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55
Photobook: ContentBased Manipulation of Image Databases
, 1995
"... We describe the Photobook system, which is a set of interactive tools for browsing and searching images and image sequences. These query tools differ from those used in standard image databases in that they make direct use of the image content rather than relying on text annotations. Direct search o ..."
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Cited by 462 (0 self)
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We describe the Photobook system, which is a set of interactive tools for browsing and searching images and image sequences. These query tools differ from those used in standard image databases in that they make direct use of the image content rather than relying on text annotations. Direct search on image content is made possible by use of semanticspreserving image compression, which reduces images to a small set of perceptuallysignificant coefficients. We describe three types of Photobook descriptions in detail: one that allows search based on appearance, one that uses 2D shape, and a third that allows search based on textural properties. These image content descriptions can be combined with each other and with textbased descriptions to provide a sophisticated browsing and search capability. In this paper we demonstrate Photobook on databases containing images of people, video keyframes, hand tools, fish, texture swatches, and 3D medical data.
Periodicity, directionality, and randomness: Wold features for image modeling and retrieval
 IEEE Trans. Pattern Analysis and Machine Intelligence
, 1996
"... One of the fundamental challenges in pattern recognition is choosing a set of features appropriate to a class of problems. In applications such as database retrieval, it is important that image features used in pattern comparison provide good measures of image perceptual similarities. In this paper, ..."
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Cited by 115 (5 self)
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One of the fundamental challenges in pattern recognition is choosing a set of features appropriate to a class of problems. In applications such as database retrieval, it is important that image features used in pattern comparison provide good measures of image perceptual similarities. In this paper, we present an image model with a new set of features that address the challenge of perceptual similarity. The model is based on the 2D Wold decomposition of homogeneous random fields. The three resulting mutually orthogonal subfields have perceptual properties which can be described as "periodicity", "directionality ", and "randomness", approximating what are indicated to be the three most important dimensions of human texture perception. The method presented here improves upon earlier Woldbased models in its tolerance to a variety of local inhomogeneities which arise in natural textures and its invariance under image transformation such as rotation. An image retrieval algorithm based on ...
Asymptotic enumeration of spanning trees
 Combin. Probab. Comput
, 2005
"... Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that ..."
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Cited by 28 (6 self)
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Note: Theorem numbers differ from the published version. Abstract. We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy”, which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Tree entropy is also expressed using random walks. We relate tree entropy to the metric entropy of the uniform spanning forest process on quasitransitive amenable graphs, extending a result of Burton and Pemantle (1993). §1. Introduction. Methods of enumeration of spanning trees in a finite graph G and relations to various areas of mathematics and physics have been investigated for more than 150 years. The number of spanning trees is often called the complexity of the graph, denoted here by τ(G). The best known formula for the complexity, proved in every basic text on graph
A WoldLike Decomposition of 2D Discrete Homogeneous Random Fields
"... this paper we consider the structure of twodimensional discrete homogeneous random fields. We extend the results of Helson and Lowdenslager (1962), Korezlioglu and Loubaton (1986), Kallianpur et al. (1990), and Chiang (1991), to show that the two, three, and fourfold Woldtype decompositions are ..."
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Cited by 19 (13 self)
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this paper we consider the structure of twodimensional discrete homogeneous random fields. We extend the results of Helson and Lowdenslager (1962), Korezlioglu and Loubaton (1986), Kallianpur et al. (1990), and Chiang (1991), to show that the two, three, and fourfold Woldtype decompositions are special cases of the countablyinfinitefold decomposition presented in this paper. The countablyinfinitefold decomposition arises from a set of new totalorder and nonsymmetricalhalf plane (NSHP) definitions imposed on the random field. These order definitions are obtained by rotating the NSHP support by angles of rational tangent, rather than considering only the vertical and horizontal orientations. A family of real, zeromean, random variables fy(n; m); (n; m) 2 Z
Stationary determinantal processes: phase multiplicity
 Bernoullicity, entropy, and domination, Duke Math. Journal
, 2003
"... We study a class of stationary processes indexed by Z d that are defined via minors of ddimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Pha ..."
