RICE UNIVERSITY

Coherent Multiscale Image Processing

using Quaternion Wavelets

by

Wai Lam Chan

A Thesis Submitted

in Partial Fulfillment of the

Requirements for the Degree

Master of Science

Approved, Thesis Committee:

Richard Baraniuk, ChairProfessor of Electrical and ComputerEngineering, Rice University

Michael OrchardProfessor of Electrical and ComputerEngineering, Rice University

Deceased

Hyeokho ChoiAdjunct Assistant Professor of Electricaland Computer Engineering,Rice University

Houston, Texas

May, 2006

ABSTRACT

Coherent Multiscale Image Processing

using Quaternion Wavelets

by

Wai Lam Chan

This thesis develops a quaternion wavelet transform (QWT) as a new multiscale

analysis tool for geometric image features. The QWT is a near shift-invariant tight

frame representation whose coefficients sport a magnitude and three phases: two

phases encode local image shifts while the third contains textural information. The

QWT can be computed using a dual-tree filter bank with linear computational com-

plexity. We develop the QWT by applying an alternative theory of 2-D Hilbert trans-

forms and analytic signals to the 1-D complex wavelet transform. To demonstrate

the properties of the QWT’s coherent magnitude/phase representation, we develop a

efficient and accurate algorithm for estimating the local geometrical structure of im-

ages and a multiscale algorithm for estimating the disparity between a pair of images.

The latter algorithm has potentials for various applications in image registration and

flow estimation. It uses an interesting multiscale phase unwrapping procedure and

features linear complexity and sub-pixel accuracy.

Acknowledgments

My sincere thanks to Dr. Hyeokho Choi for his valuable insights and guidance; to

Dr. Richard Baraniuk for his encouragement and research support; to Dr. Michael

Orchard for being part of my thesis committee; to my friends, Vinay Ribeiro and

Gang Hua, for their help in preparation for my defense; and to many others for their

constant prayer, support and friendship; and last but not the least, to my family for

their love and patience.

Contents

Abstract ii

Acknowledgments iii

List of Illustrations vi

1 Introduction 1

2 Quaternion Wavelet Transform 8

2.1 Real and Complex Wavelet Transforms . . . . . . . . . . . . . . . . . 8

2.2 Quaternion Wavelet Transform (QWT) . . . . . . . . . . . . . . . . . 14

2.2.1 Quaternion Hilbert Transform . . . . . . . . . . . . . . . . . . 14

2.2.2 QWT Construction . . . . . . . . . . . . . . . . . . . . . . . . 16

3 QWT Properties 18

3.1 Tight Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Relationship to the 2-D CWT . . . . . . . . . . . . . . . . . . . . . . 19

3.3 QWT of Real Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 QWT Phase Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 QWT Applications 24

4.1 Edge Geometry Estimation . . . . . . . . . . . . . . . . . . . . . . . . 24

v

4.2 QWT-based Image Disparity Estimation . . . . . . . . . . . . . . . . 30

5 Conclusions 39

A QFT Plancharel Theorem 41

B Continuous QWT of 2-D edge signal 42

B.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

B.2 Calculation of (4.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

B.3 Calculation of QWT Phase Angles for a Step Edge . . . . . . . . . . 44

C Theoretical Analysis of Leakage Effect 46

Bibliography 49

Illustrations

1.1 Three real wavelets (from the vertical, horizontal, and diagonal subbands,

respectively) from the 2-D DWT basis generated from the length-14

Daubechies filter. The color map sets large negative values to blue, zero to

green, and large positive values to red. . . . . . . . . . . . . . . . . . . . 2

1.2 Six complex wavelets from the 2-D dual-tree CWT frame generated from

the orthogonal near-symmetric filters [1] in the first stage and the Q-filters

[2] in subsequent stages. (a) Real parts; (b) imaginary parts. . . . . . . . 4

1.3 Three quaternion wavelets from the 2-D dual-tree QWT frame. Each

quaternion wavelet comprises four components that are 90◦ phase shifts of

each other in the vertical, horizontal, and both directions. (a) Diagonal

subband, from left to right: ψh(x)ψh(y) (a usual, real DWT tensor

wavelet), ψh(x)ψg(y), ψg(x)ψh(y), ψg(x)ψg(y). The image on the far right

is the quaternion wavelet magnitude |ψq(x, y)|, a non-oscillating function.

(b) Horizontal subband, from left to right: φh(x)ψh(y) (a usual, real DWT

tensor wavelet), φh(x)ψg(y), φg(x)ψh(y), φg(x)ψg(y). (c) Vertical subband,

from left to right: ψh(x)φh(y) (a usual, real DWT tensor wavelet),

ψh(x)φg(y), ψg(x)φh(y), ψg(x)φg(y). . . . . . . . . . . . . . . . . . . . 6

vii

2.1 (a) Magnitude of the 1-D step response of length-8 Daubechies DWT

coefficients d`,p(t0) at four consecutive scales. The horizontal axis

represents the location of the step edge, and the vertical axis represents

the magnitude of the corresponding wavelet coefficient. The vertical line

passes through wavelet coefficient values for one particular location of an

edge. Wavelet magnitudes vary significantly across both scales and various

step locations, indicating the shift-variance of the DWT. (b) Step response

of the CWT coefficients with the same locales. The non-oscillatory

response indicates the near shift-invariance of the CWT. . . . . . . . . . 10

2.2 The 1-D dual-tree CWT is implemented using a pair of filter banks

operating on the same data simultaneously. Outputs of the filter banks are

the dual-tree scaling coefficients, chpand cgp , and the wavelet coefficients,

dh`,pand dg`,p

, at scale ` and shift p. The CWT coefficients are then

obtained as dh`,p+ j dg`,p

. . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 (a) The CWT coefficient’s single phase angle responds linearly to image

shifts d in a direction orthogonal to the wavelet’s orientation. (b) Two of

the QWT coefficient’s three phase angles respond linearly to image shifts d

in an absolute horizontal/vertical coordinate system. . . . . . . . . . . . 13

2.4 Complex Fourier domain relationships among the four quadrature

components of a quaternion wavelet ψq(x, y) in the diagonal subband. . . 17

viii

3.1 Effect of varying θ3 on the structure of the corresponding weighted

quaternion wavelet from the diagonal subband (left to right):

θ3 = −π4 ,−π

8 , 0,π8 ,

π4 respectively. The corresponding wavelet changes from

textured (θ3 = 0) to oriented (θ3 = ±π4 ). . . . . . . . . . . . . . . . . . 23

4.1 (a) Parameterization of a single edge in a dyadic image block

(wedgelet [3]). (b) QFT spectrum of the edge; shaded squares represent

the quaternion wavelets in the vertical, horizontal, and diagonal subbands.

The energy of the edge is concentrated along the two dark lines crossing at

the origin and is captured by the horizontal subband with spectral center

at quaternion frequency (u, v). The region bounded by the dashed line

demonstrates the spectral support of the QWT basis “leaking” into the

neighboring quadrant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Magnitude of a QWT coefficient as a function of edge orientation β

and offset d for the (a) horizontal φ(x)ψ(y) subband and (b) diagonal

ψ(x)ψ(y) subband. (c), (d) Absolute error for estimating d by the

phase angles (θ1,θ2) from the same two subbands, relative to the

normalized edge length of the dyadic block (= 1 unit). Estimates are

accurate in the regions where the coefficient magnitudes are large. . . 26

ix

4.3 Local edge geometry estimation using the QWT. (a) Several edgy regions

from the “cameraman” image are shown on the left; (b)–(e) on the right

are edge estimates from the corresponding QWT coefficients. The upper

row shows the original image region, the lower row shows a wedgelet (see

Figure 4.1(a)) having the edge parameter estimates (β, d). (No attempt is

made to capture the texture within the block.) . . . . . . . . . . . . . . 30

4.4 Multiscale QWT phase-based disparity estimation results. (a), (b)

Reference and target images from the “Rubik’s cube” image sequence [4].

(c) Disparity estimates between two images in the sequence, shown as

arrows overlaid on top of the reference image (zoomed in for better

visualization). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.5 Comparison of multiscale QWT phase-based disparity estimation with two

motion estimation algorithms, Gradient Correlation (GC) [5] and

exhaustive search (ES). Performance measure is the PSNR (in dB) between

the motion-compensated image and the target image of three test image

sequences (“Rubik”, “Taxi” and “Mobcal”). (a) Frame-by-frame PSNR

performance comparison in the “Rubik” sequence. (b) Table of average

PSNR performance (over all frames) for each test sequence. Multiscale

QWT phase-based method demonstrates the best performance among the

three test algorithms for the “Rubik” sequence, and shows comparable

performance to the other algorithms for the “Taxi” and “Mobcal” sequences. 36

1

Chapter 1

Introduction

The encoding and estimation of the relative locations of image features play an im-

portant role in many image processing applications, ranging from feature detection

and target recognition to image compression. In edge detection, for example, the

goal is to locate object boundaries in an image. In image denoising or compression,

state-of-the-art techniques achieve significant performance improvements by exploit-

ing information on the relative locations of large transform coefficients [6, 7, 8, 9].

An efficient way to compute and represent relative location information in signals

is through the phase of the Fourier transform. The Fourier shift theorem provides a

simple linear relationship between the signal shift and the Fourier phase. When only

a local region of the signal is of interest, the short-time Fourier transform (STFT)

provides a local Fourier phase for each windowed portion of the signal. The use of the

Fourier phase to decipher the relative locations of image features is well established in

the image processing and computer vision communities for applications such as stereo

matching and image registration [10, 11, 12]. Indeed, the classic experiment of Lim

and Oppenheim [13] demonstrated that for natural images the Fourier phase contains

a wealth of information, often more than the magnitude. Estimating location infor-

mation using local phase provides more robust estimates with sub-pixel accuracy and

requires less computational effort compared to purely amplitude-based approaches.

