Infimal Convolution of Oscillation Total Generalized Variation for the Recovery of Images with Structured Texture

Abstract

We propose a new type of regularization functional for images called oscillation total generalized variation (TGV) which can represent structured textures with oscillatory character in a specified direction and scale. The infimal convolution of oscillation TGV with respect to several directions and scales is then used to model images with structured oscillatory texture. Such functionals constitute a regularizer with good texture preservation properties and can flexibly be incorporated into many imaging problems. We give a detailed theoretical analysis of the infimal-convolution-type model with oscillation TGV in function spaces. Furthermore, we consider appropriate discretizations of these functionals and introduce a first-order primaldual algorithm for solving general variational imaging problems associated with this regularizer. Finally, numerical experiments are presented which show that our proposed models can recover textures well and are competitive in comparison to existing state-of-the-art methods.Mathematics subject classification: 94A08, 68U10, 26A45, 90C90. component u belonging to BV(Ω) and a noise or small-scale texture component in L 2 (Ω). In [42], Y. Meyer pointed out some limitations of the above model and introduced a new space G(Ω) = W −1,∞ (Ω), which is larger than L 2 (Ω), to model oscillating patterns that are typical for structured texture: minwhere G(Ω) denotes the spaceand again, J(u) = Ω |Du|. In this model, u represents a piecewise constant function, consisting of homogeneous regions with sharp boundaries and is usually called cartoon component. The other part v in G(Ω) contains oscillating patterns, like textures and noise. This model can be considered as the original cartoon-texture decomposition model. However, the G-space may be difficult to handle in implementations. Vese and Osher [52] approximated the G-space by a Sobolev space of negative differentiability order W −1,p (Ω), which can, in practice, more easily be implemented by partial differential equations. Later, Aujol et al. [4] made a modification to the G-norm term in (2) which is replaced by constrainingAs a consequence, the problem can be solved alternatingly by Chambolle's projection method [23] with respect to the two variables u and v. Certainly, the aim of above models is to find an appropriate norm to describe textures. However, in general, the above norms can represent all kinds of oscillatory parts. As a consequence, since noise can be regarded as small-scale oscillating texture, these models do not deal well with noise. In order to tackle this disadvantage, in [47], Schaeffer and Osher incorporated robust principal component analysis (PCA) [22] into cartoon-texture decomposition models based on the patch method. For the texture part, the authors suggest to decompose the image into small, non-overlapping patches, written as vectors. The collection of the patch vectors is assumed to be (highly) linearly dependent and thus to have low rank. This suggests to minimize the nuclear norm of the patch-vector matrix. Howeve...

Modern regularization methods for inverse problems

“…For the combination of first-and second-order derivatives the nullspace naturally consists of piecewise affine functions (thus M = d + 1). For a further discussion and advanced aspects we refer to [53,29,311,52,54,55,81,82,36,340,223,181,37]. In certain cases it is also interesting to use total variation regularization on some transform of the image.…”

“…(2010), Benning et al. (2013), Ranftl, Pock and Bischof (2013), Bredies and Holler (2014, 2015 a , 2015 b ), Burger, Papafitsoros, Papoutsellis and Schönlieb (2015 b , 2016 c ), Bergounioux (2016), Setzer, Steidl and Teuber (2011), Holler and Kunisch (2014), Gao and Bredies (2017) and Bergounioux and Papoutsellis (2018).…”

Higher-order total variation approaches and generalisations

“…}u ´Rω u} d{pd´1q ď C TGV osci α,ω puq for all u P BVpΩq, see [89]. The functional can therefore be used as a regulariser in all cases where TV is applicable.…”

A Nonlocal Laplacian-Based Model for Bituminous Surfacing Crack Recovery and its MPI Implementation

“…We thus want to discriminate and separate the texture component v that catches redundant and oscillatory patterns from the simplified version of the image denoted by u in which useless information has been removed, while keeping the crack. We therefore introduce a joint decomposition/segmentation model in which the thin-structure recognition task operates on the component u. Texture modelling consists in finding the best functional space to represent the oscillatory patterns and has been extensively studied, either on its own or as a close counterpart of image denoising (see [25], [26], [29], [31], [32], [33], [34], [37], [39], [38] or [44]). In this work, to model the texture, we propose to use the space G(R 2 ) introduced by Meyer ( [32]), space of distributions v that can be written as v = divg where 2 and endowed with the norm defined by…”