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Cited by 19 (6 self)
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We study a class of stationary processes indexed by Z d that are defined via minors of ddimensional (multilevel) Toeplitz matrices. We obtain necessary and sufficient conditions for phase multiplicity (the existence of a phase transition) analogous to that which occurs in statistical mechanics. Phase uniqueness is equivalent to the presence of a strong Kproperty, a particular strengthening of the usual K (Kolmogorov) property. We show that all of these processes are Bernoulli shifts (isomorphic to independent identically distributed (i.i.d.) processes in the sense of ergodic theory). We obtain estimates of their entropies, and we relate these processes via stochastic domination
ENTROPY AND MULTIVARIABLE INTERPOLATION
, 2003
"... We define a new notion of entropy for operators on Fock spaces and positive definite multiToeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multiToeplitz, multianalytic, etc.) operators on Fock spaces. These results lead to entropy inequalities an ..."
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Cited by 14 (2 self)
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We define a new notion of entropy for operators on Fock spaces and positive definite multiToeplitz kernels on free semigroups. This is studied in connection with factorization theorems for (multiToeplitz, multianalytic, etc.) operators on Fock spaces. These results lead to entropy inequalities and entropy formulas for positive definite multiToeplitz kernels on free semigroups (resp. multiToeplitz operators) and consequences concerning the extreme points of the unit ball of the noncommutative analytic Toeplitz algebra F ∞ n. We obtain several geometric characterizations of the multivariable central intertwining lifting, a maximum principle, and a permanence principle for the noncommutative commutant lifting theorem. Under certain natural conditions, we find explicit forms for the maximal entropy solution (and its entropy) for this multivariable commutant lifting theorem. All these results are used to solve maximal entropy interpolation problems in several variables. We obtain explicit forms for the maximal entropy solution (as well as its entropy) of the Sarason, CarathéodorySchur, and NevanlinnaPick type interpolation problems for the noncommutative (resp. commutative) analytic Toeplitz algebra F ∞ n (resp. W ∞ n) and their tensor products with B(H, K). In particular, we provide explicit forms for the maximal entropy solutions of several interpolation (resp. optimization) problems on the unit ball of C n.
Factorization of Almost Periodic Matrix Functions of Several Variables and Toeplitz Operators
 Math.Notes45 (1989), no. 5–6, 482–488. MR 90k:47033
"... this paper. We let (AP ..."
Positive extensions, FejérRiesz factorization and autoregressive filters in two variables
 Ann. of Math
, 2004
"... In this paper we treat the twovariable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate st ..."
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Cited by 10 (5 self)
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In this paper we treat the twovariable positive extension problem for trigonometric polynomials where the extension is required to be the reciprocal of the absolute value squared of a stable polynomial. This problem may also be interpreted as an autoregressive filter design problem for bivariate stochastic processes. We show that the existence of a solution is equivalent to solving a finite positive definite matrix completion problem where the completion is required to satisfy an additional low rank condition. As a corollary of the main result a necessary and sufficient condition for the existence of a spectral FejérRiesz factorization of a strictly positive twovariable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Tools in the proofs include a specific twovariable Kronecker theorem based on certain elements from algebraic geometry, as well as a twovariable ChristoffelDarboux like formula. The key ingredient is a matrix valued polynomial that appears in a parameterized version of the SchurCohn test for stability. The results also have consequences in the theory of twovariable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doublyindexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem. Key Words: autoregressive filter, bivariate stochastic processes, twovariable positive extension, structured matrix completions, doublyindexed Toeplitz matrix, twovariable
Spectral Factorizations and Sums of Squares Representations via Semidefinite Programming
 SIAM J. Matrix Anal. Appl
"... In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding a sum of squares representation is presented as well. In the onevariable case, the ..."
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Cited by 7 (2 self)
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In this paper we find a characterization for when a multivariable trigonometric polynomial can be written as a sum of squares. In addition, the truncated moment problem is addressed. A numerical algorithm for finding a sum of squares representation is presented as well. In the onevariable case, the algorithm finds a spectral factorization. The latter may also be used to find innerouter factorizations. Key words: Spectral factorization, innerouter factorization, sums of squares, multivariable trigonometric polynomial, truncated moment problem, semidefinite programming MR Classification: 15A48, 42B05, 93B36 1 Introduction The classical RieszFejer factorization theorem states that a trigonometric polynomial q(z) = m X i=\Gammam q i z i ; jzj = 1; on the torus that solely takes on nonnegative values can be written as q(z) = jp(z)j 2 ; jzj = 1; (1.1) where p(z) is a polynomial that has no zeroes in the disk D = fz 2 C : jzj ! 1g. A proof of this result based on the fundame...