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ter Haar Romeny, FEV
Geometry-driven diffusion: nonlinear scale-space – adaptive scale-space
Original scale 9
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
- 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.
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
Original scale 9
GDD is particularly useful for ultrasound edge-preserving speckle removal
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
Coherence enhancing diffusion
J. Weickert, 2001
ter Haar Romeny, FEV
Interestingly, the S/N ratio increasesduring the evolution.
Signal = mean differenceNoise = variance
SNR = S/N
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
ter Haar Romeny, FEV
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:
ter Haar Romeny, FEV
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
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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.
ter Haar Romeny, FEV
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