Augmented Lagrangian Method for Total Variation Based Image Restoration and Segmentation Over Triangulated Surfaces

Abstract

Recently total variation (TV) regularization has been proven very successful in image restoration and segmentation. In image restoration, TV based models offer a good edge preservation property. In image segmentation, TV (or vectorial TV) helps to obtain convex formulations of the problems and thus provides global minimizations. Due to these advantages, TV based models have been extended to image restoration and data segmentation on manifolds. However, TV based restoration and segmentation models are difficult to solve, due to the nonlinearity and non-differentiability of the TV term. Inspired by the success of operator splitting and the augmented Lagrangian method (ALM) in 2D planar image processing, we extend the method to TV and vectorial TV based image restoration and segmentation on triangulated surfaces, which are widely used in computer graphics and computer vision. In particular, we will focus on the following problems. First, several Hilbert spaces will be given to describe TV and vectorial TV based variational models in the discrete setting. Second, we present ALM applied to TV and vectorial TV image restoration on mesh surfaces, leading to efficient algorithms for both gray and color image restoration. Third, we discuss ALM for vectorial TV based multi-region image segmentation, which also works for both gray and color images. The proposed method benefits from fast solvers for

Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space

“…Due to its good edge-preserving property, TV immediately received much attention. It has many extensions such as vectorial TV and high order TV; see, e.g., [3], [4], [5], [6], [7], [8], [9], [10]. TV and its extensions are widely used in image restoration and segmentation problems.…”

“…In recent years, piecewise linear function space, as a basic finite element space in numerical PDE [20], has achieved great successes in many computer graphics applications like mesh smoothing, image smoothing and segmentation on meshes, derivation of discrete differential operators and texture generation; see, e.g., [21], [22], [23], [24], [25], [26], [10], [27]. In particular, piecewise linear function space combined with TV has been studied [10] for image restoration and segmentation on meshes.…”

“…In particular, piecewise linear function space combined with TV has been studied [10] for image restoration and segmentation on meshes. However, we found that, in feature-preserving mesh denoising, piecewise linear function space has some limitations, compared to piecewise constant function space which is related to piecewise constant finite element method in numerical PDE.…”

“…2. A basis function of piecewise linear function space used in [26] and [10] is a hat function, which is defined vertex-by-vertex. These basis can be used to interpolate data defined on mesh vertices, such as vertex color and normal fields.…”

Group-Valued Regularization for Analysis of Articulated Motion

“…Our notion of smoothness, however, should be defined in terms of the 3D surface tangent plane. Thus, the regularization term we seek is intimately linked to the problem of image processing for images defined on parametric surfaces [34,19,40]. The measure of smoothness of the associated locally-rigid motion should take into account the geometry of the 3D surface.…”