Regularization on Discrete Spaces

Zhou, Dengyong; Schölkopf, Bernhard

doi:10.1007/11550518_45

Cited by 114 publications

(124 citation statements)

References 6 publications

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“…Among them, the NL-means regularization term leads to very good denoising results. The idea of nonlocal means goes back to [12] and was incorporated into the variational framework in [35,36,37,67]. We refer to these papers for further information on NL-means.…”

Section: Review Of Modelsmentioning

confidence: 99%

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

2009

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In this paper, we consider a variational restoration model consisting of the I-divergence as data fitting term and the total variation semi-norm or nonlocal means as regularizer for removing multiplicative Gamma noise. Although the I-divergence is the typical data fitting term when dealing with Poisson noise we substantiate why it is also appropriate for cleaning Gamma noise. We propose to compute the minimizers of our restoration functionals by applying Douglas-Rachford splitting techniques, resp. alternating direction methods of multipliers. For a particular splitting, we present a semi-implicit scheme to solve the involved nonlinear systems of equations and prove its Q-linear convergence. Finally, we demonstrate the performance of our methods by numerical examples.

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Section: Review Of Modelsmentioning

confidence: 99%

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

2009

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“…The edge connecting two vertices and represents the similarity between both pixels, expressed as a weight function ( , ) > 0. Employing such graph representation and special calculus on graphs [89,90], several regularisation models for general data living on discrete spaces have been recently proposed. In the context of image denoising Weickert [82] developed a space-discrete theory for diffusion filtering that is directly applicable to functions defined on graphs, and Chan et al [17] introduced the digital TV filter as a discrete version of the continues ROF model [66].…”

Section: Nds and Graph Regularisationmentioning

confidence: 99%

“…In the context of image denoising Weickert [82] developed a space-discrete theory for diffusion filtering that is directly applicable to functions defined on graphs, and Chan et al [17] introduced the digital TV filter as a discrete version of the continues ROF model [66]. Following the ideas from graph theory presented in [89,90], Gilboa and Osher [38] proposed the use of nonlocal operators to extend some known PDEs and variational techniques in image processing to a nonlocal framework. In particular, they use discretised differential operators such as gradient and divergence.…”

Section: Nds and Graph Regularisationmentioning

confidence: 99%

Generalised Nonlocal Image Smoothing

Pizarro

Pavel²,

Didas

et al. 2010

Int J Comput Vis

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We propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness contraints that penalise general dissimilarity measures defined on image patches. One of such dissimilarity measures is the weighted 2 distance between patches. In such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain. It can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels, but also the similarity of their corresponding neighbours. This leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure. The suggested approach includes two recently proposed filters as special cases: The NLmeans filter of Buades et al. and the NDS filter of Mrázek et al. In fact, the approach introduced here can be considered as a generalisation of the latter filter. We evaluate our method for the task of denoising greyscale and colour images degraded by Gaussian and impulse noise, demonstrating that it compares very well to other more sophisticated patch-based approaches.

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“…They reformulate image segmentation tasks into semi-supervised classification approaches by label propagation strategies [8,9,29]. Other applications of these label diffusion methods can be found in [30,5]. Our previously presented discrete regularization framework (Section 3.2.1) can be naturally adapted to address this learning problem for semisupervised segmentation.…”

Section: Discrete Semi-supervised Clusteringmentioning

confidence: 99%

Graph-based tools for microscopic cellular image segmentation

Ta¹,

Lézoray²,

Elmoataz³

et al. 2009

Pattern Recognition

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We propose a framework of graph based tools for the segmentation of microscopic cellular images. This framework is based on an object oriented analysis of imaging problems in pathology. Our graph tools rely on a general formulation of discrete functional regularization on weighted graphs of arbitrary topology. It leads to a set of useful tools which can be combined together to address various image segmentation problems in pathology. To provide fast image segmentation algorithms, we also propose an image simplification based on graphs as a pre processing step. The abilities of this set of image processing discrete tools are illustrated through automatic and interactive segmentation schemes for color cytological and histological images segmentation problems.

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Regularization on Discrete Spaces

Cited by 114 publications

References 6 publications

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

Removing Multiplicative Noise by Douglas-Rachford Splitting Methods

Generalised Nonlocal Image Smoothing

Graph-based tools for microscopic cellular image segmentation

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