Showing papers by "Stanley Osher published in 2004"

Convergence rates of convex variational regularization

Abstract: The aim of this paper is to provide quantitative estimates for the minimizers of non-quadratic regularization problems in terms of the regularization parameter, respectively the noise level. As usual for ill-posed inverse problems, these estimates can be obtained only under additional smoothness assumptions on the data, the so-called source conditions, which we identify with the existence of Lagrange multipliers for a limit problem. Under such a source condition, we shall prove a quantitative estimate for the Bregman distance induced by the regularization functional, which turns out to be the natural distance measure to use in this case. We put a special emphasis on the case of total variation regularization, which is probably the most important and prominent example in this class. We discuss the source condition for this case in detail and verify that it still allows discontinuities in the solution, while imposing some regularity on its level sets.

Decomposition of images by the anisotropic Rudin-Osher-Fatemi model

Abstract: The total-variation-based image denoising model of Rudin, Osher, and Fatemi can be generalized in a natural way to favor certain edge directions. We consider the resulting anisotropic energies and study properties of their minimizers. © 2004 Wiley Periodicals, Inc.

Noise removal using smoothed normals and surface fitting

Abstract: 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.

Image Denoising and Decomposition with Total Variation Minimization and Oscillatory Functions

Abstract: In this paper, we propose a new variational model for image denoising and decomposition, witch combines the total variation minimization model of Rudin, Osher and Fatemi from image restoration, with spaces of oscillatory functions, following recent ideas introduced by Meyer. The spaces introduced here are appropriate to model oscillatory patterns of zero mean, such as noise or texture. Numerical results of image denoising, image decomposition and texture discrimination are presented, showing that the new models decompose better a given image, possible noisy, into cartoon and oscillatory pattern of zero mean, than the standard ones. The present paper develops further the models previously introduced by the authors in Vese and Osher (Modeling textures with total variation minimization and oscillating patterns in image processing, UCLA CAM Report 02-19, May 2002, to appear in Journal of Scientific Computing, 2003). Other recent and related image decomposition models are also discussed.

Fast Sweeping Methods for Static Hamilton--Jacobi Equations

Abstract: We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in two and three dimensions.

Kernel density estimation and intrinsic alignment for knowledge-driven segmentation: Teaching level sets to walk

Abstract: We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose a novel multi-modal statistical shape prior which allows to encode multiple fairly distinct training shapes. This prior is based on an extension of classical kernel density estimators to the level set domain. Secondly, we propose an intrinsic registration of the evolving level set function which induces an invariance of the proposed shape energy with respect to translation. We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person.

Kernel Density Estimation and Intrinsic Alignment for Knowledge-Driven Segmentation: Teaching Level Sets to Walk

Abstract: We address the problem of image segmentation with statistical shape priors in the context of the level set framework. Our paper makes two contributions: Firstly, we propose a novel multi-modal statistical shape prior which allows to encode multiple fairly distinct training shapes. This prior is based on an extension of classical kernel density estimators to the level set domain. Secondly, we propose an intrinsic registration of the evolving level set function which induces an invariance of the proposed shape energy with respect to translation. We demonstrate the advantages of this multi-modal shape prior applied to the segmentation and tracking of a partially occluded walking person.

Inverse Problem Techniques for the Design of Photonic Crystals

Abstract: This paper provides a review on the optimal design of photonic bandgap structures by inverse problem techniques. An overview of inverse problems techniques is given, with a special focus on topology design methods. A review of first applications of inverse problems techniques to photonic bandgap structures and waveguides is given, as well as some model problems, which provide a deeper insight into the structure of the optimal design problems.

A level set method for three-dimensional paraxial geometrical optics with multiple point sources ⁄

Abstract: We apply the level set method to compute the three dimensional multivalued ge- ometrical optics term in a paraxial formulation. The paraxial formulation is obtained from the 3-D stationary eikonal equation by using one of the spatial directions as the artiflcial evolution direction. The advection velocity fleld used to move level sets is obtained by the method of char- acteristics; therefore the motion of level sets is deflned in phase space. The multivalued travel-time and amplitude-related quantity are obtained from solving advection equations with source terms. We derive an amplitude formula in a reduced phase space which is very convenient to use in the level set framework. By using a semi-Lagrangian method in the paraxial formulation, the method has O(N 2 ) rather than O(N 4 ) memory storage requirement for up to O(N 2 ) multiple point sources in the flve dimensional phase space, where N is the number of mesh points along one direction. Although the computational complexity is still O(MN 4 ), where M is the number of steps in the ODE solver for the semi-Lagrangian scheme, this disadvantage is largely overcome by the fact that up to O(N 2 ) multiple point sources can be treated simultaneously. Three dimensional numerical examples demonstrate the e-ciency and accuracy of the method.