2

Figure 1.1 : Three real wavelets (from the vertical, horizontal, and diagonal subbands,respectively) from the 2-D DWT basis generated from the length-14 Daubechies filter. Thecolor map sets large negative values to blue, zero to green, and large positive values to red.

Another, more far-flung application of Fourier phase is in radar interferometry, which

infers a topographic map from the phase differences between signals at two (or more)

receivers [14].

For signals containing isolated singularities, such as piecewise smooth functions,

the discrete wavelet transform (DWT) has proven to be more efficient than the STFT.

The locality and zooming properties of the wavelet basis functions leads to a sparse

representation of such signals that compacts the signal energy into a small number

of coefficients. Wavelet coefficient sparsity is the key enabler of algorithms such as

wavelet-based denoising by shrinkage [15]. Many natural images consist of smooth

or textured regions separated by edges and are well-suited to wavelet analysis and

representation. Other advantages of wavelet analysis include its multiscale structure,

invertibility, and efficient filter-bank implementation. 2-D DWT basis functions are

easily formed as the tensor products of 1-D DWT basis function along the vertical

and horizontal directions; see Figure 1.1.

The conventional, real-valued DWT, however, suffers from two major drawbacks.

The first drawback is shift-variance: a small shift of the signal causes significant

fluctuations in wavelet coefficient energy, making it difficult to extract or model signal

3

information from the coefficient values. The second drawback is the lack of a notion

of phase to encode signal location information as in the Fourier case.

Complex wavelet transforms (CWTs) provide an avenue to remedy these two

drawbacks of the DWT. It is interesting to note that the earliest modern wavelets,

those of Grossmann and Morlet [16], where in fact complex, and Grossman continu-

ally emphasized the power of the CWT phase for signal analysis and representation.

Subsequent researchers have developed orthogonal or biorthogonal CWTs; see, for

example [17, 18, 19, 20, 21, 22].

A productive line of research has developed over the past decade on the dual-tree

CWT, which in 1-D combines two orthogonal or biorthogonal wavelet bases using

complex algebra into a single system, with one basis corresponding to the “real part”

of the complex wavelet and the other to the “imaginary part” [23]. Ideally, the real

and imaginary wavelets are a Hilbert transform pair (90◦ out of phase) and form an

analytic wavelet supported on only the positive frequencies in the Fourier domain,

just like the cosine and sine components of a complex sinusoid. The 1-D dual-tree

CWT is a slightly (2×) redundant tight frame, and the magnitudes of its coefficients

are nearly shift invariant [23]. There also exists an approximate linear relationship

between the dual-tree CWT phase and the locations of 1-D signal singularities [24]

as in the Fourier shift theorem.

The 2-D dual-tree CWT for images is based on the theory of the 2-D Hilbert

transform (HT) and 2-D analytic signal first suggested by Hahn [25]. In particular,

a 2-D dual-tree complex wavelet is formed by the 1-D HT of a usual 2-D real DWT

wavelet in either or both of the horizontal or/and vertical directions. The result is

4

(a)

(b)

Figure 1.2 : Six complex wavelets from the 2-D dual-tree CWT frame generated from theorthogonal near-symmetric filters [1] in the first stage and the Q-filters [2] in subsequentstages. (a) Real parts; (b) imaginary parts.

a 4× redundant tight frame with six directional subbands; see Figure 1.2 [23, 26].

The 2-D CWT is near shift-invariant, and its magnitude-phase representation has a

complex phase component that encodes shifts of local 1-D structures in images such

as edges and ridges [27]. As a result, the 2-D dual-tree CWT has proved useful for a

variety of tasks in image processing [8, 27, 28, 29].

Each 2-D dual-tree CWT basis function has one phase angle, which encodes the

1-D shift of image features perpendicular to its orientation. This may be sufficient

for analyzing local 1-D structures such as edges. However, when the feature under

analysis is intrinsically 2-D [4] — for example an image T-junction — then its rel-

ative location is defined in both the horizontal and vertical directions. This causes

ambiguity in the CWT phase shift; we cannot resolve the image shifts in both the

horizontal and vertical directions from the change of only one CWT coefficient phase.

To overcome this ambiguity, we must conduct a joint analysis with two CWT phases

from differently oriented subbands, which can complicate image analysis and model-

ing considerably.

5

In this thesis, we explore an alternative theory of the 2-D HT and analytic signal

due to Bulow [4, 30] and show that it leads to an alternative to the dual-tree CWT. In

Bulow’s HT, the 2-D analytic signal is defined by limiting the 2-D Fourier spectrum to

a single quadrant. Applying this theory within the dual tree framework, we develop

and study in this thesis a new dual-tree quaternion wavelet transform (QWT), where

each quaternion wavelet consists of a real part (a usual real DWT wavelet) and three

imaginary parts that are organized by quaternion algebra; see Figure 1.3. The QWT,

is a 4× redundant tight frame with three subbands (horizontal, vertical and diagonal).

It is also near shift-invariant.

The QWT inherits a quaternion magnitude-phase representation from the quater-

nion Fourier transform (QFT). The first two QWT phases (θ1,θ2) encode the shifts of

image features in the absolute horizontal/vertical coordinate system, while the third

phase θ3 encodes edge orientation mixtures and textural information.

To illustrate the power of coherent processing using both the QWT magnitude

and phase, we consider two image processing applications. In the first application,

edge estimation, we demonstrate the efficiency of the QWT magnitude-phase repre-

sentation for encoding the orientation and offset of edges in local image blocks. Our

algorithm is entirely based on the QWT shift theorem and the interpretation of the

QWT as a local QFT analysis. In the second application, image registration, we design

a new multiscale image disparity estimation algorithm. The QWT provides a natu-

ral multiscale framework for measuring and adjusting local disparities and performing

phase unwrapping from coarse to fine scales with linear computational efficiency. The

convenient QWT encoding of location information in the absolute horizontal/vertical

6

(a)

(b)

(c)

Figure 1.3 : Three quaternion wavelets from the 2-D dual-tree QWT frame. Each quater-nion wavelet comprises four components that are 90◦ phase shifts of each other in the ver-tical, horizontal, and both directions. (a) Diagonal subband, from left to right: ψh(x)ψh(y)(a usual, real DWT tensor wavelet), ψh(x)ψg(y), ψg(x)ψh(y), ψg(x)ψg(y). The image onthe far right is the quaternion wavelet magnitude |ψq(x, y)|, a non-oscillating function. (b)Horizontal subband, from left to right: φh(x)ψh(y) (a usual, real DWT tensor wavelet),φh(x)ψg(y), φg(x)ψh(y), φg(x)ψg(y). (c) Vertical subband, from left to right: ψh(x)φh(y)(a usual, real DWT tensor wavelet), ψh(x)φg(y), ψg(x)φh(y), ψg(x)φg(y).

coordinate system facilitates averaging across subband estimates for more robust per-

formance. Our algorithm offers sub-pixel estimation accuracy and runs faster than

existing disparity estimation algorithms like block matching and phase correlation [5].

When the underlying image disparity field is smooth, our method also has superior

performance over these existing techniques.

Previous work in quaternions and the theory of 2-D HT and analytic signals for

image processing includes Bulow’s extension of the Fourier transform and complex

Gabor filters to a quaternion Fourier transform (QFT) [4]. Our QWT can be inter-

7

preted as a local QFT and thus inherits many of its interesting and useful theoretical

properties such as the quaternion phase representation, symmetry properties, and

shift theorem. In addition, our QWT sports a linear-time and invertible computa-

tional algorithm based on filter banks. There are also interesting connections between

our QWT and the (non-redundant) quaternion wavelet representations of Ates and

Orchard [31] and Gang and Orchard [32]. It is also worthwhile to note a third al-

ternative 2-D HT called the Riesz transform and its associated analytic signal called

the monogenic signal [33]. The monogenic signal, generated by spherical quadrature

filters, has a vector-valued phase that encodes both the orientations of intrinsically

1-D (edge-like) image features and their shifts normal to the edge orientation. While

useful for image processing applications such as edge detection and stereo correspon-

dence [33, 34], this transform is not invertible and thus can only be used as an analysis

tool.

This thesis is organized as follows. We start by briefly reviewing the DWT and

dual-tree CWT in Chapter 2, which then develops the QWT. Chapter 3 discusses

some important properties of the new transform, in particular, its phase response to

singularities. We develop and demonstrate the QWT phase-based image registration

algorithm in Chapter 4. Chapter 5 concludes the thesis and discusses the potential

of the QWT for future applications. The appendix contains detailed derivations and

proofs of some QWT properties and theorems from Chapter 3 and 4.

8

Chapter 2

Quaternion Wavelet Transform

2.1 Real and Complex Wavelet Transforms

This section overviews the real DWT and the dual-tree CWT. We also develop a new

formulation for the 2-D CWT [23, 26] using the theory of 2-D HTs.

