the sampling distance is given by sθ = 2π/No.

A.5.1 Implementation of SE(2)-convolution via Discretization

The most straightforward implementation of SE(2)-convolution is obtained by replacing theintegrals in (3.35) by finite sums. We index the samples of the SE(2) kernel, Ψ(x, y, l), withx, y ∈ −KH

s , . . . ,KHs , KH

s = bKs/2c and l ∈ −KHo , . . . ,K

Ho , KH

o = bKo/2c. Here ithas been assumed that the sampled Ψ has odd dimensions and the central value Ψ(0, 0, 0)is represented by the middle sample. The samples of the orientation scores are indexed byU(x, y, l) with x, y ∈ 0, 1, . . . , Ns − 1 and l ∈ 0, 1, . . . , No − 1. We arrive at,

R(x, y, l) =

l+KHo∑

l′=l−KHo

x+KHs∑

x′=x−KHs

y+KHs∑

y′=y−KHs

Rl′·sθ [Ψ](x− x′, y − y′, l − l′)Uext(x′, y′, l′), (A.8)

where Rl′.sθ [Ψ] denotes the rotated SE(2) kernel and Uext denotes the sampled orientationscore U on the extended domain, i.e. Uext is defined on Z × Z × Z, taking into account theappropriate boundary conditions. For the orientation dimensions, the boundary condition isperiodic. The choice of spatial boundary conditions is a more delicate issue since any boundarycondition would create undesired discontinuities on the spatial boundary leading to boundaryartefacts. In the implementation used during this study,a zero-padding boundary condition isemployed.

Implementation of G-convolution, G = R2 n SO(2) 85

The rotated kernel Rl′·sθ [Ψ] can be obtained in various ways. If an analytical formula for Ψis known, the rotated kernel is obtained by sampling it for No required rotations. Note thatwhile this method is quite accurate, it takes a large amount of memory. Alternatively, Ψ issampled for a single orientation, and it’s rotated versions are obtained using interpolation. Inthe implementation used for this study, B-spline interpolation proposed in [38] is employed. 1

A.5.2 Implementation of SE(2)-convolution via Steerability

Steerable filters, which find widespread use in image analysis, were introduced by Freeman etal. [22]. For a complete treatment along with implementation details of the steerable SE(2)-convolution, see [21, Chap. 3].

The idea is that steerable convolution kernels ψ ∈ L1(R2) ∩ L2(R2), which by definition canwritten as

ψ(x) =

N∑m=−N

ψm(x) =

N∑m=−N

fm(r)e−imφ, N ∈ N,

for some fm ∈ C1(R,C), are easily rotated (without interpolation artefacts on a rectangularpixel grid) by means of

Rθψ(x) =

N∑m=−N

ψm(r)eimθe−imφ,

with the additional advantage (regarding computation time, in case multiple orientations areneeded) that

(Rθψ ∗ f)(x) =

N∑m=−N

eimθ(ψ ∗ f)(x), x ∈ R2, θ ∈ [0, 2π).

Though it may seem that any convolution kernel within L2(R2) is steerable after expansion inthe complete orthonormal base of eigenfunctions of the Harmonic oscillator, note that, in inthis case we have a finite expansion (N <∞).

The input equals a function U : G→ C (for example an orientation score Uf of an image f) anda convolution kernel K : G→ C. The output equals W = K ∗G U , where the G-convolution isgiven by (3.35) and is related to the G-correlation as W = K ?G U , and is explicitly in termsof G-correlation is given by

W (b, θ) = (K ∗G U)(b, θ) = (K ?G U)(b, θ)

=

∫R2

2π∫0

K(R−1θ (b′ − b), θ′ − θ)U(b′, θ′)dθ′db′. (A.9)

Expand K in steerable components (with respect to both angle and space):

K(b, θ) =∑m∈Z

∑n∈Z

Kmn(b)eimθ, b = (r cosφ, r sinφ) (A.10)

such that Kmn(R−1θ b) = einθKmn(b) and

Kmn(b) =1

4π2

2π∫0

2π∫0

K(R−1θ′ b, eiθ)eimθeinθ

′dθdθ′. (A.11)

1The implementations of SE(2)-convolution employed throughout this study have been taken from the workdone by Erik Franken [21, Chap. 3] and Markus van Almsick [39].

