Xue-Cheng Tai

Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time

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.

A binary level set model and some applications to Mumford-Shah image segmentation

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.

Scale Space and Variational Methods in Computer Vision

Segmentation and Detection.- Graph Cut Optimization for the Piecewise Constant Level Set Method Applied to Multiphase Image Segmentation.- Tubular Anisotropy Segmentation.- An Unconstrained Multiphase Thresholding Approach for Image Segmentation.- Extraction of the Intercellular Skeleton from 2D Images of Embryogenesis Using Eikonal Equation and Advective Subjective Surface Method.- On Level-Set Type Methods for Recovering Piecewise Constant Solutions of Ill-Posed Problems.- The Nonlinear Tensor Diffusion in Segmentation of Meaningful Biological Structures from Image Sequences of Zebrafish Embryogenesis.- Composed Segmentation of Tubular Structures by an Anisotropic PDE Model.- Extrapolation of Vector Fields Using the Infinity Laplacian and with Applications to Image Segmentation.- A Schrodinger Equation for the Fast Computation of Approximate Euclidean Distance Functions.- Semi-supervised Segmentation Based on Non-local Continuous Min-Cut.- Momentum Based Optimization Methods for Level Set Segmentation.- Optimization of Divergences within the Exponential Family for Image Segmentation.- Convex Multi-class Image Labeling by Simplex-Constrained Total Variation.- Geodesically Linked Active Contours: Evolution Strategy Based on Minimal Paths.- Validation of Watershed Regions by Scale-Space Statistics.- Adaptation of Eikonal Equation over Weighted Graph.- A Variational Model for Interactive Shape Prior Segmentation and Real-Time Tracking.- Image Enhancement and Reconstruction.- A Nonlinear Probabilistic Curvature Motion Filter for Positron Emission Tomography Images.- Finsler Geometry on Higher Order Tensor Fields and Applications to High Angular Resolution Diffusion Imaging.- Bregman-EM-TV Methods with Application to Optical Nanoscopy.- PDE-Driven Adaptive Morphology for Matrix Fields.- On Semi-implicit Splitting Schemes for the Beltrami Color Flow.- Multi-scale Total Variation with Automated Regularization Parameter Selection for Color Image Restoration.- Multiplicative Noise Cleaning via a Variational Method Involving Curvelet Coefficients.- Projected Gradient Based Color Image Decomposition.- A Dual Formulation of the TV-Stokes Algorithm for Image Denoising.- Anisotropic Regularization for Inverse Problems with Application to the Wiener Filter with Gaussian and Impulse Noise.- Locally Adaptive Total Variation Regularization.- Basic Image Features (BIFs) Arising from Approximate Symmetry Type.- An Anisotropic Fourth-Order Partial Differential Equation for Noise Removal.- Enhancement of Blurred and Noisy Images Based on an Original Variant of the Total Variation.- Coarse-to-Fine Image Reconstruction Based on Weighted Differential Features and Background Gauge Fields.- Edge-Enhanced Image Reconstruction Using (TV) Total Variation and Bregman Refinement.- Nonlocal Variational Image Deblurring Models in the Presence of Gaussian or Impulse Noise.- A Geometric PDE for Interpolation of M-Channel Data.- An Edge-Preserving Multilevel Method for Deblurring, Denoising, and Segmentation.- Fast Dejittering for Digital Video Frames.- Sparsity Regularization for Radon Measures.- Split Bregman Algorithm, Douglas-Rachford Splitting and Frame Shrinkage.- Anisotropic Smoothing Using Double Orientations.- Image Denoising Using TV-Stokes Equation with an Orientation-Matching Minimization.- Augmented Lagrangian Method, Dual Methods and Split Bregman Iteration for ROF Model.- The Convergence of a Central-Difference Discretization of Rudin-Osher-Fatemi Model for Image Denoising.- Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering.- L 0-Norm and Total Variation for Wavelet Inpainting.- Total-Variation Based Piecewise Affine Regularization.- Image Denoising by Harmonic Mean Curvature Flow.- Motion Analysis, Optical Flow, Registration and Tracking.- Tracking Closed Curves with Non-linear Stochastic Filters.- A Multi-scale Feature Based Optic Flow Method for 3D Cardiac Motion Estimation.- A Combined Segmentation and Registration Framework with a Nonlinear Elasticity Smoother.- A Scale-Space Approach to Landmark Constrained Image Registration.- A Variational Approach for Volume-to-Slice Registration.- Hyperbolic Numerics for Variational Approaches to Correspondence Problems.- Surfaces and Shapes.- From a Single Point to a Surface Patch by Growing Minimal Paths.- Optimization of Convex Shapes: An Approach to Crystal Shape Identification.- An Implicit Method for Interpolating Two Digital Closed Curves on Parallel Planes.- Pose Invariant Shape Prior Segmentation Using Continuous Cuts and Gradient Descent on Lie Groups.- A Non-local Approach to Shape from Ambient Shading.- An Elasticity Approach to Principal Modes of Shape Variation.- Pre-image as Karcher Mean Using Diffusion Maps: Application to Shape and Image Denoising.- Fast Shape from Shading for Phong-Type Surfaces.- Generic Scene Recovery Using Multiple Images.- Scale Space and Feature Extraction.- Highly Accurate PDE-Based Morphology for General Structuring Elements.- Computational Geometry-Based Scale-Space and Modal Image Decomposition.- Highlight on a Feature Extracted at Fine Scales: The Pointwise Lipschitz Regularity.- Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2).- Spatio-Featural Scale-Space.- Scale Spaces on the 3D Euclidean Motion Group for Enhancement of HARDI Data.- On the Rate of Structural Change in Scale Spaces.- Transitions of a Multi-scale Image Hierarchy Tree.- Local Scale Measure for Remote Sensing Images.