The DWT represents a 1-D real signal f(t) in terms of shifted versions of a scaling

function φ(t) and shifted and scaled versions of a wavelet function ψ(t) [35]. The

functions φL,p(t) = 2Lφ(2Lt − p) and ψ`,p(t) = 2`ψ(2`t − p), ` ≥ L, p ∈ Z form an

orthonormal basis, and we can represent any f(t) ∈ L2(R) as

f(t) =∑

p∈Z

cL,pφL,p(t) +∑

`≥L,p∈Z

d`,pψ`,p(t), (2.1)

where cL,p =∫

f(t)φL,p(t)dt and d`,p =∫

f(t)ψ`,p(t)dt are the scaling and wavelet co-

efficients, respectively. The parameter L sets the coarsest scale space that is spanned

by φL,p(t). Behind each wavelet transform is a filterbank based on lowpass and high-

pass filters.

The standard 2-D DWT is obtained using tensor products of 1-D DWTs over the

horizontal and vertical dimensions. The result is the scaling function φ(x)φ(y) and

three subband wavelets ψ(x)ψ(y), φ(x)ψ(y), and ψ(x)φ(y) that are oriented in the

diagonal, horizontal, and vertical directions, respectively [35] (see Figure 1.1).

The DWT suffers from shift variance; that is, a small shift in the signal can

9

greatly perturb the magnitude of wavelet coefficients around singularities. Consider

a simple 1-D example: a piecewise smooth signal f(t − t0) such as a step function

u(t) analyzed by a DWT basis having a sufficient number of vanishing moments. Its

wavelet coefficients are samples of the step response of the wavelet

d`,p(t0) = ∆

∫ 2`t0−p

−∞

ψ(t) dt (2.2)

where ∆ is the height of the jump. Since ψ(t) is a bandpass function that oscillates

around zero, so does its step response d`,p as a function of p. Moreover, the factor 2`

in the upper limit (` ≥ 0) amplifies the sensitivity of d`,p to the time shift t0, leading

to strong shift variance. Figure 2.1(a) demonstrates this shift-variance of the DWT.

The wavelet coefficients representing the edge oscillate significantly both across scales

and across shifts within the same scale. Besides shift-variance, the DWT also lacks

a notion of phase to encode signal location information. These problems complicate

modeling and information extraction in the wavelet domain.

The 1-D dual-tree CWT expands a real signal in terms of two sets of wavelet

and scaling functions obtained from two independent filterbanks [23], as shown in

Figure 2.2. We will use the notation φh(t) and ψh(t) to denote the scaling and

wavelet functions and chL,pand dh`,p

to denote their corresponding coefficients, where

h specifies a particular set of wavelet filters. The wavelet functions ψh(t) and ψg(t)

from the two trees play the role of the real and imaginary parts of a complex analytic

wavelet ψc(t) = ψh(t)+jψg(t). The imaginary wavelet ψg(t) is the 1-D HT of the real

wavelet ψh(t). The combined system is a 2× redundant tight frame that, by virtue

10

(a) D8 magnitude step response (b) complex magnitude step response

Figure 2.1 : (a) Magnitude of the 1-D step response of length-8 Daubechies DWT coeffi-cients d`,p(t0) at four consecutive scales. The horizontal axis represents the location of thestep edge, and the vertical axis represents the magnitude of the corresponding wavelet coef-ficient. The vertical line passes through wavelet coefficient values for one particular locationof an edge. Wavelet magnitudes vary significantly across both scales and various step loca-tions, indicating the shift-variance of the DWT. (b) Step response of the CWT coefficientswith the same locales. The non-oscillatory response indicates the near shift-invariance ofthe CWT.

of the fact that |ψc(t)| is non-oscillating, is shift-invariant.∗ Figure 2.1(b) shows the

smooth magnitude step responses of the CWT coefficients at four consecutive scales.

Compared to the oscillatory step response in the DWT, the CWT step response

magnitude varies monotonically with the distance of the edge from the location of the

wavelet basis function. Thus the CWT magnitude is always large around singularities,

which is beneficial in many applications [36].

It is useful to recall that the Fourier transform of the imaginary wavelet Ψg(ω)

equals −jΨh(ω) when ω > 0 and jΨh(ω) when ω < 0. Thus, the Fourier transform of

the complex wavelet function Ψh(ω) + jΨg(ω) = Ψc(ω) has no (or little in practice)

∗In practice, in order to have finite-length wavelets, the HT is only approximately satisfied, ψc(t)

is only approximately analytic, and the CWT is only approximately shift-invariant [23, 26].

11

2

2g (n)

g (n)0

1

2

2

h (n)

h (n)

0

1

2

2

h (n)

h (n)

0

1

2

2g (n)

g (n)0

1

2

2

2

2g (n)

g (n)0

1

h (n)

h (n)

0

1 h1ψ

g1

ψ

g2

ψ

h2ψ

h3ψ

φh

g3

ψ

φg(n)

(n)

(n)

(n)

(n)

(n)

(n)

(n)

Figure 2.2 : The 1-D dual-tree CWT is implemented using a pair of filter banks operatingon the same data simultaneously. Outputs of the filter banks are the dual-tree scalingcoefficients, chp

and cgp , and the wavelet coefficients, dh`,pand dg`,p

, at scale ` and shift p.The CWT coefficients are then obtained as dh`,p

+ j dg`,p.

energy in the negative frequency region, making it an analytic signal [23].

Extending the 1-D CWT to 2-D requires an extension of the HT and analytic

signal. There exist not one but several different definitions that each zero out a

different portion of the 2-D frequency plane [30]. We will consider two definitions.

The first, proposed in [25], employs complex algebra and zeros out frequencies on all

but a single quadrant (kx, ky > 0, for example). In this formulation, the complete

2-D analytic signal consists of two parts: one having spectrum on the upper right

quadrant (kx, ky > 0) and the other on the upper left quadrant (kx < 0, ky > 0) [30].

Definition 1 [25] Let f be a real, 2-D function. The complete 2-D complex analytic

signal is defined in the space domain, x = (x, y), as the pair of complex signals

fA1(x) = (f(x)− fHi(x)) + j(fHi1(x) + fHi2(x)), (2.3)

fA2(x) = (f(x) + fHi(x)) + j(fHi1(x)− fHi2(x)), (2.4)

12

where

fHi1(x) = f(x) ∗ ∗ δ(y)πx

, (2.5)

fHi2(x) = f(x) ∗ ∗ δ(x)πy

, (2.6)

fHi(x) = f(x) ∗ ∗ 1

π2xy. (2.7)

The function fHi is the total HT; the functions fHi1 and fHi2 are the partial HTs; δ(x)

and δ(y) are impulse sheets along the y-axis and x-axis respectively; and ∗∗ denotes

2-D convolution.

The 2-D complex analytic signal in (2.3)–(2.4) is the notion behind the 2-D dual-

tree CWT [23, 26]. Each 2-D CWT basis function is a 2-D complex analytic signal

consisting of a standard DWT tensor wavelet plus three additional real wavelets

obtained by 1-D HTs along either or both coordinates. For example, starting from

real DWT’s diagonal-subband tensor product wavelet f(x) = ψh(x)ψh(y) from above

we obtain from equations (2.5)-(2.7) its partial and total HTs

(fHi1 , fHi2 , fHi) = (ψg(x)ψh(y), ψh(x)ψg(y), ψg(x)ψg(y)).

¿From Definition 1, we then obtain the two complex wavelets

ψc1(x, y) = (ψh(x)ψh(y)− ψg(x)ψg(y)) + j(ψh(x)ψg(y) + ψg(x)ψh(y)), (2.8)

ψc2(x, y) = (ψh(x)ψh(y) + ψg(x)ψg(y)) + j(ψh(x)ψg(y)− ψg(x)ψh(y)), (2.9)

having orientations, 45◦ and −45◦, respectively. Similar expressions can be obtained

for the other two subbands (±15◦ and ±75◦) based on ψh(x)φh(y) and φh(x)ψh(y).

Each 2-D CWT coefficient has only one phase angle, which encodes the 1-D shift

of image features perpendicular to its subband direction. Figure 2.3(a) illustrates this

13

(a) (b)

Figure 2.3 : (a) The CWT coefficient’s single phase angle responds linearly to image shiftsd in a direction orthogonal to the wavelet’s orientation. (b) Two of the QWT coefficient’sthree phase angles respond linearly to image shifts d in an absolute horizontal/verticalcoordinate system.

phase-shift property. This encoding may be sufficient for local 1-D structures such

as edges, since we can define shifts of an edge uniquely by a single value, say d, in

the direction perpendicular to the edge. However, even in this case, the analysis is

not so straightforward when the edge does not align with the six orientations of the

CWT subbands. And for intrinsically 2-D (non-edge-like) image features such as in

Figure 2.3(a), we must define its relative location using two values (d1,d2) in the x and

y directions, respectively. This creates ambiguity in the CWT phase shift. We can re-

solve this ambiguity with the orthogonal CWT subband, but this complicates the use

of the CWT for image analysis, modeling, and other image processing applications.

In contrast, Figure 2.3(b) illustrates a more convenient encoding of image shifts in

absolute x, y-coordinates (with two phase angles) using the quaternion phases of our

new QWT, to which we now turn our attention.

14

2.2 Quaternion Wavelet Transform (QWT)

2.2.1 Quaternion Hilbert Transform

There are several alternatives to the 2-D analytic signal of Definition 1; we focus on

one here due to Bulow [30]. It combines the partial and total HTs from (2.5)–(2.7) to

form an analytic signal comprising a real part and three imaginary components that

are manipulated using quaternion algebra [4].

