Chunlin Wu

Augmented Lagrangian Method, Dual Methods, and Split Bregman Iteration for ROF, Vectorial TV, and High Order Models

In image processing, the Rudin-Osher-Fatemi (ROF) model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268] based on total variation (TV) minimization has proven to be very useful. So far many researchers have contributed to designing fast numerical schemes and overcoming the nondifferentiability of the model. Methods considered to be particularly efficient for the ROF model include the Chan-Golub-Mulet (CGM) primal-dual method [T.F. Chan, G.H. Golub, and P. Mulet, SIAM J. Sci. Comput., 20 (1999), pp. 1964-1977], Chambolles dual method [A. Chambolle, J. Math. Imaging Vis., 20 (2004), pp. 89-97], the splitting and quadratic penalty-based method [Y. Wang, J. Yang, W. Yin, and Y. Zhang, SIAM J. Imaging Sci., 1 (2008), pp. 248-272], and the split Bregman iteration [T. Goldstein and S. Osher, SIAM J. Imaging Sci., 2 (2009), pp. 323-343], as well as the augmented Lagrangian method [X.C. Tai and C. Wu, Lecture Notes in Comput. Sci. 5567, Springer-Verlag, Berlin, 2009, pp. 502-513]. In this paper, we first review the augmented Lagrangian method for the ROF model and then provide some convergence analysis and extensions to vectorial TV and high order models. All the algorithms and analysis will be presented in the discrete setting, which is much clearer for practical implementation than the continuous setting as in Tai and Wu, above. We also present, in the discrete setting, the connections between the augmented Lagrangian method, the dual methods, and the split Bregman iteration. Using our extensions and observations, we can easily figure out CGM and the split Bregman iteration for vectorial TV and high order models, which, to the best of our knowledge, have not been presented in the literature. Numerical examples demonstrate the efficiency and accuracy of our method, especially in the image deblurring case.

Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model

In the recent decades the ROF model (total variation (TV) minimization) has made great successes in image restoration due to its good edge-preserving property. However, the non-differentiability of the minimization problem brings computational difficulties. Different techniques have been proposed to overcome this difficulty. Therein methods regarded to be particularly efficient include dual methods of CGM (Chan, Golub, and Mulet) [7] Chambolle [6] and split Bregman iteration [14], as well as splitting-and-penalty based method [28] [29]. In this paper, we show that most of these methods can be classified under the same framework. The dual methods and split Bregman iteration are just different iterative procedures to solve the same system resulted from a Lagrangian and penalty approach. We only show this relationship for the ROF model. However, it provides a uniform framework to understand these methods for other models. In addition, we provide some examples to illustrate the accuracy and efficiency of the proposed algorithm.

Variational mesh decomposition

The problem of decomposing a 3D mesh into meaningful segments (or parts) is of great practical importance in computer graphics. This article presents a variational mesh decomposition algorithm that can efficiently partition a mesh into a prescribed number of segments. The algorithm extends the Mumford-Shah model to 3D meshes that contains a data term measuring the variation within a segment using eigenvectors of a dual Laplacian matrix whose weights are related to the dihedral angle between adjacent triangles and a regularization term measuring the length of the boundary between segments. Such a formulation simultaneously handles segmentation and boundary smoothing, which are usually two separate processes in most previous work. The efficiency is achieved by solving the Mumford-Shah model through a saddle-point problem that is solved by a fast primal-dual method. A preprocess step is also proposed to determine the number of segments that the mesh should be decomposed into. By incorporating this preprocessing step, the proposed algorithm can automatically segment a mesh into meaningful parts. Furthermore, user interaction is allowed by incorporating the users inputs into the variational model to reflect the users special intention. Experimental results show that the proposed algorithm outperforms competitive segmentation methods when evaluated on the Princeton Segmentation Benchmark.

