Proximity algorithms for image models: denoising

Micchelli, Charles A.; Shen, Lixin; Xu, Yuesheng

doi:10.1088/0266-5611/27/4/045009

Cited by 220 publications

(258 citation statements)

References 38 publications

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“…Therefore, D Φ * (w|x) is strictly convex and coercive in terms of w by Theorem 4 (2). In addition, TV is a composition of · 1 • D, where D is a linear matrix (i.e, first order difference matrix) [44] and a 1 = ∑ i |a i |. Hence, we have…”

Section: Remarkmentioning

confidence: 99%

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

Woo

2017

Entropy

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Abstract:In image and signal processing, the beta-divergence is well known as a similarity measure between two positive objects. However, it is unclear whether or not the distance-like structure of beta-divergence is preserved, if we extend the domain of the beta-divergence to the negative region. In this article, we study the domain of the beta-divergence and its connection to the Bregman-divergence associated with the convex function of Legendre type. In fact, we show that the domain of beta-divergence (and the corresponding Bregman-divergence) include negative region under the mild condition on the beta value. Additionally, through the relation between the beta-divergence and the Bregman-divergence, we can reformulate various variational models appearing in image processing problems into a unified framework, namely the Bregman variational model. This model has a strong advantage compared to the beta-divergence-based model due to the dual structure of the Bregman-divergence. As an example, we demonstrate how we can build up a convex reformulated variational model with a negative domain for the classic nonconvex problem, which usually appears in synthetic aperture radar image processing problems.

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Section: Remarkmentioning

confidence: 99%

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

Woo

2017

Entropy

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“…Now we want to show that these optimal coefficients c opt can be computed by a fixed point iteration method similar as in [17]. Suppose that L(x, y, ·) ∈ C 1 (C) for all x ∈ Ω 1 and all y ∈ C, and R ∈ C 1 ([0, ∞)).…”

Section: The Minimization Problem (42) Is Equivalent To Min F ∈B T Dmentioning

confidence: 99%

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

Fasshauer

Hickernell

2015

Applied and Computational Harmonic Analysis

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In this paper we solve support vector machines in reproducing kernel Banach spaces with reproducing kernels defined on nonsymmetric domains instead of the traditional methods in reproducing kernel Hilbert spaces. Using the orthogonality of semi-innerproducts, we can obtain the explicit representations of the dual (normalized-duality-mapping) elements of support vector machine solutions. In addition, we can introduce the reproduction property in a generalized native space by Fourier transform techniques such that it becomes a reproducing kernel Banach space, which can be even embedded into Sobolev spaces, and its reproducing kernel is set up by the related positive definite function. The representations of the optimal solutions of support vector machines (regularized empirical risks) in these reproducing kernel Banach spaces are formulated explicitly in terms of positive definite functions, and their finite numbers of coefficients can be computed by fixed point iteration. We also give some typical examples of reproducing kernel Banach spaces induced by Matérn functions (Sobolev splines) so that their support vector machine solutions are well computable as the classical algorithms. Moreover, each of their reproducing bases includes information from multiple training data points. The concept of reproducing kernel Banach spaces offers us a new numerical tool for solving support vector machines.

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“…The Tikhonov regularization method [9], the truncated singular value decomposition (TSVD) method [10], the modified TSVD (MTSVD) method [11], the Chebyshev interpolation method [12], the collocation method [13,14], the projected Tikhonov regularization method [15], and so on, are applied to obtain approximate continuous solutions of Equation (2). The total variation (TV) regularization method [16][17][18][19][20][21][22], adaptive TV methods [23][24][25][26][27], the piecewise-polynomial TSVD (PP-TSVD) method [28], and so on, are applied to obtain approximate piecewise-continuous solutions.…”

Section: Introductionmentioning

confidence: 99%

A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

Lin¹,

Yang²

2017

Mathematics

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A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f * TV . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f * TV as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.

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Proximity algorithms for image models: denoising

Cited by 220 publications

References 38 publications

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

A Characterization of the Domain of Beta-Divergence and Its Connection to Bregman Variational Model

Solving support vector machines in reproducing kernel Banach spaces with positive definite functions

A Two-Stage Method for Piecewise-Constant Solution for Fredholm Integral Equations of the First Kind

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