Mumford and Shah Model and Its Applications to Image Segmentation and Image Restoration

Bar, Leah; Chan, Tony F.; Chung, Ginmo; Jung, Mi Sook; Kiryati, Nahum; Sochen, Nir; Vese, Luminita A.

doi:10.1007/978-3-642-27795-5_25-5

Cited by 14 publications

(16 citation statements)

References 69 publications

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“…Here, the length of Γ can be written as H 1 (Γ), the 1-dimensional Hausdorff measure in R 2 ; see [4]. Because model (1.1) is nonconvex, it is very challenging to find or approximate its minimizer.…”

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confidence: 99%

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A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding

Cai¹,

Chan²,

Zeng³

2013

SIAM J. Imaging Sci.

190

243

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Abstract. The Mumford-Shah model is one of the most important image segmentation models and has been studied extensively in the last twenty years. In this paper, we propose a two-stage segmentation method based on the Mumford-Shah model. The first stage of our method is to find a smooth solution g to a convex variant of the Mumford-Shah model. Once g is obtained, then in the second stage the segmentation is done by thresholding g into different phases. The thresholds can be given by the users or can be obtained automatically using any clustering methods. Because of the convexity of the model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle-Pock method. We prove that our method is convergent and that the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any K-phase segmentations by choosing (K − 1) thresholds after g is found in the first stage, and in the second stage there is no need to recompute g if the thresholds are changed to reveal different segmentation features in the image. Experimental results show that our two-stage method performs better than many standard two-phase or multiphase segmentation methods for very general images, including antimass, tubular, MRI, noisy, and blurry images.

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confidence: 99%

“…In [32], the continuous multiclass labeling approaches were discussed. Interested readers can read the references therein or see [4] for more details.…”

mentioning

confidence: 99%

A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding

Cai¹,

Chan²,

Zeng³

2013

SIAM J. Imaging Sci.

190

243

View full text Add to dashboard Cite

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“…In subsequent formalisations of the Mumford-Shah problem, I is constrained to the set C 1 (Ω\Γ) of continuously differentiable functions on Ω\Γ, where Γ is a closed set of Hausdorff dimension 1. Ennio De Giorgi proposed a relaxed Mumford-Shah problem in which I is constrained to the set SBV(R 2 ) of special bounded total variation functions and Γ = SI is the jump set of I (for detailed presentations of the different formulations of the Mumford-Shah problem and their connections, see [28,5]). When µ → ∞, the smoothness term forces I to be constant on the connected components of Ω\Γ.…”

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confidence: 99%

“…A working set algorithm for total variation regularization. In this section, we consider the problem of solving the minimization of a convex, differentiable function f regularized by a weighted total variation of the form TV(x) = 1 2 (i,j)∈E w ij |x i − x j | with w ij some nonnegative weights 5 .…”

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confidence: 99%

Cut Pursuit: Fast Algorithms to Learn Piecewise Constant Functions on General Weighted Graphs

Landrieu¹,

Obozinski²

2017

SIAM J. Imaging Sci.

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“…An application is color image quantization [8,9]: one looks for the palette of K colors representing at best a given image; in that case, the points are the pixel values in R 3 , corresponding to the coordinates in some color space. A fundamental problem in image processing and vision, which is even more difficult, is image segmentation: one wants to decompose an image of N pixels into K regions, corresponding to the objects of the scene, by favoring, in addition to intra-region similarity, some notion of spatial homogeneity [10,11].…”

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confidence: 99%

A Convex Approach to K-Means Clustering and Image Segmentation

Condat

2018

Lecture Notes in Computer Science

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Abstract.A new convex formulation of data clustering and image segmentation is proposed, with fixed number K of regions and possible penalization of the region perimeters. So, this problem is a spatially regularized version of the K-means problem, a.k.a. piecewise constant Mumford-Shah problem. The proposed approach relies on a discretization of the search space; that is, a finite number of candidates must be specified, from which the K centroids are determined. After reformulation as an assignment problem, a convex relaxation is proposed, which involves a kind of l1,∞ norm ball. A splitting of it is proposed, so as to avoid the costly projection onto this set. Some examples illustrate the efficiency of the approach.

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Mumford and Shah Model and Its Applications to Image Segmentation and Image Restoration

Cited by 14 publications

References 69 publications

A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding

A Two-Stage Image Segmentation Method Using a Convex Variant of the Mumford--Shah Model and Thresholding

Cut Pursuit: Fast Algorithms to Learn Piecewise Constant Functions on General Weighted Graphs

A Convex Approach to K-Means Clustering and Image Segmentation

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