86

Similarly expanding U into steerable components (with respect to angle only we obtain,

W (b, eiθ) =

∫R2

2π∫0

K(b′ − b, eiθ′)dθ′db′

=

∫R2

2π∫0

∑m∈Z

∑n∈Z

∑m∈Z

Kmn(b′ − b)Um(b′)ei(n+m)θe−i(m+m)θ

=

∫R2

∑m∈Z

∑n∈Z

∑m∈Z

Kmn(b′ − b)Um(b′)ei(n+m)θ

2π∫0

e−i(m+m)θdθ′

db′

= 2π∑m∈Z

∑n∈Z

ei(n+m)θ(Kmn ? U−m)(b),

where (Kmn ? U−m)(b) are just ordinary 2D-correlations in R2, i.e.

(Kmn ? U−m)(b) =

∫R2

Kmn(b′ − b)U−m(b′)db′.

So in practice when the series is truncated at say |m| ≤ Mmax, a G-convolution can beimplemented by means of (1 + 2 ∗ Nmax)(1 + 2 ∗Mmax) 2D-convolutions. The evaluation ofW is sped up by using a fast Fourier Transform instead of a discrete Fourier Transform.

Appendix B

B.1 Regularized Derivatives and Local Features

In this section we present a brief synopsis of the ill-posed nature of numerical differentiation ofan image (in the sense of Hadamard). This question has been dealt with in detail by variousauthors, such as in the context of edge detection, see [37]. We then make this operation wellposed by adding regularization.

In machine vision, as well as in most numerical problems, the data is noisy. Sensor noise arisesatleast in part from quantum fluctuations in the number of absorbed photons per sensor andunit time. This represents a fundamental limitation for real-time imagery when integrationtime and size of the sensors are limited by the need of high temporal and spatial resolution. Itis therefore important, that the results of numerical operations performed on the data are nottoo sensitive to noise. Even a small amount of noise may disrupt differentiation. For instance,consider a function f(x) and f(x) = f(x) + εsin(ωx). f(x) may be close to f(x) according tostandard norms (L2, L∞, · · · ), provided ε is sufficiently small. On the other hand, f ′(x) may

be quite different from f ′(x) if ω is large enough.

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard,[24]. Differentiation of the function f is a typically ill-posed problem, since it can be seen thatthe solution of the inverse problem

g(x) = Af(x)

where Af is the integral operator

x∫−∞

f(x)dx =

+∞∫−∞

h(x− x)f(x)dx,

where h is the step function. It is well known that inverse linear problems in which g and fbelong to Hilbert spaces are ill-posed, [36]. Various rigorous methods have been proposed fortransforming ill-posed problems into well-posed problems, the chief of which are regulariza-tion techniques, see [36, Chap II] for more details.Gaussian derivatives,[19], is an often usedtechnique to regularize derivatives of images.

Recall that the general diffusion equation for orientation scores using left-invariant differentialoperators is

∂tu =

∂θ∂ξ∂η∂τ

T

D11 D12 D13 D14

D21 D22 D23 D24

D31 D32 D33 D34

D41 D42 D43 D44

∂θ∂ξ∂η∂τ

u = Q(A) (B.1)

87

88

with u(x, τ, θ; 0)) = Uf (x, τ, θ) where U(x, τ, θ) is the scale-OS of an image f . The solution of(B.1) can be written as, u(·, ·, ·; t) = etAU . Our aim is to find local features using well posedleft-invariant derivative operators. We therefore wish to operationalize D(etAU) where D isany derivative constructed from ∂θ, ∂ξ, ∂η, ∂τ and etA is the left-invariant diffusion operatordescribed above. Note that due to the noncommutative nature of left-invariant derivatives,the order of regularization operator and differential operator matters. We provide two im-plementation of the regularized derivatives. In the first one we consider the use of diffusionwith a diagonal diffusion tensor as regularizer. Subsequently we consider the case of isotropicdiffusion over the spatial derivatives as regulaizer.

