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 semantics-preserving image compression, which reduces images to a small set of perceptually-significant coefficients. We describe three types of Photobook descriptions in detail: one that allows search based on appearance, one that uses 2-D 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 3-D medical data.

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 2-D 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 Wold-based 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 ...

this paper we consider the structure of two-dimensional 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 four-fold Wold-type decompositions are special cases of the countably-infinite-fold decomposition presented in this paper. The countably-infinite-fold decomposition arises from a set of new total-order and non-symmetricalhalf -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, zero-mean, random variables fy(n; m); (n; m) 2 Z

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 quasi-transitive 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

We study a class of stationary processes indexed by Z d that are defined via minors of d-dimensional (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 K-property, 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

In this paper we treat the two-variable 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 Fejér-Riesz factorization of a strictly positive two-variable trigonometric polynomial is given in terms of the Fourier coefficients of its reciprocal. Tools in the proofs include a specific two-variable Kronecker theorem based on certain elements from algebraic geometry, as well as a two-variable Christoffel-Darboux like formula. The key ingredient is a matrix valued polynomial that appears in a parameterized version of the Schur-Cohn test for stability. The results also have consequences in the theory of two-variable orthogonal polynomials where a spectral matching result is obtained, as well as in the study of inverse formulas for doubly-indexed Toeplitz matrices. Finally, numerical results are presented for both the autoregressive filter problem and the factorization problem. Key Words: autoregressive filter, bivariate stochastic processes, two-variable positive extension, structured matrix completions, doubly-indexed Toeplitz matrix, two-variable

this paper. We let (AP

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 one-variable case, the algorithm finds a spectral factorization. The latter may also be used to find inner-outer factorizations. Key words: Spectral factorization, inner-outer factorization, sums of squares, multivariable trigonometric polynomial, truncated moment problem, semidefinite programming MR Classification: 15A48, 42B05, 93B36 1 Introduction The classical Riesz-Fejer 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...

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We consider the problem of sequential decision making on random fields corrupted by noise. In this scenario, the decision maker observes a noisy version of the data, yet judged with respect to the clean data. In particular, we first consider the problem of sequentially scanning and filtering noisy random fields. In this case, the sequential filter is given the freedom to choose the path over which it traverses the random field (e.g., noisy image or video sequence), thus it is natural to ask what is the best achievable performance and how sensitive this performance is to the choice of the scan. We formally define the problem of scanning and filtering, derive a bound on the best achievable performance and quantify the excess loss occurring when non-optimal scanners are used, compared to optimal scanning and filtering. We then discuss the problem of sequential scanning and prediction of noisy random fields. This setting is a natural model for applications such as restoration and coding of noisy im-ages. We formally define the problem of scanning and prediction of a noisy multidimensional array and relate the optimal performance to the clean scandictability defined by Merhav and Weissman. Moreover, bounds on the excess loss due to sub-optimal scans are derived, and a universal prediction algorithm is suggested.