ter haar romeny, fev geometry-driven diffusion: nonlinear scale-space – adaptive scale-space

ter Haar Romeny, FEV

Geometry-driven diffusion: nonlinear scale-space – adaptive scale-space

Original scale 9


The advantage of selecting a larger scale is • the improved reduction of noise, • the appearance of more prominent structure,

but the price to pay for this is reduced localization accuracy.

Linear, isotropic diffusion cannot preserve the position of the differential invariant features over scale.

A solution is to make the diffusion, i.e. the amount of blurring, locally adaptive to the structure of the image.


1. Nonlinear partial differential equations (PDE's), i.e. nonlinear diffusion equations

which evolve the luminance function as some function of a flow;

2. Curve evolution of the isophotes (curves in 2D, surfaces in 3D) in the image;

3. Variational methods that minimize some energy functional on the image.

It takes geometric reasoning to come up with the right nonlinearity for the task, to include knowledge.

We call this field evolutionary computing of image structure, or the application of evolutionary operations.


It is a divergence of a flow. We also call the flux function. Withc = 1 we have normal linear, isotropic diffusion: the divergence of the gradient flow is the Laplacian.

A conductivity coefficient (c) is introduced in the diffusion equation:

L

s

.c

L c cL,

L

x,

2 L

x2, ...

c

L

L s

2 L

x 2

2 L

y 2


- linear diffusion, equivalent to isotropic diffusion;

- geometry-driven diffusion, the most general naming;

- variable conductance diffusion, the 'Perona and Malik' type of gradient magnitude

controlled diffusion;

- inhomogeneous diffusion: the diffusion is different for different locations in the image;

this is the most general naming;

- anisotropic diffusion: the diffusion is different for different directions;

- tensor driven diffusion: the diffusion coefficient is a tensor, not a scalar.

- coherence enhancing diffusion: the direction of the diffusion is governed by the direction

of local image structure, for example the eigenvectors of the structure matrix or structure

tensor (the outer product of the gradient with itself, explained in chapter 6), or the local

ridgeness.


L

s

.cLL

The Perona & Malik equation (1991):

c1 eL2

k2

c2 11 L2

k2

1 L2

k2 L42 k4

O L51 L2

k2 L4

k4 O L5

The conductivity terms are equivalent to first order.

Paper P&M


k 5 k 10 k 20

L

s

.eL2

k2

LThe function k determines the weightof the gradient squared, i.e. how muchis blurred at the edges.

For the limit of k to infinity, we get linear diffusion again.


Lx2Ly

2

k2 k2 2 Lx2Lxx 4 Lx Lxy Ly k2 2 Ly

2Lyyk2

Working out the differentiations, we get a strongly nonlineardiffusion equation:

The solution is not known analytically, so we have to relyon numerical methods, such as the forward Euler method:

L s .c LLs

This process is an evolutionary computation.

Nonlinear scale-space


10 20 30 40 50 60

50

100

150

200

250

300

350

10 20 30 40 50 60

50

100

150

200

250

300

350

Test on a small test image:

Note the preserved steepnessof the edges with thestrongly reduced noise.


Original scale 9

GDD is particularly useful for ultrasound edge-preserving speckle removal


Locally adaptive elongation of the diffusion kernel:Coherence Enhancing Diffusion

Reducing to a small kernel is often not effective:

1

2 2E x.D.x


Coherence enhancing diffusion

J. Weickert, 2001


Interestingly, the S/N ratio increasesduring the evolution.

Signal = mean differenceNoise = variance

SNR = S/N


5 10 15 20

evolutiontimein iterations

0.2

0.4

0.6

0.8

1

1.2

SNR

Interestingly, the S/N ratio has a maximumduring the evolution. Blurring too long maylead to complete blurring (always some leak).

Signal = mean differenceNoise = variance

SNR = S/N


The Perona & Malik equation is ill-posed.

Ls x

cL

xL

x c ' Lx c Lx x

Suppose that the flow c Lx is decreasing with respect to Lx at some point x0.

Then Lx

c Lx c c 'Lx a with a 0. Now c ' cLx

.

So Ls a Lx x 0, from which we get Ls a Lx x .

Locally we have an inverse heat equation which is well known to be ill-posed. This

heat equation locally blurs or deblurs, dependent on the condition of c.


The function c1Lx decreases for Lx 12

2k and c2Lx decreases for Lx k .

k determines the turnover point.

1 2 3 4 5Lx

0.20.4

0.60.8

1flowc2 Lx

1 2 3 4 5Lx

0.20.40.60.8

1Lxc2 Lx

deblurring

blurring

1 2 3 4 5Lx

0.20.40.60.8

1Lxc2 Lx1 2 3 4 5

Lx

0.2

0.4

0.6

0.8

flowc1 Lx

1 2 3 4 5Lx

0.40.2

0.20.40.60.8

1Lxc1 Lx

deblurring

blurring

1 2 3 4 5Lx

0.40.2

0.20.40.60.8

1Lxc1 Lx


1068klusters T1w_TSE PDw T1w_TFE T2w_TSE T2w_FSE

1068klusters_eucl T1w_TSE PDw T1w_TFE T2w_TSE T2w_FSE

1070klusters T1w_TSE PDw T1w_TFE T2w_TSE T2w_FSE

1070klusters_eucl T1w_TSE PDw T1w_TFE T2w_TSE T2w_FSE

Atherosclerotic plaque classification – P&M, k = 50, s = 5

J. Wijnen, TUE-BME, 2003


Stability of the numerical implementation

R t

x2 2 s 2

The maximal step size is limited due to the Neumann stability criterion (1/4). The maximal step size when using Gaussian derivatives is substantially larger:


Test image and its blurredversion.

Note the narrow rangeof the allowable time step maximum.

timestep 1.82857 timestep 1.77778 timestep 1.72973 timestep 1.68421 timestep 1.64103

The critical timestep s 2 2.1718 x 0.82 1.74


L

sL

.

LLEuclidean shortening flow:

Lyy Lx2 2 Lxy Ly Lx Lxx Ly

2

Lx2 Ly

2

Working out the derivatives, this is Lvv in gauge coordinates,i.e. the ridge detector.

So: L

s Lv v

L Lx x Ly y Lv v Lw w

L

s L Lw w

Because the Laplacian is we get

We see that we have corrected the normal diffusion with a factor proportional to the second order derivative in the gradient direction (in gauge coordinates: ). This subtractive term cancels the diffusion in the direction of the gradient.


This diffusion scheme is called Euclidean shortening flow due to the shortening of the isophotes, when considered as a curve-evolution scheme.

Advantage: there is no parameter k.Disadvantage: rounding of corners.


Name of

flow

Luminance

evolution

Curve

evolution

Timestep

N.N.

Timestep

Gauss der.

Linear Lt

L Ct

LL x22D

2s

Variable

conductance

Lt

.cL C

t

.cLL x2

2D2s

Normal or

constant motion

Lt

cLwCt

cN x

c

Euclidean

shortening

Lt

LvvCt

N x2

22s

Affine

shortening

Lt

Lvv13 Lw

23 C

t

13 N

Affine shortening

modified

Lt

Lvv 13 Lw

23

1Lwk23 2

Ct

13 Lw

k 2

3 N

Entropy Lt

0Lw 1LvvCt

0 1N,x2

2, 2s

ter haar romeny, fev geometry-driven diffusion: nonlinear scale-space – adaptive scale-space

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