B.1.1 General Regularized Derivatives in Scale-OS

We wish to use anisotropic diffusion as regularizer to calculate left-invariant derivatives inSIM(2),

∂tu(g, t) = (D11∂2θ +D22∂

2ξ +D44∂

2τ )u(g, t), g ∈ SIM(2), t ≥ 0,

u(g, t = 0) = U(g).(B.2)

Note that for proper regularization in all directions we require D11 6= 0, D22 6= 0 and D44 6=0. However D33 6= 0 is not a requirement because non-zero diffusion in θ and ξ directionsinduces smoothing in η direction due to the commutator relationships, recall (3.18). AlsoD22 6= 0 and D33 = 0 implies diffusion tangent to the contours and not orthogonal to it. Recallthat an approximative analytical expression was derived for the Green’s function of (B.2) inSection(4.5),

|Kq=1,Dt (g)| ≤ 1

4πt2D11D22D44exp

(−1

4t

(θ2

D11+

(c2)2

D22+

τ2

D44+

|c3|√D11D22D44

))(B.3)

where,

c2 =(yθ − xτ) + (−θη + τξ)

t (1 + e2τ − 2eτ cosθ), c3 =

−(xθ + yτ) + (θξ + τη)

t (1 + e2τ − 2eτ cosθ).

Note that this approximation renders a non-differentiable approximation for the Green’s func-tion and is therefore not usable for the purpose of regularized derivatives. A smooth approxi-mation of (B.3) was provided in Section4.5,

|Kq=1,Dt (g)| ≤ 1

4πt2D11D22D44exp

(−1

4t

([θ2

D11+

(c2)2

D22+

τ2

D44

]2

+|c3|2

D11D22D44

)). (B.4)

To arrive at regularized derivatives we sample the required derivatives of this approximateGreen’s function and then apply a SIM(2)-convolution to the scale-OS. Since the formulaefor such kernels is rather cumbersome below we provide a different approach to regularizedderivatives.

B.1.2 Gaussian Derivatives in Scale-OS

The implementation in the previous section is rather complicated. However, the implementa-tion becomes much simpler if we restrict to the case of isotropic diffusion (D22 = D33) with nodiffusion over scaling (D44 = 0), i.e.

∂tu = (D11∂2θ +D22(∂2

ξ + ∂2η))u = (D11∂

2θ +D22e

2τ (∂2x + ∂2

y))u. (B.5)

Horizontal Curves 89

Since at a fixed scale (B.5) can be described using commutative ∂x, ∂y and ∂θ derivatives, thisequation simplifies to the diffusion equation in R3, except the 2π periodicity of the θ dimension.Therefore, the Green’s function is approximated by the Gaussian kernel

Gts,to(x, y, θ) =1

8√π3t2sto

e−x2+y2

4ts− θ2

4to , (B.6)

where to = tD11 and ts = t(e2τD22). Though in this case we can use standard separableimplementations of Gaussian derivatives we have to take into account the non-commutativenature of the derivatives. A (i, j, k)th order isotropic Gaussian derivative implementation fora 3D image f satisfies the following relation,

∂ix∂jy∂

kz et(∂2

x+∂2y+∂2

z)f = ∂ixet∂2x

(∂jye

t∂2y

(∂kz e

t∂2zf))

. (B.7)

where the equality implies separability along the three dimensions. We want to make use ofsimilar implementations to construct Gaussian derivatives in the orientation scores, meaningthat we have to ensure that the same permutation of differential operators and regularizationoperators is allowed. By noting that