The set of quaternions H = {a+ j1b+ j2c+ j3d | a, b, c, d ∈ R} has multiplication

rules j1j2 = −j2j1 = j3 and j21 = j2

2 = −1 as well as component-wise addition and

multiplication by real numbers [37]. Additional multiplication rules include: j23 = −1,

j2j3 = −j3j2 = j1 and j3j1 = −j1j3 = j2. Note that the quaternionic multiplication is

not commutative. The conjugate q∗ of a quaternion q = a+ j1b+ j2c+ j3d is defined

by q∗ = a− j1b− j2c− j3d while the magnitude is defined as |q| = √qq∗.

An alternative representation for a quaternion is through its magnitude and three

phase angles: q = |q| ej1θ1ej3θ3ej2θ2 [4], where (θ1, θ2, θ3) are the quaternion phase

angles, with each angle uniquely defined within the range (θ1, θ2, θ3) ∈ [−π, π) ×

[−π2, π

2)× [−π

4, π

4].

The operation of conjugation in the set of complex numbers, C = a+ jb|a, b ∈ R

where j2 = −1, is a so-called algebra involution that fulfills the two following prop-

erties. Let z, w ∈ C → (z∗)∗ = z and (wz)∗ = w∗z∗. In H, there are three nontrivial

15

algebra involutions:

α(q) = a+ j1b− j2c− j3d, (2.10)

β(q) = a− j1b+ j2c− j3d, (2.11)

γ(q) = a− j1b− j2c+ j3d. (2.12)

Using these involutions we can extend the definition of Hermitian symmetry. A

function f : R2 → H is called quaternionic Hermitian if, for each (x, y) ∈ R2,

f(−x, y) = β(f(x, y)) and f(x,−y) = α(f(x, y)). (2.13)

This leads directly to an alternative definition of 2-D analytic signal.

Definition 2 [30] Let f be a real 2-D signal. The 2-D quaternion analytic signal is

defined as

fqA(x) = f(x) + j1fHi1(x) + j2fHi2(x) + j3fHi(x), (2.14)

where the functions fHi1 , fHi2 and fHi are defined as in (2.5)–(2.7).

This 2-D quaternion analytic signal forms the core of the quaternion Fourier

transform (QFT) [4]. The QFT of a real 2-D signal f is given by

F q(u) =

∫

R2

e−j12πuxf(x)e−j22πvydx, (2.15)

where u = (u, v) indexes the QFT domain, x = (x, y) indexes the space domain, and

the quaternion exponential

e−j12πuxe−j22πvy (2.16)

is the QFT basis function (and itself a 2-D quaternion analytic signal). We employ

the notation u = (u, v) for the QFT domain, as opposed to (kx, ky) for the usual

complex Fourier transform domain.

16

2.2.2 QWT Construction

Our new 2-D dual-tree QWT rests on the quaternion definition of 2-D analytic signal.

By organizing the four quadrature components of a 2-D wavelet (the real wavelet and

its 2-D HTs) as a quaternion, we obtain a 2-D analytic wavelet and its associated

quaternion wavelet transform (QWT). For example, for the diagonal subband, with

(f, fHi1, fHi2 , fHi) = (ψh(x)ψh(y), ψg(x)ψh(y), ψh(x)ψg(y), ψg(x)ψg(y)), we obtain the

quaternion wavelet

ψq(x, y) = ψh(x)ψh(y)− j1ψg(x)ψh(y)− j2ψh(x)ψg(y) + j3ψg(x)ψg(y). (2.17)

Figure 1.3(a) illustrates the four components of a quaternion wavelet and its

quaternion magnitude for the diagonal subband. The partial and total HT com-

ponents resemble ψh(x)ψh(y) but are phase-shifted by 90◦ in the horizontal, vertical,

and both directions, respectively. The magnitude of each quaternion wavelet (square

root of the sum-of-squares of all four components) is a bell-shaped function. We can

also interpret the four components of ψq(x, y) in the Fourier domain as multiplying

the quadrants of the Fourier transform of ψh(x)ψh(y) by ±j and ±1, as shown in

Figure 2.4.

The construction and properties are similar for the other two subband quater-

nion wavelets based on φh(x)ψh(y) and ψh(x)φh(y) (see the horizontal and vertical

subbands in Figures 1.3(b) and (c), respectively). In summary, in contrast with the

six complex pairs of CWT wavelets (12 functions in total), the QWT sports three

quaternion sets of four QWT wavelets (12 functions in total).

Finally, note that the quaternion wavelet transform is approximately a windowed

17

+1

+1 +1

+1

(a) Ψh(kx)Ψh(ky)

+j −j

+j −j

(b) Ψg(kx)Ψh(ky)

−j −j

+j+j

(c) Ψh(kx)Ψg(ky)

+1 −1

+1−1

(d) Ψg(kx)Ψg(ky)

Figure 2.4 : Complex Fourier domain relationships among the four quadrature componentsof a quaternion wavelet ψq(x, y) in the diagonal subband.

quaternion Fourier transform (QFT) [4]. In contrast to the QFT in (2.15), the basis

functions for the QWT are the scales and shifts of the quaternion wavelet (ψh(x) −

j1ψg(x))(ψh(y)− j2ψg(y)) plus the wavelets for the other two subbands.

18

Chapter 3

QWT Properties

Since the QWT is based on combining 1-D CWT functions, it preserves many of

the attractive properties of the CWT. Furthermore, the quaternion organization and

manipulation provide new features not present in either the 2-D DWT or CWT. In

this section, we discuss some of the key properties of the QWT with special emphasis

on the QWT phase.

3.1 Tight Frame

The QWT contains four orthonormal basis sets and thus forms a 4× redundant tight

frame. The components of the QWT wavelets can be organized in matrix form as

G =

ψh(x)ψh(y) ψh(x)φh(y) φh(x)ψh(y)

ψg(x)ψh(y) ψg(x)φh(y) φg(x)ψh(y)

ψh(x)ψg(y) ψh(x)φg(y) φh(x)ψg(y)

ψg(x)ψg(y) ψg(x)φg(y) φg(x)ψg(y)

. (3.1)

More precisely, the frame contains shifted and scaled versions of the functions in G

plus the scaling function φh(x)φh(y) and its 2-D HTs. Each column of the matrix G

contains the four components of the quaternion wavelet corresponding to a subband of

the QWT. For example, the first column contains the quaternion wavelet components

in Figure 1.3(a), that is, the tensor product wavelet ψh(x)ψh(y) and its 2-D partial and

19

total HTs from Section 2.2. Each row of G contains the wavelet functions necessary

to form one orthonormal basis set; that is the sum-of-squares of the inner product

between an arbitrary function f ∈ L2(R) and each function in one row of G across

all shifts and scales equals ‖f‖2. Since G has four rows, a similar summation for

all functions in G equals 4‖f‖2, thus satisfying the tight-frame property with 4×

redundancy. An important consequence is that the QWT is stably invertible. The

wavelet coefficients corresponding to the projections onto the functions in G can be

computed using a 2-D dual-tree filter bank with linear computational complexity.

3.2 Relationship to the 2-D CWT

There exists a unitary transformation linking the QWT coefficients and the 2-D CWT

coefficients. The components of the CWT can be written in matrix form as

C =1√2

ψh(x)ψh(y) + ψg(x)ψg(y) ψh(x)φh(y) + ψg(x)φg(y) φh(x)ψh(y) + φg(x)ψg(y)

ψg(x)ψh(y) + ψh(x)ψg(y) ψg(x)φh(y) + ψh(x)φg(y) φg(x)ψh(y) + φh(x)ψg(y)

ψg(x)ψh(y)− ψh(x)ψg(y) ψg(x)φh(y)− ψh(x)φg(y) φg(x)ψh(y)− φh(x)ψg(y)

ψh(x)ψh(y)− ψg(x)ψg(y) ψh(x)φh(y)− ψg(x)φg(y) φh(x)ψh(y)− φg(x)ψg(y)

.

(3.2)

The columns of C contain the complex wavelets oriented at ±45◦, ±15◦ and ±75◦,

respectively. We obtain the CWT wavelets by multiplying the matrix G in (3.1) by

20

the unitary matrix

U =1√2

1 0 0 1

0 1 1 0

0 1 −1 0

1 0 0 −1

. (3.3)

Since G = UC, the CWT also satisfies the tight-frame property with the same 4×

redundancy factor. As we will see in the next section, both the CWT phase and the

QWT phases encode 2-D image feature shifts; however there exists no straightforward

relationship between the phase angles of the QWT and CWT coefficients.

3.3 QWT of Real Signals

For real signals and the DWT, we have the Plancharel theorem that allows us to

interpret wavelet analysis as local Fourier analysis with a multiscale frequency tiling

[38]. We seek a similar interpretation for the QWT as local QFT analysis in the first

quadrant of the QFT domain. This interpretation is not immediate, since quaternion

multiplication in the QFT domain is non-commutative, and the QFT convolution

theorem is not as simple as in the complex case. We first prove the QFT Plancharel

theorem and then provide an inner product formula in the QFT domain.

QFT Plancharel Theorem. Let f(x) and w(x) be two real 2-D signals, and

let F q(u) and W q(u) be their respective QFTs. Then the inner product of f(x) and

w(x) in the space domain can be stated equivalently in the QFT domain as

∫

R2

f(x)w(x)dx =

∫

R2

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))du, (3.4)

where γ(·) and α(·) are the algebra involutions defined in (2.10)–(2.12).

21

The functions F q·e and F q

·o are respectively the even and odd components of F q

with respect to the spatial coordinate y, as defined in the QFT convolution theorem

[4]

F q·e(u) =

∫

R2

e−j12πux f(x) cos(−2πvy)dx, (3.5)

F q·o(u) =

∫

R2

e−j12πux f(x) j2 sin(−2πvy)dx. (3.6)

We call the right side of (3.4) the QFT inner product between F q(u) and W q(u).

