We present spectral matting: a new approach to natural image matting that automatically computes a set of fundamental fuzzy matting components from the smallest eigenvectors of a suitably defined Laplacian matrix. Thus, our approach extends spectral segmentation techniques, whose goal is to extract hard segments, to the extraction of soft matting components. These components may then be used as building blocks to easily construct semantically meaningful foreground mattes, either in an unsupervised fashion, or based on a small amount of user input. 1.

Network analysis begins with data that describes the set of relationships among the members of a system. The goal of analysis is to obtain from the low-level relational data a higher-level description of the structure of the system which identifies various kinds of patterns in the set of relationships. These patterns will be based on the way individuals are related to other individuals in the network. Some approaches to network analysis look for clusters of individuals who are tightly connected to one another; some look for sets of individuals who have similar patterns of relations to the rest of the network. Other methods don’t “look for ” anything in particular — instead, they construct a continuous multidimensional representation of the network in which the coordinates of the individuals can be further analyzed to obtain a variety of kinds of information about them and their relation to the rest of the network. One approach to this is to choose a set of axes in the multidimensional space occupied by the network and rotate them so that the first axis points in the direction of the greatest variability in the data; the second axis, orthogonal to the first, points in the direction of greatest remaining variability, and so on. This set of axes is a coordinate system that can be used to describe the relative positions

This thesis presents research into parallel linear solvers for block-diagonal-bordered sparse matrices. The block-diagonal-bordered form identifies parallelism that can be exploited for both direct and iterative linear solvers. We have developed efficient parallel block-diagonal-bordered sparse direct methods based on both LU factorization and Choleski factorization algorithms, and we have also developed a parallel block-diagonal-bordered sparse iterative method based on the Gauss-Seidel method. Parallel factorization algorithms for block-diagonal-bordered form matrices require a specialized ordering step coupled to an explicit load balancing step in order to generate this matrix form and to distribute the computational workload uniformly for an irregular matrix throughout a distributed-memory multi-processor. Matrix orderings are performed using a diakoptic technique based on node-tearing-nodal analysis. Parallel Gauss-Seidel algorithms for block-diagonal-bordered form matrices require a two-part matrix ordering technique -- first to partition the matrix into block-diagonal-bordered form, again, using the node-tearing diakoptic techniques and then to multi-color the data in the last diagonal block using graph coloring techniques. The ordered matrices have extensive parallelism, while maintaining the strict precedence relationships in the Gauss-Seidel algorithm. Empirical

A survey of published methods for partitioning sparse arrays is presented. These include early attempts to describe the partitioning properties of eigenvectors of the adjacency matrix. More direct methods of partitioning are developed by introducing the Laplacian of the adjacency matrix via the directed (signed) edge-vertex incidence matrix. It is shown that the Laplacian solves the minimization of total length of connections between adjacent nodes, which induces clustering of connected nodes by partitioning the underlying graph. Another matrix derived from the adjacency matrix is also introduced via the unsigned edge-vertex matrix. This (the Normal) matrix is not symmetric, and it also is shown to solve the minimization of total length in its own non-Euclidean metric. In this case partitions are induced by clustering the connected nodes. The Normal matrix is closely related to Correspondence Analysis.

In this paper, we present a measure associated with detection and inference of statistically anomalous clusters of a graph based on the likelihood test of observed and expected edges in a subgraph. This measure is adapted from spatial scan statistics for point sets and provides quantitative assessment for clusters. We discuss some important properties of this statistic and its relation to modularity and Bregman divergences. We apply a simple clustering algorithm to find clusters with large values of this measure in a variety of real-world data sets, and we illustrate its ability to identify statistically significant clusters of selected granularity. 1 Introduction. Numerous techniques have been proposed for identifying clusters in large networks, but it has proven difficult to

Internet topology and BGP datasets Power-laws and spectrum of a graph Power-laws analysis

This paper presents research into parallel direct methods for block-diagonal-bordered sparse matrices --- LU factorization and Choleski factorization algorithms developed with special consideration for irregular sparse matrices from the electrical power systems community. Direct block-diagonal bordered sparse linear solvers exhibit distinct advantages when compared to general direct parallel sparse algorithms for irregular matrices. Task assignments for numerical factorization on distributedmemory multi-processors depend only on the assignment of data to blocks, and data communications are significantly reduced with uniform and structured communications. Factorization algorithms for block-diagonal-bordered form matrices require a specialized ordering step coupled to an explicit load balancing step in order to generate this matrix form and to uniformly distribute the computational workload for an irregular matrix throughout a distributed-memory multi-processor. This ordering relates to m...

: We describe in detail some algorithms currently in use for unstructured mesh partitioning, with some emphasis on spectral methods, that is, those methods which involve eigenvector computations. When applied to 3-regular graphs, previous methods can be theoretically improved, should a stated conjecture proves true. Key-words: unstructured meshes, graphs, partitioning, bisections (R'esum'e : tsvp) Fabio.Guerinoni@irisa.fr Unite de recherche INRIA Rennes IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) Telephone : (33) 99 84 71 00 -- Telecopie : (33) 99 84 71 71 Partitionnement des maillages et nouvelles observations pour les "graphs 3-r'eguliers" R'esum'e : Nous d'ecrivons queques algorithmes courants pour le partitionnement des maillages non structur'es. L'accent est mis sur les m'ethodes spectrales, demandant le calcul des vecteurs propres. Appliqu'ees aux "graphes 3-r'eguliers," une conjecture est propos'ee permettant d'am'eliorer en th'eorie les algorith...

All rights reserved. However, in accordance with the Copyright Act of Canada, this work may be reproduced, without authorization, under the conditions for Fair Dealing. Therefore, limited reproduction of this work for the purposes of private study, research, criticism, review and news reporting is likely to be in accordance with the law, particularly if cited appropriately. APPROVAL Name: Degree:

The discovery of power-laws and spectral properties of the Internet topology illustrates a complex underlying network infrastructure that carries a variety of the Internet applications. Analysis of spectral properties of the Internet topology is based on matrices of graphs capturing Internet structure on the Autonomous System (AS) level. The analysis of data collected from the Route Views and RIPE projects confirms the existence of power-laws and certain historical trends in the development of the Internet topology. While values of various powerlaws