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ter Haar Romeny, Computer Vision 2014 Geometry-driven diffusion: nonlinear scale-space – adaptive scale-space

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ter Haar Romeny, Computer Vision 2014 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.

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ter Haar Romeny, Computer Vision 2014 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.

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ter Haar Romeny, Computer Vision 2014 It is a divergence of a flow. We also call the flux function. With c = 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:

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ter Haar Romeny, Computer Vision 2014

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The Perona & Malik equation (1991): The conductivity terms are equivalent to first order. Paper P&M

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ter Haar Romeny, Computer Vision 2014 The function k determines the weight of the gradient squared, i.e. how much is blurred at the edges. For the limit of k to infinity, we get linear diffusion again.

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ter Haar Romeny, Computer Vision 2014 Working out the differentiations, we get a strongly nonlinear diffusion equation: The solution is not known analytically, so we have to rely on numerical methods, such as the forward Euler method: This process is an evolutionary computation. Nonlinear scale-space

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ter Haar Romeny, Computer Vision 2014 Test on a small test image: Note the preserved steepness of the edges with the strongly reduced noise.

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ter Haar Romeny, Computer Vision 2014 GDD is particularly useful for ultrasound edge-preserving speckle removal

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ter Haar Romeny, Computer Vision 2014 Locally adaptive elongation of the diffusion kernel: Coherence Enhancing Diffusion Reducing to a small kernel is often not effective:

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ter Haar Romeny, Computer Vision 2014 Coherence enhancing diffusion J. Weickert, 2001

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ter Haar Romeny, Computer Vision 2014 Interestingly, the S/N ratio increases during the evolution. Signal = mean difference Noise = variance SNR = S/N

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ter Haar Romeny, Computer Vision 2014 Interestingly, the S/N ratio has a maximum during the evolution. Blurring too long may lead to complete blurring (always some leak). Signal = mean difference Noise = variance SNR = S/N

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ter Haar Romeny, Computer Vision 2014 The Perona & Malik equation is ill-posed.

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ter Haar Romeny, Computer Vision 2014

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Atherosclerotic plaque classification – P&M, k = 50, s = 5 J. Wijnen, TUE-BME, 2003

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ter Haar Romeny, Computer Vision 2014

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Stability of the numerical implementation 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:

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ter Haar Romeny, Computer Vision 2014 Test image and its blurred version. Note the narrow range of the allowable time step maximum.

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ter Haar Romeny, Computer Vision 2014 Euclidean shortening flow: Working out the derivatives, this is L vv in gauge coordinates, i.e. the ridge detector. So: 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.

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ter Haar Romeny, Computer Vision 2014 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.

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ter Haar Romeny, Computer Vision 2014

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Ter Haar Romeny, EMBS Berder 2004 Gaussian derivative kernels The algebraic expressions for the first 4 orders:

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