Chromaticity Denoising using Solution to the Skorokhod Problem

Borkowski, Dariusz

doi:10.1007/978-3-540-33267-1_9

Cited by 7 publications

(7 citation statements)

References 16 publications

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“…However, we emphasize that the literature devoted to Skorohod problems, their extensions and applications is much larger than what is conveyed above and, in fact, many more researcher have contributed to this rich field. In particular, applied areas where Skorohod problems occur include heavy traffic analysis of queueing networks (see, e.g, [2,20,24,40,47,48,51,52]), control theory, game theory and mathematical economics (see, e.g., [3,37,49,50,58]), image processing (see, e.g., [8]) and molecular dynamics (see, e.g., [54][55][56]). For further results concerning Skorohod problems, as well as applications of Skorohod problems, we also refer to [1,4,5,14,23,25,29,31,35,38,44,46] and [60].…”

Section: Introductionmentioning

confidence: 99%

The Skorohod oblique reflection problem in time-dependent domains

Nyström¹,

Önskog²

2010

Ann. Probab.

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The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka [61] and further investigated by P.-L. Lions and A.-S. Sznitman [42] in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of c\`{a}dl\`{a}g solutions $(x,\lambda)$ to the Skorohod problem, with oblique reflection, for $(D,\Gamma,w)$ assuming, in particular, that $D$ is a time-dependent domain (Theorem 1.2). In addition, we prove that if $w$ is continuous, then $x$ is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.Comment: Published in at http://dx.doi.org/10.1214/10-AOP538 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Introductionmentioning

confidence: 99%

The Skorohod oblique reflection problem in time-dependent domains

Nyström¹,

Önskog²

2010

Ann. Probab.

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“…In image processing one can find theoretical results [1] and some practical aspects [4,6,10,11,13,16] of BSDE-based applications. In [4] the problem of reconstruction of the noisy chromaticity is considered.…”

Section: Introductionmentioning

confidence: 99%

Implementation of Image Denoising based on Backward Stochastic Differential Equations

Borkowski

2023

Image Processing on Line

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In this paper, we give the implementation of an image denoising algorithm based on backward stochastic differential equations. In our algorithm, we consider two stochastic processes. One of them has values in the image domain and determines pixels that will be involved in the reconstruction, the second one has values in the image codomain and gives weights to values of pixels. The reconstructed image is characterized by smoothing noisy pixels and at the same time enhancing edges. Our experiments show that the new approach gives very good results and can be successfully used to reconstruct images. Source CodeThe reviewed source code and documentation for this algorithm are available at the IPOL web page of this article 1 . Compilation and usage instructions are included in the README.txt file of the archive.

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“…In image processing one can find only theoretical results and some practical aspects [3,4,6] of BSDE-based applications. The papers [3,4,6] explore the inverse problem in the following form:…”

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confidence: 99%

“…where the blurring operator is omitted. In [3] the problem of reconstruction of the noisy chromaticity is considered. Presented model of denoising is expressed in terms of Skorokhod problem associated with the solution of BSDE and an epsilon neighbourhood of two dimensional sphere.…”

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confidence: 99%

“…In [4] problems of reconstruction of the noisy grayscale image (smoothing filters) and enhancing of the blurred grayscale image (enhancing filters) are presented. Smoothing filter, similarly as in [3], models on anisotropic forward diffusion with BSDEs with f ≡ 0. Enhancing filters presented in [4] are driven by BSDE with non trivial drift function and correspond to inverse heat equation.…”

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confidence: 99%

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Forward and backward filtering based on backward stochastic differential equations

Borkowski¹

2016

IPI

Self Cite

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In this paper we explore the problem of reconstruction of blurred and noisy images. The idea presented here provides a new methodology based on advanced tools of stochastic analysis which can be successfully used to solve the inverse problem. In order to solve this problem we use backward stochastic differential equations. The reconstructed image is characterized by smoothing noisy pixels and at the same time enhancing and sharpening edges. Our experiments show that the new approach gives very good results and compares favourably with deterministic partial differential equation methods.

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Chromaticity Denoising using Solution to the Skorokhod Problem

Cited by 7 publications

References 16 publications

The Skorohod oblique reflection problem in time-dependent domains

The Skorohod oblique reflection problem in time-dependent domains

Implementation of Image Denoising based on Backward Stochastic Differential Equations

Forward and backward filtering based on backward stochastic differential equations

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