Details of the proof are included in Appendix A.

The QFT Plancharel Theorem holds for two real functions. To better interpret

the QWT, we can compute a formula for the inner product between a real function

and a quaternion analytic function.

Theorem 1 Let f(x) be a real 2-D signal and let wqA(x) be a 2-D quaternion analytic

function whose real component is w(x). Let F q(u) and W q(u) be the QFTs of f(x)

and w(x), respectively. Then the inner product of f(x) and wqA(x) in the space domain

can be expressed as the QFT inner product in a single quadrant of the (u, v)-plane

∫

R2

f(x)wqA(x)dx =

∫

S1

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))du, (3.7)

where S1 = {(u, v) : u ≥ 0, v ≥ 0}.

Note the integration limit of the QFT inner product: S1 in (3.7), as opposed to

R2 in (3.4). Details of the proof for Theorem 1 are included in Appendix B.1.

22

3.4 QWT Phase Properties

Recall from Section 2.2.1 that each QWT coefficient can be expressed in terms of its

magnitude and phase as q = |q| ej1θ1ej3θ3ej2θ2 . We seek a shift theorem for the QWT

phase that is analogous to that for the CWT. Since the QWT performs a local QFT

analysis, the shift theorem for the QFT [4] holds approximately for the QWT. When

we shift an image from f(x) to f(x − d), the QFT phase undergoes the following

transformation

(θ1(u), θ2(u), θ3(u))→ (θ1(u)− 2πud1, θ2(u)− 2πvd2, θ3(u)) (3.8)

where d = (d1, d2) denotes the shift in the horizontal and vertical directions respec-

tively.

To transfer the shift theorem from the QFT to the QWT, we exploit the fact

that the QWT is approximately a windowed QFT. Thus, each quaternion wavelet

coefficient can be interpreted as the inner product between the real image and a

windowed version of the quaternion exponential (2.16). The scale of analysis controls

the center frequency (u′, v′) of the windowed exponential (that is, the quaternion

wavelet) in the QFT plane. The magnitude and phase of the resulting coefficient are

determined by two factors: the spectral content of the image and the center frequency

(u′, v′) of the wavelet. These two factors determine the frequency parameters (u, v)

we should use in the shift theorem for the QFT (3.8). We call this (u, v) (which

is different from (u′, v′) in general due to the scale of analysis and the quaternion

wavelet) the effective center frequency for the corresponding wavelet coefficient. Note

that the effective center frequency (u, v) should always lie in the first quadrant of the

23

Figure 3.1 : Effect of varying θ3 on the structure of the corresponding weighted quaternionwavelet from the diagonal subband (left to right): θ3 = −π

4 ,−π8 , 0,

π8 ,

π4 respectively. The

corresponding wavelet changes from textured (θ3 = 0) to oriented (θ3 = ±π4 ).

QFT domain (that is, u, v ≥ 0).

Thus, before we can apply the shift theorem for QWT image analysis, we must

estimate the effective center frequency for each QWT coefficient. Fortunately, for

images having a smooth spectrum over the support of the quaternion wavelet in the

QFT domain, the wavelet’s center frequency (u′, v′) is a close approximation to the

effective center frequency (u, v). Thus, using the relationship between the QWT phase

and image shifts, we can estimate the shift (d1, d2) of one image relative to a second

image from the phase change (∆θ1,∆θ2). Conversely, we can estimate the phase shift

once we know the image shift.

We can interpret the third QWT phase angle θ3 as the relative amplitude of image

energy along two orthogonal directions as in [4]. By adjusting only θ3 of the QWT

ψ(x)ψ(y) subband coefficients, we observe a gradual change in appearance of the

corresponding wavelets from oriented to texture-like and back (see Figure 3.1). This

property could prove useful for the analysis of images with rich textures [4]. As we

describe below in Section 4.1, this third phase also relates to the orientation of a

single edge.

24

Chapter 4

QWT Applications

The QWT, with its shift-invariance and quaternion phase representation, opens up

new image processing capabilities unattainable with either the 2-D DWT or CWT. As

in the Fourier case, the shift theorem establishes the relationship between the QWT

phases and relative location information in images. In this section, we demonstrate

two applications of the QWT, including edge structure estimation in Section 4.1, and

registration in Section 4.2.

4.1 Edge Geometry Estimation

Edge structures (see Figure 4.1(a)) are the fundamental building blocks of many real-

world images. A typical image can be considered as a piecewise smooth signal in

2-D containing blocks of smooth regions and edge singularities. Wavelet transforms

are very amenable to processing these singularities. Previous efforts to incorporate

the behavior of wavelet coefficients around edge structures into wavelet-based image

models have achieved considerable success in applications such as edge extraction and

image compression [39, 27]. Therefore, understanding the QWT phase response of

edges is key to applying the new tool to real-world images.

First consider an image block containing a single step edge, as in Figure 4.1(a).

Its QWT is the QFT inner product between the quaternion wavelets and the QFT

25

d

β

(a) single edge model (wedgelet)

β

β

( u , v )

u

v positive quadrant

leakage leakage quadrant

(b) edge QFT spectrum

Figure 4.1 : (a) Parameterization of a single edge in a dyadic image block (wedgelet [3]). (b)QFT spectrum of the edge; shaded squares represent the quaternion wavelets in the vertical,horizontal, and diagonal subbands. The energy of the edge is concentrated along the twodark lines crossing at the origin and is captured by the horizontal subband with spectralcenter at quaternion frequency (u, v). The region bounded by the dashed line demonstratesthe spectral support of the QWT basis “leaking” into the neighboring quadrant.

spectrum of the edge image. For an edge oriented at angle β, any shift (d1, d2) in the

(x, y) directions satisfying the constraint

d1 cosβ + d2 sin β = d (4.1)

corresponds to the same shift as moving the edge by d perpendicularly. The QFT

of this step edge f(x) with orientation and offset (β, d) (see the parameterization in

Figure 4.1(a)) is given by the following expression:

F q(u) = e−j12πud1

(

δ(u cosβ − v sin β)

2(u sinβ + v cos β)(−j1 − j2)

+δ(u cosβ + v sin β)

2(−u sin β + v cosβ)(j1 − j2)

)

e−j22πvd2 . (4.2)

The details of this calculation are included in Appendix B.2.

Thus, the QFT spectrum of the edge lies on two lines through the origin having

orientations 90◦ − β and β − 90◦ in the QFT domain, as shown in Figure 4.1(b).

26

0 45 900

0.50

600

β

dM

agni

tude

(a) Magnitudes

0 45 900

0.50

600

Mag

nitu

de

βd

(b) Magnitudes

0 45 900

0.50

0.2

0.4

0.6

0.8

1

Est

imat

ion

erro

r

β

d

(c) Estimation errors

0 20 40 60 800

0.50

0.2

0.4

0.6

0.8

1

Est

imat

ion

erro

r

βd

(d) Estimation errors

Figure 4.2 : Magnitude of a QWT coefficient as a function of edge orientation β andoffset d for the (a) horizontal φ(x)ψ(y) subband and (b) diagonal ψ(x)ψ(y) subband.(c), (d) Absolute error for estimating d by the phase angles (θ1,θ2) from the sametwo subbands, relative to the normalized edge length of the dyadic block (= 1 unit).Estimates are accurate in the regions where the coefficient magnitudes are large.

Based on our theoretical analysis in Section 3.3, the effective center frequency of a

quaternion wavelet must lie in the first quadrant and, therefore, can be expressed as

(u, v) = c(| cosβ|, | sinβ|), where c is a positive constant that depends on β and the

location of the spectral tile corresponding to each QWT basis. Using the relations

(u, v) = c(| cosβ|, | sinβ|) in (4.1) and noting that 2πud1 = ∆θ1 and 2πvd2 = ∆θ2,

we obtain the expression

d =∆θ1 ±∆θ2

2πc, (4.3)

where we choose ∆θ1 + ∆θ2 when tan β > 0, and ∆θ1 −∆θ2 when tanβ < 0.

To verify this relationship experimentally, we apply the QWT to the edge image

27

in Figure 4.1(a) and analyze the QWT magnitudes and phases corresponding to a

32 × 32 sub-block. Figures. 4.2(a) and (b) verify that the coefficient magnitudes are

almost invariant to signal shift d. Note that the magnitude is maximum when the

edge orientation matches the orientation of the basis function (0◦ for the φ(x)ψ(y)

subband, 45◦ for the ψ(x)ψ(y) subband, and 90◦ for the ψ(x)φ(y) subband). We

experimentally study the linear relationships between ∆θ1 ± ∆θ2 and d for various

β in all subbands to obtain the slope estimates 12πc

in (4.3). We obtain an estimate

of c ≈ 0.7 for the ψ(x)φ(y) subband using the vertical edge (β = 0) which gives

the largest QWT magnitude for this subband; similarly for the ψ(x)φ(y) subband.

Using the 45◦ edge, we obtain an estimate of c ≈ 1 for the ψ(x)ψ(y) subband. This

simplification of using the same c in each subband to estimate d is valid because c is

sensitive to the location of the spectral tile of the corresponding QWT basis but not

to β. Now, we can use equation (4.3) to estimate the edge offset d from the QWT

phase using d = 0 as the reference image. Figures 4.2(c) and (d) show the estimation

accuracy for two subbands across various β and d values. In small-magnitude regions,

the phase angles are very sensitive to noise, and thus estimation errors are large. Using

only the phase change of the QWT subband with large magnitude, we can estimate

d with a maximum error of approximately 0.02 relative to the normalized unit edge

length of the dyadic block under analysis.

We also analyze the behavior of the third phase angle θ3 for the same edge block

in Figure 4.1(a). Our theoretical analysis in Appendix B.3 yields the property that

the quaternion wavelet coefficient for a step edge with arbitrary (β, d) always has

θ3 = ±π4, if we assume that all quaternion wavelets have a perfect single-quadrant

28

QFT spectrum (analytic signal). This special case corresponds to the “singular case”

in the quaternion phase calculation, under which we can uniquely define only the sum

or the difference of θ1 and θ2 [4]. This is consistent with (4.3), which always gives a

unique sum (or difference) of the first two phase angles for a given edge offset d.

However, practical quaternion wavelets will not have a perfect single-quadrant

QFT spectral support, but rather will leak somewhat into neighboring quadrants.

Then, we obtain an alternative formula for the phase angle

θ3 = −1

2arcsin

(

1− ε1 + ε

)

, (4.4)

where ε is a rough measure of the ratio of wavelet or signal energy in the main

quadrant to energy in the leakage quadrant (see Appendix C for the proof). The sum

(or difference) between θ1 and θ2 is the same as in the case without leakage. In other

words, the linear relationship between θ1±∆θ2 and edge offset d in (4.3) is unaffected

by leakage. These theoretical derivations are consistent with our experimental results.

Based on our experimental analysis, we propose a hybrid algorithm to estimate the

edge geometry (β, d) based on the QWT phase (θ1, θ2, θ3) and the magnitude ratios

between the three subbands. Our algorithm is reminiscent of the edge estimation

scheme in [27].

We first obtain the three QWT coefficients corresponding to a dyadic image block

of interest. To estimate the orientation β of the edge block, we choose the QWT sub-

band with the largest magnitude. This subband tells us the approximate orientation

of the edge (±45◦ for diagonal, ±15◦ for horizontal and ±75◦ for vertical), which the

sign of θ3 tells us whether the direction is positive or negative. We can then use either

29

the magnitude ratios among the three subbands or θ3 to make a more accurate esti-

mation of the orientation. We experimentally analyzed the QWT magnitude ratios

and θ3 corresponding to changing edge orientations β by multiples of 5◦ and, using

standard curve-fitting techniques, developed a simple relationship between these pa-

rameters and the edge orientation. The resulting orientation estimation algorithm

achieves a maximum error of only β ± 3◦ in practice.

To estimate the offset d of the edge block, we only need (θ1, θ2) in the horizontal

and vertical subbands. We choose (θ1, θ2) from the subband with the larger magnitude

and apply the same offset estimation scheme as in [40]. This offset estimation has a

maximum error of approximately ±0.02 relative to the normalized unit edge length

of the dyadic block under analysis. This demonstrates the sub-pixel accuracy of this

estimation algorithm. Such accuracy in our orientation and offset estimation schemes

is obtained with the straight edge model in Figure 4.1(a), but we will see that our

algorithm also works well for real-world images.

According to (4.3), within a 2π-range of ∆θ1±∆θ2, the range of d is limited to an

interval of length 1c≈ 1.43, which ensures that the edge stays within the image block

under analysis. Therefore, in our offset estimation, we need only consider one 2π-

range of ∆θ1 ±∆θ2 and do not need to perform any “phase-unwrapping”. However,

we need to obtain an offset estimate for each polarity of the edge with orientation β

estimated above, say d+ and d−. Then, we use the inner product between the image

block and two wedgelets [3] with the estimated edge parameters (β, d+) and (β, d−)

to determine the correct polarity. We apply the algorithm described above to the

popular “cameraman” image in Figure 4.3.

30

t

Figure 4.3 : Local edge geometry estimation using the QWT. (a) Several edgy regions fromthe “cameraman” image are shown on the left; (b)–(e) on the right are edge estimates fromthe corresponding QWT coefficients. The upper row shows the original image region, thelower row shows a wedgelet (see Figure 4.1(a)) having the edge parameter estimates (β, d).(No attempt is made to capture the texture within the block.)

Our results demonstrate the close relationship between edge geometry and QWT

magnitude and phases, in particular, the encoding of edge location in the QWT phases

(θ1,θ2) and the encoding of edge orientation in the QWT magnitude and third phase

θ3.

4.2 QWT-based Image Disparity Estimation

The relative location information encoded in the phase angles of the QWT is particu-

larly useful for image registration [11]. The image registration problem arises in many

applications such as in video processing to estimate motion between successive frames,

in time-lapse seismic imaging to study changes in a reservoir over time, in medical

imaging to monitor a patient’s body, in super-resolution, etc. In these applications,

we often need to estimate the local shifts of a target image (or several) compared

to a reference image. We can then use this disparity information to build a warp-

ing function to align the images. In this section, we describe an efficient multiscale

31

scheme to estimate local shifts in an image based on the QWT phase. Compared to

other recent phase-based image registration approaches [4, 41, 42, 5], we demonstrate

the simplicity of our approach and the effectiveness of the QWT representation for

encoding location information.

Recall that the QWT phase property states that a shift (d1, d2) in an image changes

the QWT phase from (θ1, θ2, θ3) to (θ1− 2πud1, θ2− 2πvd2, θ3). Thus, for each QWT

coefficient, if we know (u, v), the effective center frequency of the local image region

analyzed by the corresponding QWT basis functions, we can estimate the local image

shifts (d1, d2) from the phase differences.

Suppose we are given a reference image A(x, y) and a target image B(x, y) for

which we wish to estimate the local shifts, First, by manually translating the reference

image A(x, y) by known small amounts both horizontally and vertically, we obtain

two images A(x, y) and A(x − d1, y − d2). After computing QWTs of A(x, y) and

A(x − d1, y − d2), we can use the phase differences (∆θ1,∆θ2) between the QWT

coefficients to obtain estimates for the effective spectral center (u, v) for each dyadic

block across all scales as u = ∆θ1

2πd1and v = ∆θ2

2πd2. The range of QWT phase angles

limits our estimates (u, v) to [− 12D, 1

2D) and [− 1

4D, 1

4D) for horizontal and vertical

shifts, respectively, where D is the length of one side of the dyadic block corresponding

to each coefficient.

Once we know the effective center frequency (u, v) for each QWT coefficient, we

can estimate the local image shifts by measuring the difference between the QWT

phase corresponding to the same local blocks in the images A(x, y) and B(x, y). The

main challenge here is the phase wrap-around due to the limited range of phase angles;

32

that is, each observed phase difference can be mapped to more than one disparity

estimate. Specifically, for QWT phase differences (∆θ1,∆θ2) between the reference

and target images, we can express the possible image shifts of each dyadic block as

d1 =∆θ1 + π(2n+ k)

2πu, d2 =

∆θ2 +mπ

2πv, (4.5)

where n,m ∈ Z and k ∈ {0, 1}. Depending on m, k is chosen such that it equals 0

when m is even and equals 1 when m is odd. This special wrap-around effect in (4.5)

is due to the limited range in θ1 and θ2 (to [−π, π) and [−π2, π

2) respectively) and the

periodicity of intrinsically 2-D signals [4].

In our multiscale disparity estimation technique, we use coarse scale shift estimates

to unwrap the phase in the finer scales. If we assume the true image shift is small

compared to the size of dyadic squares at the coarsest scale L, then we can set

m = n = k = 0 in (4.5) at this scale (no phase wrap-around) and obtain correct

estimates for d1 and d2. Effectively, this assumption of no phase wrap-around at the

coarsest scale limits the maximum image shift that we can estimate correctly. Once

we have shift estimates at scale L, for each dyadic block at scale ` = L − 1, we

estimate the shifts in the following way:

1. bilinearly interpolate estimates from previous scale(s) to obtain predicted esti-

mates (dp1, d

p2),

2. substitute the phase differences in (4.5) and determine (n, k,m) so that (d1, d2)

are closest to (dp1, d

p2),

3. remove unreliable (d1, d2),

33

4. repeat Steps 1–3 for the finer scales ` = L− 2, L− 3, . . .

In Step 3, we use a similar reliability measure as in the confidence mask [4] to

threshold unreliable phase and offset estimates. We also threshold based on the

magnitude of the QWT coefficients. We iterate the above process until a fine enough

scale (` = 2), since estimates become unreliable at this scale and below. The QWT

coefficients for the small dyadic blocks have small magnitudes and their phase angles

are very sensitive to noise.

We improve this basic iterative algorithm by combining estimates across subbands

and scales. First, with proper interpolation, we can average over estimates from all

scales containing the same image block. Second, we can average estimates from the

three QWT subbands for the same block to yield more accurate estimates, but we

need to discard some unreliable subband estimates (for example, horizontal disparity

d1 in the horizontal subband and d2 in the vertical subband). We incorporate these

subband/scale averaging steps into each iteration of Steps 1–4.∗

Figure 4.4 shows the result of our QWT phase-based disparity estimation scheme

for two selected frames in a rotating “Rubik’s cube” video sequence [4]. The arrows

indicate both the directions and magnitudes of the local shifts, with the magnitudes

stretched for better visibility. We can clearly see the rotation of the Rubik’s cube on

the circular stage. Our experiments have indicated that the algorithm that averages

over both subbands and scales gives the most robust shift estimates. This finding is

consistent with our hypothesis, since the averaging procedure effectively utilizes all

∗Matlab software is available at http://www.dsp.rice.edu/software/qwt.shtml.

