Showing papers by "Stanley Osher published in 2006"

Structure-Texture Image Decomposition--Modeling, Algorithms, and Parameter Selection

Abstract: This paper explores various aspects of the image decomposition problem using modern variational techniques. We aim at splitting an original image f into two components u and ?, where u holds the geometrical information and ? holds the textural information. The focus of this paper is to study different energy terms and functional spaces that suit various types of textures. Our modeling uses the total-variation energy for extracting the structural part and one of four of the following norms for the textural part: L2, G, L1 and a new tunable norm, suggested here for the first time, based on Gabor functions. Apart from the broad perspective and our suggestions when each model should be used, the paper contains three specific novelties: first we show that the correlation graph between u and ? may serve as an efficient tool to select the splitting parameter, second we propose a new fast algorithm to solve the TV ? L1 minimization problem, and third we introduce the theory and design tools for the TV-Gabor model.

Kernel Density Estimation and Intrinsic Alignment for Shape Priors in Level Set Segmentation

Abstract: In this paper, we make two contributions to the field of level set based image segmentation. Firstly, we propose shape dissimilarity measures on the space of level set functions which are analytically invariant under the action of certain transformation groups. The invariance is obtained by an intrinsic registration of the evolving level set function. In contrast to existing approaches to invariance in the level set framework, this closed-form solution removes the need to iteratively optimize explicit pose parameters. The resulting shape gradient is more accurate in that it takes into account the effect of boundary variation on the object's pose. Secondly, based on these invariant shape dissimilarity measures, we propose a statistical shape prior which allows to accurately encode multiple fairly distinct training shapes. This prior constitutes an extension of kernel density estimators to the level set domain. In contrast to the commonly employed Gaussian distribution, such nonparametric density estimators are suited to model aribtrary distributions. We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person in a video sequence, and on the segmentation of the left ventricle in cardiac ultrasound images. We give quantitative results on segmentation accuracy and on the dependency of segmentation results on the number of training shapes.

Nonlinear inverse scale space methods

Abstract: In this paper we generalize the iterated refinement method, introduced by the authors in a recent work, to a time-continuous inverse scale-space formulation. The iterated refinement procedure yields a sequence of convex variational problems, evolving toward the noisy image. The inverse scale space method arises as a limit for a penalization parameter tending to zero, while the number of iteration steps tends to infinity. For the limiting flow, similar properties as for the iterated refinement procedure hold. Specifically, when a discrepancy principle is used as the stopping criterion, the error between the reconstruction and the noise-free image decreases until termination, even if only the noisy image is available and a bound on the variance of the noise is known. The inverse flow is computed directly for one-dimensional signals, yielding high quality restorations. In higher spatial dimensions, we introduce a relaxation technique using two evolution equations. These equations allow fast, accurate, efficient and straightforward implementation. We investigate the properties of these new types of flows and show their excellent denoising capabilities, wherein noise can be well removed with minimal loss of contrast of larger objects.

The Total Variation Regularized L1 Model for Multiscale Decomposition

Abstract: : This paper studies the total variation regularization model with an L1 fidelity term (TV-L1) for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry sub-problems. Using this result we show that the TV-L1 model is able to separate image features according to their scales, where the scale is analytically defined by the G-value. A number of other properties including the geometric and morphological invariance of the TV-L1 model are also proved and their applications discussed.

Color Texture Modeling and Color Image Decomposition in a Variational-PDE Approach

Abstract: This paper is devoted to a new variational model for color texture modeling and color image decomposition into cartoon and texture. A given image f in the RGB space is decomposed into a cartoon part and a texture part. The cartoon part is modeled by the space of vector-valued functions of bounded variation, while the texture or noise part is modeled by a space of oscillatory functions, dual in some sense of the BV space. Examples for color image decomposition, color image denoising, and color texture discrimination and segmentation will be presented.

Denoising by BV-duality

Abstract: In this paper we apply Meyer's G-norm for image processing problems. We use a definition of the G-norm as norm of linear functionals on BV, which seems to be more feasible for numerical computation. We establish the equivalence between Meyer's original definition and ours and show that computing the norm can be expressed as an interface problem. This allows us to define an algorithm based on the level set method for its solution. Alternatively we propose a fixed point method based on mean curvature type equations. A computation of the G-norm according to our definition additionally gives functions which can be used for denoising of simple structures in images under a high level of noise. We present some numerical computations of this denoising method which support this claim.

Iterative Total Variation Regularization with Non-Quadratic Fidelity

Abstract: A generalized iterative regularization procedure based on the total variation penalization is introduced for image denoising models with non-quadratic convex fidelity terms. By using a suitable sequence of penalty parameters we solve the issue of solvability of minimization problems arising in each step of the iterative procedure, which has been encountered in a recently developed iterative total variation procedure Furthermore, we obtain rigorous convergence results for exact and noisy data. We test the behaviour of the algorithm on real images in several numerical experiments using L 1 and L 2 fitting terms. Moreover, we compare the results with other state-of-the art multiscale techniques for total variation based image restoration.

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

Abstract: We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear