Modelling Image Processing with Discrete First-Order Swarms

Bergerhoff, Leif; Weickert, Joachim

doi:10.1007/978-3-319-27400-3_23

Cited by 3 publications

(3 citation statements)

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“…The application of the proposed model to greyscale images follows the ideas presented in [4]. We define a digital greyscale image as a mapping f : {1, .…”

Section: Greyscale Imagesmentioning

confidence: 99%

“…N as samples of a smooth 1D signal u : Ω → [0, 1] over an interval Ω of the real axis, taken at sampling positions x i = x 0 + i h with grid mesh size h > 0. We consider the model(4) with w i, j := γ (|x j − x i |), where γ : R + 0 → [0, 1] is a non-increasing weighting function with compact support [0, ). Theorem 8 (Space-Continuous Energy) Equation (4) can be considered as a discretisation of E[u] = 1 ) + B(u) dx (47) with penaliser W (u 2 x ) ≈ C Ψ (u 2 x ) and barrier function B(u) ≈ D Ψ (4u 2 ), where C and D are positive constants.…”

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confidence: 99%

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Stable Backward Diffusion Models that Minimise Convex Energies

et al. 2020

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The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy-to-handle numerical scheme. So far, existing stabilisation strategies in the literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images.

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“…The application of the proposed model to greyscale images follows the ideas presented in [4]. We define a digital greyscale image as a mapping f : {1, .…”

Section: Greyscale Imagesmentioning

confidence: 99%

mentioning

confidence: 99%

Stable Backward Diffusion Models that Minimise Convex Energies

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…The application of the proposed model to greyscale images follows the ideas presented in [5]. We define a digital greyscale image as a mapping f : {1, .…”

Section: Greyscale Imagesmentioning

confidence: 99%

Stable Backward Diffusion Models that Minimise Convex Energies

Bergerhoff¹,

Cárdenas²,

Weickert³

et al. 2019

Preprint

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Backward diffusion processes appear naturally in image enhancement and deblurring applications. However, the inverse problem of backward diffusion is known to be ill-posed and straightforward numerical algorithms are unstable. So far, existing stabilisation strategies in the literature require sophisticated numerics to solve the underlying initial value problem. Therefore, it is desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with a simple numerical scheme. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images.

show abstract