A Level Set Formulation of Geodesic Curvature Flow on Simplicial Surfaces

Curvature flow (planar geometric heat flow) has been extensively applied to image processing, computer vision, and material science. To extend the numerical schemes and algorithms of this flow on surfaces is very significant for corresponding motions of curves and images defined on surfaces. In this work, we are interested in the geodesic curvature flow over triangulated surfaces using a level set formulation. First, we present the geodesic curvature flow equation on general smooth manifolds based on an energy minimization of curves. The equation is then discretized by a semi-implicit finite volume method (FVM). For convenience of description, we call the discretized geodesic curvature flow as dGCF. The existence and uniqueness of dGCF are discussed. The regularization behavior of dGCF is also studied. Finally, we apply our dGCF to three problems: the closed-curve evolution on manifolds, the discrete scale-space construction, and the edge detection of images painted on triangulated surfaces. Our method works for compact triangular meshes of arbitrary geometry and topology, as long as there are no degenerate triangles. The implementation of the method is also simple.

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

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 sparse linear systems and closed form solutions to subproblems. Experiments on both gray and color images demonstrate the efficiency of our algorithms.

Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space

Mesh surface denoising is a fundamental problem in geometry processing. The main challenge is to remove noise while preserving sharp features (such as edges and corners) and preventing generating false edges. We propose in this paper to combine total variation (TV) and piecewise constant function space for variational mesh denoising. We first give definitions of piecewise constant function spaces and associated operators. A variational mesh denoising method will then be presented by combining TV and piecewise constant function space. It is proved that, the solution of the variational problem (the key part of the method) is in some sense continuously dependent on its parameter, indicating that the solution is robust to small perturbations of this parameter. To solve the variational problem, we propose an efficient iterative algorithm (with an additional algorithmic parameter) based on variable splitting and augmented Lagrangian method, each step of which has closed form solution. Our denoising method is discussed and compared to several typical existing methods in various aspects. Experimental results show that our method outperforms all the compared methods for both CAD and non-CAD meshes at reasonable costs. It can preserve different levels of features well, and prevent generating false edges in most cases, even with the parameters evaluated by our estimation formulae.Mesh surface denoising is a fundamental problem in geometry processing. The main challenge is to remove noise while preserving sharp features (such as edges and corners) and preventing generating false edges. We propose in this paper to combine total variation (TV) and piecewise constant function space for variational mesh denoising. We first give definitions of piecewise constant function spaces and associated operators. A variational mesh denoising method will then be presented by combining TV and piecewise constant function space. It is proved that, the solution of the variational problem (the key part of the method) is in some sense continuously dependent on its parameter, indicating that the solution is robust to small perturbations of this parameter. To solve the variational problem, we propose an efficient iterative algorithm (with an additional algorithmic parameter) based on variable splitting and augmented Lagrangian method, each step of which has closed form solution. Our denoising method is discussed and compared to several typical existing methods in various aspects. Experimental results show that our method outperforms all the compared methods for both CAD and non-CAD meshes at reasonable costs. It can preserve different levels of features well, and prevent generating false edges in most cases, even with the parameters evaluated by our estimation formulae.

Mesh Snapping: Robust Interactive Mesh Cutting Using Fast Geodesic Curvature Flow

This paper considers the problem of interactively finding the cutting contour to extract components from a given mesh. Some existing methods support cuts of arbitrary shape but require careful and tedious input from the user. Others need little user input however they are sensitive to user input and need a postprocessing step to smooth the generated jaggy cutting contours. The popular geometric snake can be used to optimize the cutting contour, but it cannot deal with the topology change. In this paper, we propose a geodesic curvature flow based framework to overcome all these problems. Since in many cases the meaningful cutting contour on a 3D mesh is locally shortest in the sense of some weighted curve length, the geodesic curvature flow is an ideal tool for our problem. It evolves the cutting contour to the nearby local minimum. We should mention that the previous numerical scheme, discretized geodesic curvature flow (dGCF) is too slow and has not been applied to mesh segmentation. With a careful observation to dGCF, we devise here a fast computation scheme called fast geodesic curvature flow (FGCF), which only needs to solve a smaller and easier problem. The initial cutting contour is generated by a variant of random walks algorithm, which is very fast and gives reasonable cutting result with little user input. Experiment results on the benchmark mesh segmentation data set show that our proposed framework is robust to user input and capable of producing good results reflecting geometric features and human shape perception.

