We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We applied this approach to segmenting static images, as well as motion sequences, and found the results to be very encouraging.

/pdf/normalized-cuts-and-image-segmentation-2ru3ikxuj5.pdf

Normalized cuts and image segmentation

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.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

We propose a novel approach for solving the perceptual grouping problem in vision. Rather than focusing on local features and their consistencies in the image data, our approach aims at extracting the global impression of an image. We treat image segmentation as a graph partitioning problem and propose a novel global criterion, the normalized cut, for segmenting the graph. The normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups. We show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion. We have applied this approach to segmenting static images and found results very encouraging.

/pdf/normalized-cuts-and-image-segmentation-1q390dk21h.pdf

Many networks of interest in the sciences, including social networks, computer networks, and metabolic and regulatory networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure is one of the outstanding issues in the study of networked systems. One highly effective approach is the optimization of the quality function known as “modularity” over the possible divisions of a network. Here I show that the modularity can be expressed in terms of the eigenvectors of a characteristic matrix for the network, which I call the modularity matrix, and that this expression leads to a spectral algorithm for community detection that returns results of demonstrably higher quality than competing methods in shorter running times. I illustrate the method with applications to several published network data sets.

/pdf/modularity-and-community-structure-in-networks-l3cxahj8x9.pdf

Modularity and community structure in networks

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

A new set of benchmarks has been developed for the performance evaluation of highly parallel supercom puters. These consist of five "parallel kernel" bench marks and three "simulated application" benchmarks. Together they mimic the computation and data move ment characteristics of large-scale computational fluid dynamics applications. The principal distinguishing feature of these benchmarks is their "pencil and paper" specification-all details of these benchmarks are specified only algorithmically. In this way many of the difficulties associated with conventional bench- marking approaches on highly parallel systems are avoided.

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The Nas Parallel Benchmarks

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is, shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorith...

/pdf/partitioning-sparse-matrices-with-eigenvectors-of-graphs-38q27a5q5m.pdf

Partitioning sparse matrices with eigenvectors of graphs

An important application of graph partitioning is data clustering using a graph model - the pairwise similarities between all data objects form a weighted graph adjacency matrix that contains all necessary information for clustering. In this paper, we propose a new algorithm for graph partitioning with an objective function that follows the min-max clustering principle. The relaxed version of the optimization of the min-max cut objective function leads to the Fiedler vector in spectral graph partitioning. Theoretical analyses of min-max cut indicate that it leads to balanced partitions, and lower bounds are derived. The min-max cut algorithm is tested on newsgroup data sets and is found to out-perform other current popular partitioning/clustering methods. The linkage-based refinements to the algorithm further improve the quality of clustering substantially. We also demonstrate that a linearized search order based on linkage differential is better than that based on the Fiedler vector, providing another effective partitioning method.

http://ants.iis.sinica.edu.tw/3bkmj9ltewxtsrrvnoknfdxrm3zfwrr/87/a%20min-max%20cut%20algorithm%20for%20graph%20partition.pdf

A min-max cut algorithm for graph partitioning and data clustering

Partitioning of unstructured problems for parallel processing

The popular K-means clustering partitions a data set by minimizing a sum-of-squares cost function. A coordinate descend method is then used to find local minima. In this paper we show that the minimization can be reformulated as a trace maximization problem associated with the Gram matrix of the data vectors. Furthermore, we show that a relaxed version of the trace maximization problem possesses global optimal solutions which can be obtained by computing a partial eigendecomposition of the Gram matrix, and the cluster assignment for each data vectors can be found by computing a pivoted QR decomposition of the eigenvector matrix. As a by-product we also derive a lower bound for the minimum of the sum-of-squares cost function.

/pdf/spectral-relaxation-for-k-means-clustering-33h36n74wf.pdf

Horst D. Simon

Papers

The Nas Parallel Benchmarks

Partitioning sparse matrices with eigenvectors of graphs

A min-max cut algorithm for graph partitioning and data clustering

Partitioning of unstructured problems for parallel processing

Spectral Relaxation for K-means Clustering