∂iξ∂jη∂

kθ eto∂

2θ+ts(∂

2x+∂2

y) = ∂iξ∂jηets(∂

2x+∂2

y)∂kθ eto∂

2θ , (B.8)

∂kθ ∂iξ∂jηeto∂

2θ+ts(∂

2x+∂2

y) 6= ∂kθ eto∂

2θ∂iξ∂

jηets(∂

2x+∂2

y),

we conclude that we always should ensure a certain ordering of the derivative operators, i.e.one should first calculate the derivative ∂θ and then the commuting spatial derivatives ∂ξ, ∂η,which are calculated from the Cartesian derivatives ∂x, ∂y using ∂ξ = eτ (cosθ∂x + sinθ∂y)and ∂η = eτ (−sinθ∂x+cosθ∂y). The commutator relations [∂θ, ∂ξ] = ∂η and [∂θ, ∂η] = −∂ξ (seeSection(3.1.4)) allow to rewrite the derivatives in this order. For e.g. the derivative ∂ξ∂θ canbe calculated directly with Gaussian derivatives but the derivative ∂θ∂ξ must be constructedwith Gaussian derivatives ∂ξ∂θ + ∂η. It is very important to note that this implementation forGaussian derivatives works τ is a constant. Thus at every scale in our OS we have a differentimplementation of the regularized derivatives.

B.2 Horizontal Curves

Definition B.1. Choose L > 0. A curve γ = (x, y, θ) : [0, L]→ SE(2), with x, y ∈ C1([0, L])and θ, τ ∈ C0([0, L]) is called horizontal iff

〈A3, γ(s)〉 = 0⇔⟨− 1

eτsin θdx+

1

eτcos θdy, γ(s)

⟩= 0, (B.9)

where γ(s) ∈ Tg(SE(2)).

A vector Xg ∈ Tg(SE(2)) is called horizontal if it is the tangent vector to some horizontal curvethrough g ∈ SE(2). A vector field H is called horizontal if Xg is horizontal for all g ∈ SE(2).(B.9) implies that a smooth curve s 7→ γ(s) is horizontal iff

γ(s) ∈ spaneθ|γ(s), eξ|γ(s), for all s > 0. (B.10)

Let H be a horizontal vector field (horizontal section of the tangent bundle). Then ∀U ∈ H wehave 〈A3, U〉 = 0. Thus in H we have that − 1

eτ sin θdx+ 1eτ cos θdy = 0, i.e. in H, tan θ = dy

dx .This provides us with an alternate description of a horizontal curve.

Definition B.2. A curve s 7→ γ(s) = (x(s), y(s), θ(s)) is called horizontal iff θ(s) = arg(x(s)+iy(s)). Then γ is called the lifted curve in SE(2) of the projected curve s 7→ x(s) = (x(s), y(s))in R2.

90

A smooth horizontal curve γ = (x, θ) in SE(2) given by s 7→ (x(s), y(s), θ(s)) can be parametrizedby the arc length s > 0 of it’s projection x = PR2γ on the spatial plane. Using the spatial arclength parametrization and (B.2), along a horizontal curve γ = (x, θ) : R2 → SE(2) one has

γ = (x, θ) is horizontal ⇒ κ(s) = θ(s) (B.11)

where κ(s) = ±‖x(s)‖R2 , with ‖ · ‖R2 the euclidean norm on R2, is the curvature of the curves 7→ x(s) = PR2γ(s).

By (B.10) and Def. B.2 it follows that the horizontal part Hg ⊂ Tg(SE(2)) of each tangentspace Tg(SE(2)) is spanned by Hg = ∂θ|g, ∂ξ|g and the space of horizontal left-invariantvector fields is spanned by A1,A2 = ∂θ, ∂ξ. However it is important to note that ahorizontal curve itself s 7→ γ(s) ∈ SIM(2), can have components in all directions eθ, eξ, eηin contrast to γ(s) ∈ Tγ(s)(SIM(2)).

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