34

(a) reference image (b) target image (c) disparity estimates

Figure 4.4 : Multiscale QWT phase-based disparity estimation results. (a), (b) Referenceand target images from the “Rubik’s cube” image sequence [4]. (c) Disparity estimatesbetween two images in the sequence, shown as arrows overlaid on top of the reference image(zoomed in for better visualization).

shift information extracted by the QWT.

A performance measure for disparity/motion estimation algorithms, commonly

used in video compression, is the Peak Signal-to-Noise Ratio (PSNR) between the

motion-compensated image C(x) and the target image B(x), given by

10 log10

(

(255)2N∑

x(B(x)− C(x))2)

)

, (4.6)

where N is the number of image pixels. We first compute the motion vectors as in

Figure 4.4(c). Then, we obtain the motion-compensated image C(x) by shifting and

interpolating every image block in the reference image A(x) and calculate the PSNR

between B(x) and C(x).

For comparison, we choose a block-matching technique known as exhaustive search

(ES), which is very computationally demanding but has the best performance among

all general block-matching techniques. We also choose a phase correlation technique

known as Gradient Correlation (GC), which has been shown to have better PSNR

35

performance than other recent phase correlation methods [5]. Figure 4.5 shows the

comparison results for three image sequences, the “Rubik” and “Taxi” sequences

commonly used in the optical flow literature and the MPEG test sequence “Mobcal”

used in [5]. Figure 4.5(a) demonstrates the superior performance of our QWT phase-

based algorithm over other algorithms for the “Rubik” sequence, which has piecewise-

smooth image frames and a smooth underlying disparity flow. For the two other

sequences, which contain some discontinuities in their underlying flows, the QWT

phase-based algorithm also has comparable performance (see table in Figure 4.5(b)).

Since the multiscale averaging step in our algorithm tends to smooth out the estimated

flow, our algorithm is not expected to perform as well for discontinuous motions fields

of rigid objects moving past each other.

Additional advantages of our QWT-based algorithm include linear computational

efficiency and sub-pixel estimation accuracy. Thus, for an N -pixel image, our O(N)

algorithm is more efficient than the O(N logN) FFT-based GC and is significantly

faster than ES, which can take up to O(N2) with the search parameter on the order of

N . General block-matching techniques such as ES can also only decipher disparities

in an integer number of pixels. However, as with other phase-based algorithms, our

QWT-based algorithm can achieve estimation accuracy up to a fractional number of

pixels. Our experiments compute motion vectors only for 8 × 8 blocks. Although it

is more common to use 32 × 32 blocks for video compression, we choose a smaller

block size because, for image registration, a high-density disparity map is needed to

register elaborate image features (for example, patterns on the Rubik’s cube and the

rotation stage; see Figure 4.4).

36

0 5 10 15 2035

36

37

38

39

40

41

42

Frame Number

PS

NR

QWTGradCorrES

(a) PSNR vs. frame number

Rubik Taxi Mobcal

QWT 39.4 36.2 25.5

GC 36.7 36.6 26.9

ES 37.2 37.0 26.3

(b) average PSNR performance (in dB)

Figure 4.5 : Comparison of multiscale QWT phase-based disparity estimation with twomotion estimation algorithms, Gradient Correlation (GC) [5] and exhaustive search (ES).Performance measure is the PSNR (in dB) between the motion-compensated image and thetarget image of three test image sequences (“Rubik”, “Taxi” and “Mobcal”). (a) Frame-by-frame PSNR performance comparison in the “Rubik” sequence. (b) Table of averagePSNR performance (over all frames) for each test sequence. Multiscale QWT phase-basedmethod demonstrates the best performance among the three test algorithms for the “Rubik”sequence, and shows comparable performance to the other algorithms for the “Taxi” and“Mobcal” sequences.

Besides phase correlation, there exist other phase-based algorithms for disparity

estimation and image registration [4, 41, 42]. These approaches use phase as a feature

map Φ(x) where the phase function Φ maps 2-D x, y-coordinates to phase angles.

They assume the phase function to stay constant upon a shift from the reference

image to the target image; that is, Φ1(x) = Φ2(x+d) where Φ1 is the phase function

for the reference image and Φ2 for the target image. Then, the disparity estimation

problem is simplified to calculating the optical flow for these phase functions. In

contrast, our algorithm is entirely based on the dual-tree QWT and its shift theorem,

without using any optical flow assumptions [43].

Our approach is similar to the QFT disparity estimation algorithm of [4]in its

37

use of quaternion phase angles. However, the QFT approach requires the design

of a special filter to compute the phase derivative function in advance, while our

approach only needs to first estimate the local frequencies (u, v). Our approach is

more computationally efficient (order O(N) as opposed to O(N logN) for an N -pixel

image.). Provided a continuous underlying disparity flow, our multiscale scheme

yields a denser and more accurate disparity map, even for smooth regions within an

image.

Kingsbury et al. have also developed a multiscale displacement estimation algo-

rithm using the 2-D CWT [41, 42]. Their approach combines information from all six

CWT subbands in an optimization framework based on the optical flow assumptions.

In addition to disparity estimation, it simultaneously registers the target image to the

reference image. In comparison, both the QWT and CWT methods are multiscale

and wavelet-based and are thus, in general, best for smooth underlying disparity flow.

However, our algorithm is much simpler and easy to use because it does not involve

the tuning of many input parameters for the iterative optimization procedures as

in the CWT algorithm. Although our method only estimates local disparities with-

out warping the image, we can apply another standard warping procedure to easily

register the two images from the estimated disparities.

Thanks to the ability of the QWT to encode location information in absolute

horizontal/vertical coordinates, we can easily combine the QWT subband estimates to

yield more accurate flow estimation results. Combining subband location information

in the 2-D CWT is more complicated, since each subband encodes the disparities by

complex phase angles in a reference frame different (rotated) from other subbands.

38

Based on our experimental results and comparison of the design of our flow estimation

algorithm with previous approaches, the QWT demonstrates its ability to efficiently

represent, encode and process location information in images.

39

Chapter 5

Conclusions

We have introduced a new 2-D multiscale wavelet representation, the QWT, that is

particularly efficient for coherent processing of relative location information in images.

This tight-frame transform generalizes complex wavelets to higher dimensions and

inspires new processing and analysis methods for wavelet phase.

Our development of the QWT is based on an alternative definition of the 2-D

HT and 2-D analytic signal and on quaternion algebra. The resulting quaternion

wavelets have three phase angles; two of them encode phase shifts in an absolute hor-

izontal/vertical coordinate system while the third encodes textual information. The

QWT’s approximate shift theorem enables efficient and easy-to-use analysis of the

phase behavior around edge regions. We have developed a novel multiscale phase-

based disparity estimation scheme. Through efficient combination of disparity esti-

mates across scale and wavelet subbands, our algorithm clearly demonstrates the ad-

vantages of coherent processing in this new QWT domain. Inherited from its complex

counterpart, the QWT also features near shift-invariance and linear computational

complexity through its dual-tree implementation.

Beyond 2-D, the generalization of the Hilbert transform to n-D signals using hy-

percomplex numbers can be used to develop higher dimensional wavelet transforms

suitable for signals containing low-dimensional manifold structures [44]. The QWT

40

developed here could play an interesting role in the analysis of (n − 2)-D manifold

singularities in n-D space. This efficient hypercomplex wavelet representation could

bring us new ways to solve high-dimensional signal compression and processing prob-

lems.

41

Appendix A

QFT Plancharel Theorem

Proof: Given the 2-D real signal w(x) and its QFT W q(u), the following QFT rela-

tionships hold:

w(x, y) ←→ W q(u, v) = γ(W q(−u,−v)), (A.1)

w(−x,−y) ←→ γ(W q(u, v)), (A.2)

w(x′ − x, y′ − y) ←→ e−j12πux′

γ(W q(u, v))e−j22πvy′

, (A.3)

The quaternion Hermitian symmetry of the QFT of w(x) gives the relationship in

(A.1) (see Theorem 2.2 and Theorem 2.11 in [4]), which also implies (A.2). By first

expressing w(x′ − x, y′ − y) as w(−(x− x′),−(y − y′)), (A.3) is obtained from (A.2)

by the QFT shift theorem (see Theorem 2.8 in [4]). Starting from the expression in

the space domain, we have∫

R2

f(x′)w(x′)d2x′ =

∫

R2

f(x′)

(∫

R2

ej12πu′x′

γ(W q(−u′))ej22πv′y′

d2u′

)

d2x′

=

∫

R2

f(x′)

(∫

R2

e−j12πux′

γ(W q(u))e−j22πvy′

d2u

)

d2x′

=

∫

R2

f(x′)

∫

R2

(∫

R2

e−j12πuxw(x′ − x)e−j22πvyd2x

)

d2ud2x′

=

∫

R2

∫

R2

e−j12πux

(∫

R2

f(x′)w(x′ − x)d2x′

)

e−j22πvyd2xd2u

=

∫

R2

F q·e(u)γ(W q(u)) + F q

·o(u)γ(β(W q(u)))d2u

=

∫

R2

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))d2u. (A.4)

Step 5 uses the QFT convolution theorem (Theorem 2.6) in [4].

42

Appendix B

Continuous QWT of 2-D edge signal

B.1 Proof of Theorem 1

According to Definition 2, the 2-D quaternion analytic signal with real component

w(x) can be expressed in the space domain as:

wqA(x) = w(x) + j1wHi1(x) + j2wHi2(x) + j3wHi(x). (B.1)

For any real function g(x), we can express its CFT and QFT respectively as

Gq(u) = A(u) + j1B(u) + j2C(u) + j3D(u) and (B.2)

G(u) = (A(u)−D(u)) + j(B(u) + C(u)), (B.3)

where j =√−1. We can see this equivalence from the CFT and the QFT formula

which basically involve the same sine and cosine basis elements. Also, notice that

this proof uses u = (u, v) to denote the general frequency domain (both the CFT and

QFT domain) because both Fourier transforms are discussed simultaneously.

