We introduce a fast, multiscale algorithm for image segmentation. Our algorithm uses modern numeric techniques to nd an approximate solution to normalized cut measures in time that is linear in the size of the image with only a few dozen operations per pixel. In just one pass the algorithm provides a complete hierarchical decomposition of the image into segments. The algorithm detects the segments by applying a process of recursive coarsening in which the same minimization problem is represented with fewer and fewer variables producing an irregular pyramid. During this coarsening process we may compute additional internal statistics of the emerging segments and use these statistics to facilitate the segmentation process. Once the pyramid is completed it is scanned from the top down to associate pixels close to the boundaries of segments with the appropriate segment. The algorithm is inspired by algebraic multigrid (AMG) solvers of minimization problems of heat or electric networks. We demonstrate the algorithm by applying it to real images.

We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritz-type finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global measures that appear in existing multigrid theory. These new measures are used to determine local representations of algebraically "smooth" error components that provide the basis for constructing effective interpolation and, hence, the coarsening process for AMG. Here, we focus on the interpolation process; choice of the coarse "grids" based on these measures is the subject of current research. We develop a theoretical foundation for AMGe and present numerical results that demonstrate the efficacy of the method.

We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE finds an optimal drawing by minimizing a quadratic energy function due to Hall, using a novel algebraic multigrid technique. The algorithm exhibits an improvement of something like two orders of magnitude over the fastest algorithms we are aware of; it draws graphs of a million nodes in less than a minute. Moreover, the algorithm can deal with more general entities, such as graphs with masses and negative weights (to be defined in the text), and it appears to be applicable outside of graph drawing too.

We present a multiresolution construction for efficiently computing, compressing and applying large powers of operators that have high powers with low numerical rank. This allows the fast computation of functions of the operator, notably the associated Green’s function, in compressed form, and their fast application. Classes of operators satisfying these conditions include diffusion-like operators, in any dimension, on manifolds, graphs, and in non-homogeneous media. In this case our construction can be viewed as a far-reaching generalization of Fast Multipole Methods, achieved through a different point of view, and of the non-standard wavelet representation of Calderón-Zygmund and pseudodifferential operators, achieved through a different multiresolution analysis adapted to the operator. We show how the dyadic powers of an operator can be used to induce a multiresolution analysis, as in classical Littlewood-Paley and wavelet theory, and we show how to construct, with fast and stable algorithms, scaling function and wavelet bases associated to this multiresolution analysis, and the corresponding downsampling operators, and use them to compress the corresponding powers of the operator. This allows to extend multiscale signal processing to general spaces (such as manifolds and graphs) in a very natural way, with corresponding fast algorithms.

Texture segmentation is a difficult problem, as is apparent from camouflage pictures. A Textured region can contain texture elements of various sizes, each of which can itself be textured. We approach this problem using a bottom-up aggregation framework that combines structural characteristics of texture elements with filter responses. Our process adaptively identifies the shape of texture elements and characterize them by their size, aspect ratio, orientation, brightness, etc., and then uses various statistics of these properties to distinguish between different textures. At the same time our process uses the statistics of filter responses to characterize textures. In our process the shape measures and the filter responses crosstalk extensively. In addition, a top-down cleaning process is applied to avoid mixing the statistics of neighboring segments. We tested our algorithm on real images and demonstrate that it can accurately segment regions that contain challenging textures.

Driven by the need to solve linear sytems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigridlike performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarse-grid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [7] for the selection of coarsegrid points, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this pa...

General purely algebraic approaches for repeated coarsening of deterministic or statistical field equations are presented, including a universal way to gauge and control the quality of the coarse-level set of variables, and generic procedures for deriving the coarse-level set of equations. They apply to the equations arising from variational as well as non-variational discretizations of general, elliptic as well as non-elliptic, partial differential systems, on structured or unstructured grids. They apply to many types of disordered systems, such as those arising in composite materials, inhomogeneous ground flows, "twisted geometry" discretizations and Dirac equations in disordered gauge fields, and also to non-PDE systems. The coarsening can be inexpensive with low accuracy, as needed for multigrid solvers, or more expensive and highly accurate, as needed for other applications (e.g., once-for-all derivation of macroscopic equations). Extensions to non-local and highly indefinite (w...

Abstract not found

We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coefficient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem and an oscillatory coefficient problem. We also show using a one-dimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive definite.

Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we demonstrate that range of applicability, while describing some of the recent advances in AMG technology. Moreover, in light of the imperatives of modern computer environments, we also examine AMG in terms of algorithmic scalability. Finally, we show some of the situations in which standard AMG does not work well, and indicate the current directions taken by AMG researchers to alleviate these difficulties.