Weighted Mean Curvature

In image processing tasks, spatial priors are essential for robust computations, regularization, algorithmic design and Bayesian inference. In this paper, we introduce weighted mean curvature (WMC) as a novel image prior and present an efficient computation scheme for its discretization in practical image processing applications. We first demonstrate the favorable properties of WMC, such as sampling invariance, scale invariance, and contrast invariance with Gaussian noise model; and we show the relation of WMC to area regularization. We further propose an efficient computation scheme for discretized WMC, which is demonstrated herein to process over 33.2 giga-pixels/second on GPU. This scheme yields itself to a convolutional neural network representation. Finally, WMC is evaluated on synthetic and real images, showing its superiority quantitatively to total-variation and mean curvature.Comment: 12 page

A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page

Multigrid Algorithm Based on Hybrid Smoothers for Variational and Selective Segmentation Models

Automatic segmentation of an image to identify all meaningful parts is one of the most challenging as well as useful tasks in a number of application areas. This is widely studied. Selective segmentation, less studied, aims to use limited user specified information to extract one or more interesting objects (instead of all objects). Constructing a fast solver remains a challenge for both classes of model. However our primary concern is on selective segmentation. In this work, we develop an effective multigrid algorithm, based on a new non-standard smoother to deal with non-smooth coefficients, to solve the underlying partial differential equations (PDEs) of a class of variational segmentation models in the level set formulation. For such models, non-smoothness (or jumps) is typical as segmentation is only possible if edges (jumps) are present. In comparison with previous multigrid methods which were shown to produce an acceptable {\it mean} smoothing rate for related models, the new algorithm can ensure a small and {\it global} smoothing rate that is a sufficient condition for convergence. Our rate analysis is by Local Fourier Analysis and, with it, we design the corresponding iterative solver, improving on an ineffective line smoother. Numerical tests show that the new algorithm outperforms multigrid methods based on competing smoothers