Let F (u) and W (u) be the CFT of f(x) and w(x) respectively. We can verify that

the integrand from the CFT inner product G(u) = F (u)W ∗(u) is equivalent to the

integrand from the QFT inner product Gq(u) = F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u)).

These two integrands from the CFT and the QFT inner products are equivalent

because they both come from the real 2-D signal g(x) = (f ∗w−)(x) described in the

proof of the QFT Plancharel Theorem.

43

Denote S2 = {(u, v) : u ≤ 0, v ≥ 0} as the second quadrant, and S3 and S4

similarly in an anti-clockwise fashion. Using the CFT Plancharel theorem, the inner

products between f(x) and each of the four components in wqA(x) can be expressed in

terms of the CFT inner product between F (u) and W (u), i.e., G(u) = F (u)W ∗(u)),

multiplied by ±j or ±1 in the appropriate quadrant, as in (B.4)-(B.7):

p =

∫

S1

G(u) +

∫

S2

G(u) +

∫

S3

G(u) +

∫

S4

G(u), (B.4)

q =

∫

S1

−jG(u) +

∫

S2

jG(u) +

∫

S3

jG(u) +

∫

S4

−jG(u), (B.5)

r =

∫

S1

−jG(u) +

∫

S2

−jG(u) +

∫

S3

jG(u) +

∫

S4

jG(u), (B.6)

s =

∫

S1

−G(u) +

∫

S2

G(u) +

∫

S3

−G(u) +

∫

S4

G(u). (B.7)

As a result, the inner product between f(x) and the quaternion analytic function

wqA(x) can be expressed, using quaternion algebra, as p+ j1q+ j2r+ j3s. Substituting

(B.3) into (B.4)-(B.7), then using conjugate symmetry of G(u), yields an expression

involving only quaternions

p + j1q + j2r + j3s

= 2

∫

S1

(A(u) + j1B(u) + j2C(u) + j3D(u))− (A(u)− j1B(u)− j2C(u)− j3D(u))j3

2

∫

S2

(A(u)− j1B(u) + j2C(u) + j3D(u)) + (A(u)− j1B(u)− j2C(u) + j3D(u))j3

= 2

∫

S1

(Gq(u)− α(Gq(u))j3) + 2

∫

S2

(β(Gq(u)) + γ(Gq(u))j3), (B.8)

where α(·), β(·) and γ(·) are the algebra involutions defined as in (2.10)–(2.12). By

the quaternion Hermite symmetry of Gq(u), further simplification obtains

p+ j1q + j2r + j3s = 4

∫

S1

Gq(u)d2u

= 4

∫

S1

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))d2u, (B.9)

44

which is an integral in the single quadrant S1. Equation (B.9) establishes the QFT

inner product of any quaternion basis function wqA(x) and a general real 2-D signal

f(x) as an integral in S1 alone.

B.2 Calculation of (4.2)

First, express the step edge as a 2-D separable function (a constant function along

the x-direction multiplied by a 1-D step function along the y-direction). The QFT

of such a function is −j2 δ(u)v

. Then, apply the QFT affine theorem (Theorem 2.12 in

[4]) with the transformation matrix involving rotation β and offset d.

B.3 Calculation of QWT Phase Angles for a Step Edge

This calculation combines the results from (3.7) and (4.2). Consider the special

case when the edge signal f(x) has zero offset (d = 0). From equation (4.2), its

QFT is F q(u) = (−j1 − j2)H(u) in S1 where H(u) is the component involving the

δ-sheet in (4.2). Let the QFT of the real component of a QWT basis, w(x), be

W q(u) = a(u) + j1b(u) + j2c(u) + j3d(u). From equation (3.7),

∫

S1

Gq(u)d2u =

∫

S1

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))d2u

=

∫

S1

H(u)(b(u) + c(u))d2u + j1

∫

S1

H(u)(a(u)− d(u)))d2u

+j2

∫

S1

H(u)(a(u)− d(u)))d2u− j3∫

S1

H(u)(b(u) + c(u))d2u

= A1 + j1B1 + j2B1 − j3A1, (B.10)

45

where A1 and B1 are the simplified expressions involving a(u), b(u), etc. After

normalizing (B.10) with its magnitude 2(A21 +B2

1), compute the third phase angle as

θ3 = −1

2arcsin

(

A21 +B2

1

2(A21 +B2

1)

)

= −π4. (B.11)

Depending on the orientation β of the 2-D edge signal f(x), its QFT F q(u) can be

(j1 − j2)H(u) in S1, which gives θ3 = π4.

Moreover, this special quaternion in (B.10) with θ3 = −π4

is in the singular case;

as described in [4], its other phase angles (θ1, θ2) are non-unique but the sum θ1 + θ2

is unique with the following expression

θ1 + θ2 =1

2arcsin

(

2A1B1

A21 +B2

1

)

, (B.12)

whose value largely depends on the QFT spectrum of the basis w(x) and on the edge

orientation and offset (β,d).

46

Appendix C

Theoretical Analysis of Leakage Effect

In practice, in order to have finite-length basis functions, the 2-D Hilbert transforms

of the real QWT basis component w(x) is only approximately satisfied. Therefore,

the non-ideal quaternion basis function wqA(x) does not have a perfect single-quadrant

spectrum in S1, but has a certain amount of “leakage” into other quadrants.

Consider the inner product of the edge signal f(x) and wqA(x) in both the main

quadrant (S1) and the leakage quadrant (S2).∗ The inner product in the main quad-

rant is given by equation (B.10) while the inner product in the leakage quadrant can

be written as

∫

S2

Gq(u)d2u =

∫

S2

F q·e(u)γ(W q(u)) + F q

·o(u)α(W q(u))d2u

=

∫

S2

H(u)(b(u)− c(u))d2u + j1

∫

S2

H(u)(a(u) + d(u)))d2u

−j2∫

S2

H(u)(a(u) + d(u)))d2u + j3

∫

S2

H(u)(b(u)− c(u))d2u

= A2(u) + j1B2(u)− j2B2(u) + j3A2(u). (C.1)

Again, A2 and B2 are the simplified expressions involving a(u), b(u), etc.

Therefore, combining equation (B.10) and (C.1) gives the inner product between

∗Leakage quadrant can be either S2 or S3 depending on the spectral support of the basis element

w(x).

47

the non-ideal basis and the edge signal

∫

S1

Gq(u)d2u +

∫

S2

Gq(u)d2u = (A1(u) + A2(u)) + j1(B1(u) +B2(u))

+j2(B1(u)−B2(u)) + j3(−A1(u) + A2(u)),

whose magnitude is 2(A21 +A2

2 +B21 +B2

2). Its third phase angle can be expressed as

θ3 = −1

2arcsin

(

2[(A21 +B2

1)− (A22 +B2

2)]

2(A21 + A2

2 +B21 +B2

2)

)

= −1

2arcsin

(

1− ε1 + ε

)

, (C.2)

where ε =A2

2+B2

2

A2

1+B2

1

is an expression for the ratio of basis or signal energy in the main

quadrant to energy in the leakage quadrant. The ratio ε is always less than unity

because the basis function has higher energy in the main quadrant than in the leakage

quadrant. When ε is close to unity, the phase angle is largely affected by leakage.

When ε is close to zero, leakage has little effect and θ3 ≈ −π4

as in the ideal case.

This relationship between ε and θ3 agrees with our experimental observation of varying

edge orientation β. When β ≈ 0◦, the QFT of the edge lies on the u-axis; thus the

signal energy ratio ε ≈ 1 (same contribution from both main and leakage quadrant)

and θ3 ≈ 0. When β ≈ 45◦, there is little overlap between the QFT of the edge with

the basis spectrum in the leakage quadrant; thus ε ≈ 0 and θ3 ≈ −π4.

Let us turn our attention to the other two phase angles (θ1,θ2). Since θ3 is not

necessarily −π4

for the QFT of the edge signal, i.e., the singular case no longer holds,

there exists unique expressions for both θ1 and θ2 as follows:

θ1 =1

2arctan

(

A1B2 + A2B1

A1A2 − B1B2

)

,

θ2 =1

2arctan

(

A2B1 − A1B2

A1A2 +B1B2

)

.

48

However, when combining the two expressions above by trigonometric identity

(sum of two arctangents), the sum of θ1 and θ2 is the same as in the ideal case (without

leakage) as in (B.12), which only involves terms A1 and B1 from the main quadrant.

This result has the important implication that, in spite of leakage, the QWT shift

theorem still holds, i.e., θ1 ± θ2 varies linearly with edge offset d. Equation (4.3),

relating the QWT phase of an edge θ1 ± θ2 to edge offset d, is still valid. Besides,

θ1 − θ2 =1

2arcsin

(−2A2B2

A22 +B2

2

)

(C.3)

only involves terms A2 and B2 from the leakage quadrant. Similar to its counterpart

in (B.12), θ1− θ2 largely depends on the edge orientation and offset (β,d) and on the

QFT spectral support of w(x), but only in the leakage quadrant. However, it is much

harder to estimate the behavior of θ1 − θ2 with varying offset d since w(x) only has

a small amount of energy in the leakage quadrant.

49

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