The

We propose a new variant of the Mumford-Shah model for simultaneous bias correction and segmentation of images with intensity inhomogeneity. First, based on the model of images with intensity inhomogeneity, we introduce an L0 gradient regularizer to model the true intensity and a smooth regularizer to model the bias field. In addition, we derive a new data fidelity using the local intensity properties to allow the bias field to be influenced by its neighborhood. Second, we use a two-stage segmentation method, where the fast alternating direction method is implemented in the first stage for the recovery of true intensity and bias field and a simple thresholding is used in the second stage for segmentation. Different from most of the existing methods for simultaneous bias correction and segmentation, we estimate the bias field and true intensity without fixing either the number of the regions or their values in advance. Our method has been validated on medical images of various modalities with intensity inhomogeneity. Compared with the state-of-art approaches and the well-known brain software tools, our model is fast, accurate, and robust with initializations.

L_{0}

Discrete filtering of information over triangulated surfaces has proved very useful in computer graphics applications. This technique is based on diffusion equations and has been extensively applied to image processing, harmonic map regularization and texture generating, etc. [C. L. Bajaj and G. Xu, ACM Trans. Graph., 22 (2003), pp. 4-32], [C. Wu, J. Deng, and F. Chen, IEEE Trans. Vis. Comput. Graph., 14 (2008), pp. 666-679]. However, little has been done on analysis (especially quantitative analysis) of the behavior of these filtering procedures. Since in applications mesh surfaces can be of arbitrary topology and the filtering can be nonlinear and even anisotropic, the analysis of the quantitative behavior is a very difficult issue. In this paper, we first present the discrete linear, nonlinear, and anisotropic filtering schemes via discretizing diffusion equations with appropriately defined differential operators on triangulated surfaces, and then use concepts of discrete scale-spaces to describe these filtering procedures and analyze their properties respectively. Scale-space properties such as existence and uniqueness, continuous dependence on initial value, discrete semigroup property, grey level shift invariance and conservation of total grey level, information reduction (also known as topology simplification), and constant limit behavior have been proved. In particular, the information reduction property is analyzed by eigenvalue and eigenvector analysis of matrices. Different from the direct observation of the local filtering to the diffusion equations and other interpretation methods based on wholly global quantities such as energy and entropy, this viewpoint helps us understand the filtering both globally (information reduction as image components shrink) and locally (how the image component contributes to its shrink rate). With careful consideration of the correspondence between eigenvalues and eigenvectors and their features, differences between linear and nonlinear filtering, as well as between isotropic and anisotropic filtering, are discussed. We also get some stability results of the filtering schemes. Several examples are provided to illustrate the properties.

Regularized Mumford–Shah Model for Bias Correction and Segmentation of Medical Images

In computer graphics, triangular mesh representations of surfaces have become very popular. Compared with parametric and implicit forms of surfaces, triangular mesh surfaces have many advantages such as being easy to render, being convenient to store, and having the ability to model geometric objects with arbitrary topology. In this paper, we are interested in data processing over triangular mesh surfaces through partial differential equations (PDEs). We study several diffusion equations over triangular mesh surfaces and present corresponding numerical schemes to solve them. Our methods work for triangular mesh surfaces with arbitrary geometry (the angles of each triangle are arbitrary) and topology (open meshes or closed meshes of arbitrary genus). Besides the flexibility, our methods are efficient due to the implicit/semi-implicit time discretization. We finally apply our methods to several filtering and texture applications such as image processing, texture generation, and regularization of harmonic maps over triangular mesh surfaces. The results demonstrate the flexibility and effectiveness of our methods.