Modelling Stable Backward Diffusion and Repulsive Swarms with Convex Energies and Range Constraints

Bergerhoff, Leif; Cárdenas, M. C.; Weickert, Joachim; Welk, Martin

doi:10.1007/978-3-319-78199-0_27

Cited by 2 publications

(2 citation statements)

References 8 publications

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“…This article is a revised version of our conference contribution [3] which we extend in several aspects. First, we enhance our model for convex backward diffusion to support not only a global and weighted setting but also a localised variant.…”

Section: Diffusion and Energy Minimisationmentioning

confidence: 99%

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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Section: Diffusion and Energy Minimisationmentioning

confidence: 99%

Stable Backward Diffusion Models that Minimise Convex Energies

et al. 2020

Self Cite

View full text Add to dashboard Cite

show abstract

“…This article is a revised version of our conference contribution [4] which we extend in several aspects. First, we enhance our model for convex backward diffusion to support not only a global and weighted setting but also a localised variant.…”

Section: Introductionmentioning

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

Modelling Stable Backward Diffusion and Repulsive Swarms with Convex Energies and Range Constraints

Cited by 2 publications

References 8 publications

Stable Backward Diffusion Models that Minimise Convex Energies

Stable Backward Diffusion Models that Minimise Convex Energies

Stable Backward Diffusion Models that Minimise Convex Energies

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