Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

Compressed sensing

Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

Gmsh is an open-source 3-D finite element grid generator with a build-in CAD engine and post-processor. Its design goal is to provide a fast, light and user-friendly meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright (C) 2009 John Wiley & Sons, Ltd.

Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities

The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

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The quickhull algorithm for convex hulls

Humans perceive the three-dimensional structure of the world with apparent ease. However, despite all of the recent advances in computer vision research, the dream of having a computer interpret an image at the same level as a two-year old remains elusive. Why is computer vision such a challenging problem and what is the current state of the art? Computer Vision: Algorithms and Applications explores the variety of techniques commonly used to analyze and interpret images. It also describes challenging real-world applications where vision is being successfully used, both for specialized applications such as medical imaging, and for fun, consumer-level tasks such as image editing and stitching, which students can apply to their own personal photos and videos. More than just a source of recipes, this exceptionally authoritative and comprehensive textbook/reference also takes a scientific approach to basic vision problems, formulating physical models of the imaging process before inverting them to produce descriptions of a scene. These problems are also analyzed using statistical models and solved using rigorous engineering techniques Topics and features: structured to support active curricula and project-oriented courses, with tips in the Introduction for using the book in a variety of customized courses; presents exercises at the end of each chapter with a heavy emphasis on testing algorithms and containing numerous suggestions for small mid-term projects; provides additional material and more detailed mathematical topics in the Appendices, which cover linear algebra, numerical techniques, and Bayesian estimation theory; suggests additional reading at the end of each chapter, including the latest research in each sub-field, in addition to a full Bibliography at the end of the book; supplies supplementary course material for students at the associated website, http://szeliski.org/Book/. Suitable for an upper-level undergraduate or graduate-level course in computer science or engineering, this textbook focuses on basic techniques that work under real-world conditions and encourages students to push their creative boundaries. Its design and exposition also make it eminently suitable as a unique reference to the fundamental techniques and current research literature in computer vision.

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Computer Vision: Algorithms and Applications

The Conjugate Gradient Method is the most prominent iterative method for solving sparse systems of linear equations. Unfortunately, many textbook treatments of the topic are written so that even their own authors would be mystified, if they bothered to read their own writing. For this reason, an understanding of the method has been reserved for the elite brilliant few who have painstakingly decoded the mumblings of their forebears. Nevertheless, the Conjugate Gradient Method is a composite of simple, elegant ideas that almost anyone can understand. Of course, a reader as intelligent as yourself will learn them almost effortlessly. The idea of quadratic forms is introduced and used to derive the methods of Steepest Descent, Conjugate Directions, and Conjugate Gradients. Eigenvectors are explained and used to examine the convergence of the Jacobi Method, Steepest Descent, and Conjugate Gradients. Other topics include preconditioning and the nonlinear Conjugate Gradient Method. I have taken pains to make this article easy to read. Sixty-two illustrations are provided. Dense prose is avoided. Concepts are explained in several different ways. Most equations are coupled with an intuitive interpretation.

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An Introduction to the Conjugate Gradient Method Without the Agonizing Pain

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

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Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles Unfortunately, this problem is not always soluble A compromise is necessary A Delaunay refinement algorithm is presented that can create a mesh in which most angles are 30^o or greater and no angle is smaller than arcsin[(3/2)sin(@f/2)]~(3/4)@f, where @f=<60^ois the smallest angle separating two segments of the input domain New angles smaller than 30^o appear only near input angles smaller than 60^o In practice, the algorithm's performance is better than these bounds suggest Another new result is that Ruppert's analysis technique can be used to reanalyze one of Chew's algorithms Chew proved that his algorithm produces no angle smaller than 30^o (barring small input angles), but without any guarantees on grading or number of triangles He conjectures that his algorithm offers such guarantees His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than 265^o, Chew's algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal

Delaunay refinement algorithms for triangular mesh generation

Exact computer arithmetic has a variety of uses, including the robust implementation of geometric algorithms. This article has three purposes. The first is to offer fast software-level algorithms for exact addition and multiplication of arbitrary precision floating-point values. The second is to propose a technique for adaptive precision arithmetic that can often speed these algorithms when they are used to perform multiprecision calculations that do not always require exact arithmetic, but must satisfy some error bound. The third is to use these techniques to develop implementations of several common geometric calculations whose required degree of accuracy depends on their inputs. These robust geometric predicates are adaptive; their running time depends on the degree of uncertainty of the result, and is usually small.

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Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates,

When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This paper explains the mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these two purposes do not agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation. Several of the upper and lower bounds on interpolation errors and element stiffness matrix conditioning given here are tighter than those that have appeared in the literature before, so the quality measures are likely to be unusually precise indicators of element fitness.

Jonathan Richard Shewchuk

Papers

An Introduction to the Conjugate Gradient Method Without the Agonizing Pain

Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

Delaunay refinement algorithms for triangular mesh generation

Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates,

What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures.