Discrete Total Variation: New Definition and Minimization

Condat, Laurent

doi:10.1137/16m1075247

Cited by 120 publications

(150 citation statements)

References 39 publications

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“…• Total Variation on a Staggered Grid (TVSG) that has been recently proposed in [26], introduces a new formulation of the total variation with a more accurate adaptation of its continuous definition to the discrete domain, instead of the one based on the classical finite differences in (9). It resorts to a sophisticated gradient operator which is defined as…”

Section: Video Estimationmentioning

confidence: 99%

An alternating proximal approach for blind video deconvolution

Abboud¹,

Chouzenoux

Pesquet

et al. 2019

Signal Processing: Image Communication

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Blurring occurs frequently in video sequences captured by consumer devices, as a result of various factors such as lens aberrations, defocus, relative camerascene motion, and camera shake. When it comes to the contents of archive documents such as old films and television shows, the degradations are even more serious due to several physical phenomena happening during the sensing, transmission, recording, and storing processes. We propose in this paper a versatile formulation of blind video deconvolution problems that seeks to estimate both the sharp unknown video sequence and the underlying blur kernel from an observed video. This inverse problem is ill-posed, and an appropriate solution can be obtained by modeling it as a nonconvex minimization problem. We provide a novel iterative algorithm to solve it, grounded on the use of recent advances in convex and nonconvex optimization techniques, and having the ability of including numerous well-known regularization strategies.

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Section: Video Estimationmentioning

confidence: 99%

An alternating proximal approach for blind video deconvolution

Abboud¹,

Chouzenoux

Pesquet

et al. 2019

Signal Processing: Image Communication

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“…where TV is some discrete form of the TV [13] and z m,: = (z m,n ) n∈Ω is a scalar signal or image of same domain Ω as y. It remains to reformulate the constraint that the number of regions Ω k is at most K or, equivalently, that x takes its values in only K among the M candidates, i.e.…”

Section: Problem Reformulation By Liftingmentioning

confidence: 99%

“…In the following, we assume that the 'isotropic' form of the discrete TV [13] is used. There would be no difficulty in using instead the form proposed in [13], for better quality but higher computational complexity.…”

Section: Proposed Algorithmmentioning

confidence: 99%

“…The definition of the discrete perimeter in 2-D, based on finite differences, is much more problematic [13,14]. We can note that, if the regions Ω k are fixed, the centroids c k , solutions to the problem (1), are the means of the elements of the regions:…”

Section: Introductionmentioning

confidence: 99%

“…Conversely, if the c k are fixed, i) if λ = 0, we simply get the regions as the Voronoi cells Ω k = {n ∈ Ω : k = arg min k ′ y n − c k ′ }; ii) if λ > 0, there exist efficient methods to solve the problem, by convex relaxation [13][14][15] or by graph cuts [16], see also [17]. Therefore, a strategy consists in alternating between updating the Ω k at fixed c k , and the other way around.…”

Section: Introductionmentioning

confidence: 99%

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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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Arterial input function segmentation based on a contour geodesic model for tissue at risk identification in ischemic stroke

et al. 2022

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Perfusion parameters such as cerebral blood flow (CBF) and T max have been proven to be useful in the diagnosis and prognosis for ischemic stroke. Arterial input function (AIF) is required as an input to estimate perfusion parameters. This makes the AIF selection paradigm of clinical importance. Methods: This study proposes a new technique to address the problem of AIF selection, based on a variational segmentation model that combines geometric constraint in a distance function. The modified model uses discrete total variation in the distance term and via minimizing energy locates the arterial regions. Matrix analysis is utilized to identify the AIF with maximum peak height within the segmented region. Results: Group mean differences indicate that overall the AIF selected by the purposed method has better arterial features of higher peak position (16.7 and 26.1 a.u.) and fast attenuation (1.08 s and 0.9 s) as compared to the other stateof -the-art methods. Utilizing the selected AIF, mean CBF, and T max values were estimated higher than the traditional methods. Ischemic regions were precisely located through the perfusion maps. Conclusions: This AIF segmentation framework worked on perfusion images at levels superior to the current clinical state of the art. Consequently, the perfusion parameters derived from AIF selected by the purposed method were more accurate and reliable. The proposed method could potentially be considered as part of the calculation for perfusion imaging in general.

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Discrete Total Variation: New Definition and Minimization

Cited by 120 publications

References 39 publications

An alternating proximal approach for blind video deconvolution

An alternating proximal approach for blind video deconvolution

A Convex Approach to K-Means Clustering and Image Segmentation

Arterial input function segmentation based on a contour geodesic model for tissue at risk identification in ischemic stroke

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