A Novel Framework for Nonlocal Vectorial Total Variation Based on ℓ p,q,r  −norms

Abstract

Abstract. In this paper, we propose a novel framework for restoring color images using nonlocal total variation (NLTV) regularization. We observe that the discrete local and nonlocal gradient of a color image can be viewed as a 3D matrix/or tensor with dimensions corresponding to the spatial extend, the differences to other pixels, and the color channels. Based on this observation we obtain a new class of NLTV methods by penalizing the p,q,r norm of this 3D tensor. Interestingly, this unifies several local color total variation (TV) methods in a single framework. We show in several numerical experiments on image denoising and deblurring that a stronger coupling of different color channels -particularly, a coupling with the ∞ norm -yields superior reconstruction results.

Collaborative Total Variation: A General Framework for Vectorial TV Models

“…The collaborative norms defined in the previous section support most of pre-existing definitions of TV for vector-valued images, the most relevant of which are displayed in Table 1. For nonlocal TV based models, we refer the reader to our conference paper [14].…”

“…Although the TV was originally designed for image denoising, it has become one of the most popular regularizations for many image processing problems and has sparked a tremendous amount of research. While many extensions like anisotropic TV [17,26,49], weighted TV [11,22,25], higher order TV [2,5,9,39,51], nonlocal TV [14,20,21,40,41], or nonconvex TV [32,37] have been proposed, the general idea of penalizing image oscillations with one-homogeneous functions depending on the spatial derivatives of the image remain the same. A lot of recent research has focused on extending the classical TV model for grayscale images to vector-valued (color or multichannel) images.…”

On the Implementation of Collaborative TV Regularization: Application to Cartoon+Texture Decomposition

“…where (DP ) i,1 and (DP ) i,2 are computed analogously to (15)- (16). More concretely, we ran the algorithm by using the 1,1,1 norm for several λ's and picked up the value for which the best visual result was obtained.…”

“…We proposed in [16,17] to take a generalized viewpoint that unifies most vectorial TV based models proposed in the literature. By considering the derivatives of a color image as a linear operator, one obtains a 3D data structure of the gradient: one dimension corresponding to the pixels, one dimension corresponding to the derivatives, and one dimension corresponding to the color channels.…”

Weighted total generalised variation scheme for image restoration

“…To overcome this defect, a great deal of effort has gone into the development of variational models. For instance, higher‐order TV model [8, 9], typically, fourth‐order partial differential equation (PDE) filter [10–13], combination model of second‐order and fourth‐order PDEs [14–16], as well as non‐local TV strategy [17–20] have been researched extensively and made great successes in image processing and computer vision.…”