Foveated Non-Local Means Denoising of Color Images, with Cross-Channel Paradigm.

Foveation, a peculiarity of the HVS, is characterized by a sharp image having maximal acuity at the central part of the retina, the fovea. The acuity rapidly decreases towards the periphery of the visual field. Foveated imaging was recently investigated for the purpose of image denoising in the Foveated Non-local Means (FNLM) algorithm, and it was shown that for natural images the foveated self-similarity is a far more effective regularization prior than the conventional windowed self-similarity. Color images exhibit spectral redundancy across the R, G and B channels which can be exploited to reduce the effects of noise. We extend the FNLM algorithm to the removal of additive white Gaussian noise from color images. The proposed Color-mixed Foveated NL-means algorithm, denominated as C-FNLM, implements the concept of foveated self-similarity, along with a cross-channel paradigm to exploit the correlation between color channels. The patch similarity is measured through an updated foveated distance for color images. In C-FNLM, we derive the explicit construction of an unified operator which explores the spatially variant nature of color perception in the HVS. We develop a framework for designing the linear operator that simultaneously performs foveation and color mixing. Within this framework, we construct several parametrized families of the color-mixing operation. Our analysis shows that the color-mixed foveation is a far more effective regularity assumption than the windowing conventionally used in NL-means, especially for color image denoising where substantial improvement was observed in terms of contrast and sharpness. Moreover, the unified operator is introduced at a negligible cost in terms of the computational complexity

A Non-Local Structure Tensor Based Approach for Multicomponent Image Recovery Problems

Non-Local Total Variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the Structure Tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the non-local variations, jointly for the different components, through various $\ell_{1,p}$ matrix norms with $p \ge 1$. To facilitate the choice of the hyper-parameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented thanks to the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for multispectral and hyperspectral images. The results demonstrate the interest of introducing a non-local structure tensor regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods

Epigraphical Projection and Proximal Tools for Solving Constrained Convex Optimization Problems: Part I

We propose a proximal approach to deal with convex optimization problems involving nonlinear constraints. A large family of such constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower level set of a sum of convex functions evaluated over different, but possibly overlapping, blocks of the signal. For this class of constraints, the associated projection operator generally does not have a closed form. We circumvent this difficulty by splitting the lower level set into as many epigraphs as functions involved in the sum. A closed half-space constraint is also enforced, in order to limit the sum of the introduced epigraphical variables to the upper bound of the original lower level set. In this paper, we focus on a family of constraints involving linear transforms of l_1,p balls. Our main theoretical contribution is to provide closed form expressions of the epigraphical projections associated with the Euclidean norm and the sup norm. The proposed approach is validated in the context of image restoration with missing samples, by making use of TV-like constraints. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems