In this paper, we introduce a new method for image smoothing based on a fourth-order PDE model. The method is tested on a broad range of real medical magnetic resonance images, both in space and time, as well as on nonmedical synthesized test images. Our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail, even at poor signal-to-noise ratio. We have also compared our method with related PDE models.

Abstract. In image processing, the Rudin-Osher-Fatemi (ROF) model [L. Rudin, S. Osher, and E. Fatemi, Physica D, 60(1992), pp. 259-268] based on total variation (TV) minimization has proven to be very useful. A lot of efforts have been devoted to obtain fast numerical schemes and overcome the non-differentiability of the model. Methods considered to be particularly efficient for the ROF model include the dual methods of Chan-Golub-Mulet (CGM) [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20(1999), pp. 1964-1977 . In this paper, we propose to use augmented Lagrangian method to solve the model. Convergence analysis will be given for the method. In addition, we observe close connections between the method proposed here and some of the existing methods. We show that the augmented Lagrangian method, dual methods, and split Bregman iteration are different iterative procedures to solve the same system. Moreover, the proposed method is extended to vectorial TV and high order models. Using the approach here, we can easily obtain the CGM dual method and split Bregman iteration for vectorial TV and high order models, which, to our best of knowledge, have not been presented in the literature. Numerical examples demonstrate the efficiency and accuracy of our method.Key words. augmented Lagrangian method, dual method, split Bregman iteration, ROF model, total variation AMS subject classifications.1. Introduction. Image restoration such as denoising and deblurring are the most fundamental tasks in image processing. To preserve image edges and features in image regularization is difficult but very desired. Recently, the ROF model [33] has been demonstrated very successful in edge-preserving image restoration. The model immediately attracted much attention and has been extended to high order models [13,46,27,29,24,35] and vectorial models for color image restoration [34,2,4,14]; see [15] for an overview.However, the numerical computation of the ROF model suffers from difficulties related to its nonlinearity and non-differentiability. In [33], the authors proposed a time marching strategy to the associated Euler-Lagrange equation. This method is slow due to the constraint of stability conditions about the time step size. To find fast algorithms has been an active research area so far.There are several methods that have proven to be particularly efficient for image restoration problems based on the ROF model. One class of approaches is dual methods [12,9,11,48], which are based on dual formulation of the ROF model. The other is based on variable-splitting and equality constrained optimization, e.g., the approach proposed in [39,40,42] which uses alternative minimization of the penalized cost functional, and the method in [25] where splitting is applied to the data fidelity term, as well as split Bregman iteration [43,22]. In this paper, we use a technique related to the augmented Lagrangian method to solve the ROF model. Convergence analysis of the proposed approach will be supplied. In addition, we show that the

In this paper, we propose a PDE-based level set method. Traditionally, interfaces are represented by the zero level set of continuous level set functions. Instead, we let the interfaces be represented by discontinuities of piecewise constant level set functions. Each level set function can at convergence only take two values, i.e., it can only be 1 or -1; thus, our method is related to phase-field methods. Some of the properties of standard level set methods are preserved in the proposed method, while others are not. Using this new method for interface problems, we need to minimize a smooth convex functional under a quadratic constraint. The level set functions are discontinuous at convergence, but the minimization functional is smooth. We show numerical results using the method for segmentation of digital images.

Finite element methods for a family of systems of singular perturbation problems of a saddle point structure are discussed. The system is approximately a linear Stokes problem when the perturbation parameter is large, while it degenerates to a mixed formulation of Poisson's equation as the perturbation parameter tends to zero. It is established, basically by numerical experiments, that most of the proposed finite element methods for Stokes problem or the mixed Poisson's system are not well behaved uniformly in the perturbation parameter. This is used as the motivation for introducing a new "robust" finite element which exhibits this property.

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Abstract. In this paper we propose a variant of the level set formulation for identifying curves separating regions into different phases. In classical level set approaches, the sign of n level set functions are utilized to identify up to 2 n phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If 2 n phases should be identified, the level set function must approach 2 n predetermined constants. We just need one level set function to represent 2 n unique phases, and this gains in storage capacity. Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is locally convex and differentiable and thus avoids some of the problems with the nondifferentiability of the Delta and Heaviside functions. Numerical examples are given, and we also compare our method with related approaches.

This paper is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are tackling the optimal labeling problem in a direct manner, we first propose a novel dual model and then build up a corresponding dualitybased approach. By analyzing the dual formulation, sufficient conditions are derived which show that the relaxation is often exact, i.e. there exists optimal solutions that are also globally optimal to the original nonconvex Potts model. In order to deal with the nonsmooth dual problem, we develop a smoothing method based on the log-sum exponential function and indicate that such a smoothing approach leads to a novel smoothed primal-dual model and suggests labelings with maximum entropy. Such a smoothing method for the dual model also yields a new thresholding scheme to ob- tain approximate solutions. An expectation maximization like algorithm is proposed based on the smoothed formulation which is shown to be superior in efficiency compared to earlier approaches from continuous optimization. Numerical experiments also show that our method outperforms several competitive approaches in various aspects, such as lower energies and better visual quality.

A noise removal technique using partial differential equations (PDEs) is proposed here. It combines the Total Variational (TV) filter with a fourth-order PDE filter. The combined technique is able to preserve edges and at the same time avoid the staircase effect in smooth regions. A weighting function is used in an iterative way to combine the solutions of the TV-filter and the fourth-order filter. Numerical experiments confirm that the new method is able to use less restrictive time step than the fourth-order filter. Numerical examples using images with objects consisting of edge, flat and intermediate regions illustrate advantages of the proposed model.