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13
Random walks for image segmentation
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Abstract—A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach on ..."
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Cited by 385 (21 self)
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Abstract—A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with userdefined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a highquality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs. Index Terms—Image segmentation, interactive segmentation, graph theory, random walks, combinatorial Dirichlet problem, harmonic functions, Laplace equation, graph cuts, boundary completion. Ç 1
TMG: A MATLAB Toolbox for Generating TermDocument Matrices from Text Collections
, 2005
"... A wide range of computational kernels in data mining and information retrieval from text collections involve techniques from linear algebra. These kernels typically operate on data that is presented in the form of large sparse termdocument matrices (tdm). We present TMG, a research and teaching too ..."
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Cited by 40 (2 self)
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A wide range of computational kernels in data mining and information retrieval from text collections involve techniques from linear algebra. These kernels typically operate on data that is presented in the form of large sparse termdocument matrices (tdm). We present TMG, a research and teaching toolbox for the generation of sparse tdm’s from text collections and for the incremental modification of these tdm’s by means of additions or deletions. The toolbox is written entirely in MATLAB, a popular problem solving environment that is powerful in computational linear algebra, in order to streamline document preprocessing and prototyping of algorithms for information retrieval. Several design issues that concern the use of MATLAB sparse infrastructure and data structures are addressed. We illustrate the use of the tool in numerical explorations of the effect of stemming and different termweighting policies on the performance of querying and clustering tasks.
The Piecewise Smooth MumfordShah Functional on an Arbitrary Graph
"... Abstract—The MumfordShah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the MumfordShah functional is predominated by ..."
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Cited by 20 (8 self)
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Abstract—The MumfordShah functional has had a major impact on a variety of image analysis problems including image segmentation and filtering and, despite being introduced over two decades ago, it is still in widespread use. Present day optimization of the MumfordShah functional is predominated by active contour methods. Until recently, these formulations necessitated optimization of the contour by evolving via gradient descent, which is known for its overdependence on initialization and the tendency to produce undesirable local minima. In order to reduce these problems, we reformulate the corresponding MumfordShah functional on an arbitrary graph and apply the techniques of combinatorial optimization to produce a fast, lowenergy solution. In contrast to traditional optimization methods, use of these combinatorial techniques necessitates consideration of the reconstructed image outside of its usual boundary, requiring additionally the inclusion of regularization for generating these values. The energy of the solution provided by this graph formulation is compared with the energy of the solution computed via traditional gradient descentbased narrowband level set methods. This comparison demonstrates that our graph formulation and optimization produces lower energy solutions than the traditional gradient descent based contour evolution methods in significantly less time. Finally, we demonstrate the usefulness of the graph formulation to apply the MumfordShah functional to new applications such as point clustering and filtering of nonuniformly sampled images. Index Terms—Level sets, active contours, piecewise smooth MumfordShah, combinatorial optimization, graph reformulation I.
Isoperimetric partitioning: A new algorithm for graph partitioning
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2006
"... We present a new algorithm for graph partitioning based on optimization of the combinatorial isoperimetric constant. It is shown empirically that this algorithm is competitive with other global partitioning algorithms in terms of partition quality. The isoperimetric algorithm is easy to parallelize, ..."
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Cited by 20 (5 self)
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We present a new algorithm for graph partitioning based on optimization of the combinatorial isoperimetric constant. It is shown empirically that this algorithm is competitive with other global partitioning algorithms in terms of partition quality. The isoperimetric algorithm is easy to parallelize, does not require coordinate information, and handles nonplanar graphs, weighted graphs, and families of graphs which are known to cause problems for other methods. Compared to spectral partitioning, the isoperimetric algorithm is faster and more stable. An exact circuit analogy to the algorithm is also developed with a natural random walks interpretation. The isoperimetric algorithm for graph partitioning is implemented in our publicly available Graph Analysis Toolbox [L. Grady, Ph.D. thesis, Boston University, Boston, MA, 2004], [L. Grady and E. L. Schwartz,; Technical report TR03021, Boston University, Boston, MA, 2003] for MATLAB obtainable at
Laplace spectra as fingerprints for image recognition
 COMPUTERAIDED DESIGN
, 2007
"... In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. T ..."
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Cited by 10 (2 self)
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In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a reinterpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous Laplace–Beltrami operator. Furthermore, we introduce the new concept of solid height functions to overcome some potential limitations of the method.
Isoperimetric graph partitioning for data clustering and image segmentation
 Boston University
, 2003
"... Spectral methods of graph partitioning have been shown to provide a powerful approach to the image segmentation problem. In this paper, we adopt a different approach, based on estimating the isoperimetric constant of an image graph. Our algorithm produces the high quality segmentations and data clus ..."
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Cited by 9 (1 self)
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Spectral methods of graph partitioning have been shown to provide a powerful approach to the image segmentation problem. In this paper, we adopt a different approach, based on estimating the isoperimetric constant of an image graph. Our algorithm produces the high quality segmentations and data clustering of spectral methods, but with improved speed and stability. 1
Dual constrained TVbased regularization on graphs
 SIAM J. on Imaging Sciences
, 2013
"... Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle’s a ..."
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Cited by 8 (5 self)
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Abstract. Algorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle’s and GilboaOsher’s projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TVbased approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.
LaplaceSpectra as Fingerprints for Image Recognition
"... In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. Th ..."
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Cited by 1 (1 self)
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In the area of image retrieval from data bases and for copyright protection of large image collections there is a growing demand for unique but easily computable fingerprints for images. These fingerprints can be used to quickly identify every image within a larger set of possibly similar images. This paper introduces a novel method to automatically obtain such fingerprints from an image. It is based on a reinterpretation of an image as a Riemannian manifold. This representation is feasible for gray value images and color images. We discuss the use of the spectrum of eigenvalues of different variants of the Laplace operator as a fingerprint and show the usability of this approach in several use cases. Contrary to existing works in this area we do not only use the discrete Laplacian, but also with a particular emphasis the underlying continuous operator. This allows better results in comparing the resulting spectra and deeper insights in the problems arising. We show how the well known discrete Laplacian is related to the continuous LaplaceBeltrami operator. Furthermore we introduce the new concept of solid height functions to overcome some potential limitations of the method. Keywords Laplace spectra, image recognition, LaplaceBeltrami operator,
Scalable Constrained Clustering: A Generalized Spectral Method
"... We present a principled spectral approach to the wellstudied constrained clustering problem. It reduces clustering to a generalized eigenvalue problem on Laplacians. The method works in nearlylinear time and provides concrete guarantees for the quality of the clusters, at least for the case of 2 ..."
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Cited by 1 (1 self)
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We present a principled spectral approach to the wellstudied constrained clustering problem. It reduces clustering to a generalized eigenvalue problem on Laplacians. The method works in nearlylinear time and provides concrete guarantees for the quality of the clusters, at least for the case of 2way partitioning. In practice this translates to a very fast implementation that consistently outperforms existing spectral approaches. We support this claim with experiments on various data sets: our approach recovers correct clusters in examples where previous methods fail, and handles data sets with millions of data points two orders of magnitude larger than before. 1.