Entropy Based Cartoon Texture Separation

Separating an image into cartoon and texture components comes useful in image processing applications such as image compression, image segmentation, image inpainting. Yves Meyer’s influential cartoon texture decomposition model involves deriving an energy functional by choosing appropriate spaces and functionals. Minimizers of the derived energy functional are cartoon and texture components of an image. In this study, cartoon part of an image is separated, by reconstructing it from pixels of multi scale Total-Variation filtered versions of the original image which is sought to be decomposed into cartoon and texture parts. An information theoretic pixel by pixel selection criteria is employed to choose the contributing pixels and their scales. Introduction An image, f , is defined as a function, f : (x, y) ∈ Ω→ R, where f maps a pixel location (x, y) ∈ Ω to an intensity value I ∈ R, and Ω is the set of pixel locations. Cartoon-Texture separation problem is about decomposing f into two functions, u and v, where function u is the cartoon part and function v is the texture part of an image f , such that f = u+ v. Total Variation noise removal algorithm [1] was introduced by Rudin, Osher and Fatemi (ROF). Instead of using L2 norm in image processing related estimation problems, they used Total Variation(TV) norm, L1 norm of gradient, which is the true norm for image processing related estimation problems. They mention that the space of functions with bounded TV norm is the space of functions of bounded variations. ROF model computes, minimizer, u, of the following functional, E(u) , where E(u) = ∫ Ω |∇u|+ λ 2 ∫ Ω (f − u)dx dy In this paper, the term scale in used to denote different values parameter λ may get. While space of functions of bounded variations represent edges and piecewise smooth parts of an image, space of oscillatory functions represent texture and noise. ROF model is useful for computing an approximation to cartoon part u. Yves Meyer [2] devised a new functional, E(u, v) and introduced a second minimizer, v representing texture and noise component of an image f . In Buades et al. Meyer’s approach is defined as , 1 ar X iv :1 40 5. 17 17 v1 [ cs .C V ] 7 M ay 2 01 4 E(u, v) = inf (u,v)∈X1,X2 { f=u+ v : F1(u) + F2(v)} where F1, F2 ≥ 0 are functionals, and X1, X2 ≥ 0 are spaces of functions. Functional F1(u) penalizes texture component and functional F2(v) penalizes cartoon component, so the aim is minimizing with respect to both variables, u and v to achieve a perfect cartoon texture separation. Many related study focus on the effect of choosing different function spaces and functionals for the v component. e.g. spaces G, F, E of various generalized functions. Yin et al. [3] gives a detailed comparison of these function space approaches and their performance. Although, choices of many function spaces and functionals are possible for texture component, v, cartoon component, u, is always modelled by the space of L1 norm of the gradient. ROF functional is convex, so solution of the following Euler-Lagrange equation, u− f − div ( ∇u |∇u| ) = 0 is the minimizer, u , of the ROF energy E(u). To compute, u, gradient descent iterations on the direction − du is applied which leads to the following PDE, ∂u ∂t = λ (f − u) + div ( ∇u |∇u| ) Main problem on implementation of a numerical scheme for gradient descent iterations is when gradient component,|∇u|, is zero. A simple solution is, replacing |∇u| term with √ 2 +∇u by adding a very small and carrying on time dependent step by step iterations. However due to CFL conditions and small value these iterations are very slow to converge. Many other optimization methods, such as primal-dual methods, gradient projection methods, graph cut algorithms were proposed to minimize ROF energy. A detailed review about efficient approaches for computing ROF minimizers is given in Zhu et al. [4]. The most notable one is Chan, Golub, Mulet (CGM)Algorithm [5] which I preferred to use in my experiments. URL of a publicly available implementation is given in [6]. The reason for preference was its fast convergence rate. Entropy, H(X), of a random variable X is defined as, − ∑ i p(xi) log p(xi) where xi are the values random variable X maps to. According to this definition, uniform distribution must have the maximum entropy i.e. highest uncertainty among all continuous probability distributions since each value of uniform random variable (RV) has equal probability of occurring.