Second-order Variational Models for Image Texture Analysis

Abstract

To cite this version:Maïtine Bergounioux. Second order variational models for image texture analysis. Advances in Imaging and Electron Physics, Elsevier, 2014, 181, pp.35-124 AbstractWe present variational models to perform texture analysis and/or extraction for image processing. We focus on second order decomposition models. Variational decomposition models have been studied extensively during the past decades. The most famous one is the Rudin-Osher-Fatemi model. We first recall most classical first order models . Then we deal with second order ones : we detail the mathematical framework, theoretical models and numerical implementation. We end with two 3D applications. Eventually, an appendix includes the mathematical tools that are used to perfom this study and Matlab c codes are provided.

On the Taut String Interpretation and Other Properties of the Rudin–Osher–Fatemi Model in One Dimension

“…They have subsequently found applications in restoration of MRI [27] and image inpainting [35], to mention just two examples. A detailed account of the second-order regularization in image restoration, as well as additional references, can be found in Begrounioux [10]. The one-dimensional case has be studied in a purely discrete setting in Steidl et al [41], and onedimensional examples are found in [34,36], among others.…”

On Poincaré-Friedrichs Type Inequalities for the Broken Sobolev Space ${\rm W}^{2,1}$

“…Here, Ω is assumed to be a bounded Lipschitz domain in R 2 with boundary Γ = ∂Ω, u d ∈ L 2 (Ω) represents the noisy image and λ > 0 is a fidelity parameter. We refer to [4,5,11] for details (cf. also [15,16] for texture analysis based on first order total variation models).…”

“…also [15,16] for texture analysis based on first order total variation models). The existence of a solution u ∈ BV 2 0 (Ω) can be shown by a minimizing sequence argument (cf., e.g., [4,Theorem 3.4]).…”

“…In this paper, given a bounded Lipschitz domain Ω in R d , d ∈ N, we derive such inequalities for the broken Sobolev space W 2,1 (Ω; T h ) with respect to a geometrically conforming, simplicial triangulation T h of Ω. They are based on Poincaré-Friedrichs type inequalities for the space BV 2 (Ω) of functions of bounded second order total variation [3,4]. As an application, we consider a C 0 Discontinuous Galerkin (C 0 DG) approximation of a minimization problem in BV 2 (Ω) which is used in image processing for texture analysis and management.…”

“…We then derive two Poincaré-Friedrichs type inequalities for the broken Sobolev space W 2,1 (Ω; T h ) (Theorem 5.2). Finally, in Section 6 we consider the C 0 DG approximation of a minimization problem in BV 2 (Ω) which can be applied to texture analysis and management in image restoration [4,5] and prove that the sequence of C 0 DG approximations in W 2,1 (Ω; T h ) contains a subsequence converging weakly * in BV 2 (Ω) to a solution of the original minimization problem (Theorem 6.1).…”