G-Norm Properties of Bounded Variation Regularization

Abstract: Recently Y. Meyer derived a characterization of the minimizer of the Rudin-Osher- Fatemi functional in a functional analytical framework. In statistics the discrete version of this functional is used to analyze one dimensional data and belongs to the class of nonparametric regres- sion models. In this work we generalize the functional analytical results of Meyer and apply them to a class of regression models, such as quantile, robust, logistic regression, for the analysis of multi- dimensional data. The characterization of Y. Meyer and our generalization is based on G-norm properties of the data and the minimizer. A geometric point of view of regression minimization is provided. whereDudenotes the total variation semi-norm of u and α> 0. The minimizer is called the bounded variation regularized solution. The taut-string algorithm consists in finding a string of minimal length in a tube (with radius α) around the primitive of f . The differentiated string is the taut-string reconstruction and corresponds to the minimizer of the ROF-model. Generalizing these ideas to higher dimensions is complicated by the fact that there is no obvious analog to primitives in higher space dimensions. We overcome this difficulty by solving Laplace's equation with right hand side f (i.e. integrate twice), and differentiating. The tube with radius α around the derivative of the potential specifies all functions u which satisfyu − fGs ≤ α (see also (21)). In this paper we show that the bounded variation regularized solutions (in any number of space dimensions) are contained in a tube of radius α .F or several other regression models in statistics, such as robust, quantile, and logistic regression (reformulated in a Banach space setting for analyzing multi-dimensional data) the

Reflection in a Level Set Framework for Geometric Optics

Abstract: Geometric optics makes its impact both in mathematics and real world applications related to ray tracing, migration, and tomography. Of special importance in these problems are the wavefronts, or points of constant traveltime away from sources, in the medium. Previously in [Osher, Cheng, Kang, Shim, and Tsai (2002)], we initiated a level set approach for the construction of wavefronts in isotropic media that handled the two major algorithmic issues involved with this problem: resolution and multivalued solutions. This approach was quite general and we were able to construct wavefronts in the presence of refraction, reflection, higher dimensions, and, in [Qian, Cheng, and Osher (2003)], anisotropy as well. However, the technique proposed for handling reflections of waves off objects, an important phenomenon involved in all applications of geometric optics, was inefficient and unwieldy to the point of being unusable, especially in the presence of multiple reflections. We introduce here an alternative approach based on the foundation presented in [Osher, Cheng, Kang, Shim, and Tsai (2002)]. This reworking allows the level set method to be considered for realistic applications involving reflecting surfaces in geometric optics.

Using geometry and iterated refinement for inverse problems (1): total variation based image restoration

Abstract: We introduce a new iterative regularization procedure for inverse problems based on the use of Bregman distances, with particular focus on problems arising in image processing. We are motivated by the problem of restoring noisy and blurry images via variational methods, specifically by using the BV seminorm. Although our procedure applies in quite general situations it was obtained by geometric considerations (first discussed in [23]) associated with the Rudin-OsherFatemi procedure developed in [29] for image restoration. We obtain rigorous convergence results, and effective stopping criteria for the general procedure. The numerical results for denoising appear to be state-of-the-art and preliminary results for deblurring/denoising are very encouraging.

Inverse Problem Techniques for the Design of Photonic Crystals (INVITED)

Abstract: This paper provides a review on the optimal design of photonic bandgap structures by inverse problem techniques. An overview of inverse problems techniques is given, with a special focus on topology design methods. A review of first applications of inverse problems techniques to photonic bandgap structures and waveguides is given, as well as some model problems, which provide a deeper insight into the structure of the optimal design problems.

Level set and PDE methods for computer graphics

Abstract: Level set methods, an important class of partial differential equation (PDE) methods, define dynamic surfaces implicitly as the level set (iso-surface) of a sampled, evolving nD function. The course begins with preparatory material that introduces the concept of using partial differential equations to solve problems in computer graphics, geometric modeling and computer vision. This will include the structure and behavior of several different types of differential equations, e.g. the level set equation and the heat equation, as well as a general approach to developing PDE-based applications. The second stage of the course will describe the numerical methods and algorithms needed to actually implement the mathematics and methods presented in the first stage. The course closes with detailed presentations on several level set/PDE applications, including image/video inpainting, pattern formation, image/volume processing, 3D shape reconstruction, image/volume segmentation, image/shape morphing, geometric modeling, anisotropic diffusion, and natural phenomena simulation.

Nonlinear regularizations of TV based PDEs for image processing

Abstract: We consider both second order and fourth order TV-based PDEs for image processing in one space dimension. A general class of nonlinear regularization of the TV functional result in well-posed uniformly parabolic equations in two dimensions. However for the fourth order analogue (Osher et. al. Multiscale Methods and Simulation 1(3) 2003) based on a total variation minimization in an H 1 norm, has very different properties. In particular, nonlinear regularizations should have special structure in order to guarantee that the regularized PDE does not produce finite time singularities.

INVITED PAPER Special Section on Photonic Crystals and Their Device Applications Inverse Problem Techniques for the Design of Photonic Crystals

Abstract: SUMMARY This paper provides a review on the optimal design of photonic bandgap structures by inverse problem techniques. An overview of inverse problems techniques is given, with a special focus on topology design methods. A review of first applications of inverse problems techniques to photonic bandgap structures and waveguides is given, as well as some model problems, which provide a deeper insight into the structure of the optimal design problems.