Iterative Image Restoration Combining Total Variation Minimization and a Second-Order Functional

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.

Noise removal using smoothed normals and surface fitting

In this work, we use partial differential equation techniques to remove noise from digital images. The removal is done in two steps. We first use a total-variation filter to smooth the normal vectors of the level curves of a noise image. After this, we try to find a surface to fit the smoothed normal vectors. For each of these two stages, the problem is reduced to a nonlinear partial differential equation. Finite difference schemes are used to solve these equations. A broad range of numerical examples are given in the paper.

A study on continuous max-flow and min-cut approaches

We propose and study novel max-flow models in the continuous setting, which directly map the discrete graph-based max-flow problem to its continuous optimization formulation. We show such a continuous max-flow model leads to an equivalent min-cut problem in a natural way, as the corresponding dual model. In this regard, we revisit basic conceptions used in discrete max-flow / min-cut models and give their new explanations from a variational perspective. We also propose corresponding continuous max-flow and min-cut models constrained by priori supervised information and apply them to interactive image segmentation/labeling problems. We prove that the proposed continuous max-flow and min-cut models, with or without supervised constraints, give rise to a series of global binary solutions λ∗(x) ∊ {0,1}, which globally solves the original nonconvex image partitioning problems. In addition, we propose novel and reliable multiplier-based max-flow algorithms. Their convergence is guaranteed by classical optimization theories. Experiments on image segmentation, unsupervised and supervised, validate the effectiveness of the discussed continuous max-flow and min-cut models and suggested max-flow based algorithms.

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.

Global Minimization for Continuous Multiphase Partitioning Problems Using a Dual Approach

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 duality-based 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 obtain 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.

Identification of Discontinuous Coefficients in Elliptic Problems Using Total Variation Regularization

We propose several formulations for recovering discontinuous coefficients in elliptic problems by using total variation (TV) regularization. The motivation for using TV is its well-established ability to recover sharp discontinuities. We employ an augmented Lagrangian variational formulation for solving the output-least-squares inverse problem. In addition to the basic output-least-squares formulation, we introduce two new techniques for handling large observation errors. First, we use a filtering step to remove as much of the observation error as possible. Second, we introduce two extensions of the output-least-squares model; one model employs observations of the gradient of the state variable while the other uses the flux. Numerical experiments indicate that the combination of these two techniques enables us to successfully recover discontinuous coefficients even under large observation errors.

Global and uniform convergence of subspace correction methods for some convex optimization problems

This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.