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11
Statistics of Natural Images and Models
"... Large calibrated datasets of `random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. ..."
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Cited by 260 (8 self)
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Large calibrated datasets of `random' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some of these statistics shed light on old issues such as the near scaleinvariance of image statistics and some are entirely new. We fit mathematical models to some of the statistics and explain others in terms of local image features. 1
Statistics of range images
 CVPR
, 2000
"... The statistics of range images from natural environments is a largely unexplored eldofresearch. It closely relates to the statistical modeling of the scene geometry in natural environments, and the modeling of optical natural images. We have use d a 3D laser range nder to collect range images from ..."
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Cited by 83 (6 self)
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The statistics of range images from natural environments is a largely unexplored eldofresearch. It closely relates to the statistical modeling of the scene geometry in natural environments, and the modeling of optical natural images. We have use d a 3D laser range nder to collect range images from mixed forest scenes. The images are hereanalyzed with respect to di erent statistics. 1
Gaussian limits for determinantal random point
 Annals of Probability
, 2002
"... We prove that, under fairly general conditions, a properly rescaled determinantal random point field converges to a generalized Gaussian random process. 1. Introduction and formulation of results. Let E be a locally compact Hausdorff space satisfying the second axiom of countability, B—σalgebra of ..."
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Cited by 51 (0 self)
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We prove that, under fairly general conditions, a properly rescaled determinantal random point field converges to a generalized Gaussian random process. 1. Introduction and formulation of results. Let E be a locally compact Hausdorff space satisfying the second axiom of countability, B—σalgebra of Borel subsets and µ a σfinite measure on (E, B), such that µ(K) < ∞ for any compact K ⊂ E. We denote by X the space of locally finite configurations of particles in E: X ={ξ = (xi) ∞ i=− ∞ : xi ∈ E ∀i, and for any compact K ⊂ E #K(ξ):=
3 RIGOROUS QUANTUM FIELD THEORY FUNCTIONAL INTEGRALS OVER THE pADICS I: ANOMALOUS DIMENSIONS
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Exact simulation of timevarying fractionally differenced processes, preprint
 Journal of Computational and Graphical Statistics
, 2004
"... Timevarying fractionally differenced (TVFD) processes can serve as useful models for certain time series whose statistical properties evolve over time. The spectral density function for a TVFD process obeys a power law whose exponent can be time dependent. In contrast to locally stationary or loca ..."
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Cited by 4 (1 self)
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Timevarying fractionally differenced (TVFD) processes can serve as useful models for certain time series whose statistical properties evolve over time. The spectral density function for a TVFD process obeys a power law whose exponent can be time dependent. In contrast to locally stationary or locally selfsimilar processes, the power law exponent for a TVFD process is not restricted to certain intervals, which is of practical importance for modeling time series of, e.g., atmospheric turbulence. In this paper we construct a uniform representation for Gaussian TVFD processes that allows the power law exponent to evolve in an arbitrary manner. Even though this representation in general involves a timedependent linear combination of an infinite number of random variables from a Gaussian white noise process, we demonstrate that simulations with exactly correct statistical properties can be achieved based upon two wellknown approaches, each of which involves a finite portion of a white noise process. The first approach is based on the modified Cholesky decomposition, and the second, on circulant embedding. For either approach, the resulting algorithm for generating simulations of a TVFD process can be simply described as ‘cutting and pasting’ pieces of simulations from several different FD processes, all created from a single realization of a white noise process. Use of these exact methods will ensure that Monte Carlo studies of the statistical properties of estimators for TVFD processes are not adversely influenced by imperfections arising from the use of approximate simulation methods.
A hierarchical model of quantum anharmonic oscillators: critical point convergence
 Comm. Math. Phys
, 2004
"... Abstract. A hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the Euclidean approach, in which the local Gibbs states are constructed as measures on infinite dimensional spaces. The local states restricted to the subalgebra generated by fluctuations ..."
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Cited by 2 (2 self)
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Abstract. A hierarchical model of interacting quantum particles performing anharmonic oscillations is studied in the Euclidean approach, in which the local Gibbs states are constructed as measures on infinite dimensional spaces. The local states restricted to the subalgebra generated by fluctuations of displacements of particles are in the center of the study. They are described by means of the corresponding temperature Green (Matsubara) functions. The result of the paper is a theorem, which describes the critical point convergence
j huang @ cfm. brown.edu
"... Large calibrated datasets of 'random ' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet resp ..."
Abstract
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Large calibrated datasets of 'random ' natural images have recently become available. These make possible precise and intensive statistical studies of the local nature of images. We report results ranging from the simplest single pixel intensity to joint distribution of 3 Haar wavelet responses. Some of these statistics shed light on old issues such as the near scaleinvariance of image statistics and some are entirely new. We fit mathematical models to some of the statistics and explain others in terms of local image features. 1
SELFSIMILARITY AND DILATIVE STABILITY
"... The object of this note is to parallel two properties of stochastic processes: selfsimilarity (ss) and dilative stability (ds). Theorems on ss and functional limit theorems on convergence to ss processes, e.g. to fractional Brownian motion (FBM) have been known since [Lamperti, 1962] and [Davydov, ..."
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The object of this note is to parallel two properties of stochastic processes: selfsimilarity (ss) and dilative stability (ds). Theorems on ss and functional limit theorems on convergence to ss processes, e.g. to fractional Brownian motion (FBM) have been known since [Lamperti, 1962] and [Davydov, 1970], respectively. We will show that theorems about
RENORMALIZATION OF WEAK NOISES OF ARBITRARY SHAPE FOR ONE–DIMENSIONAL CRITICAL DYNAMICAL SYSTEMS: ANNOUNCEMENT OF RESULTS AND NUMERICAL EXPLORATIONS
"... Abstract. We study the effect of noise on one–dimensional critical dynamical systems (that is, maps with a renormalization theory). We consider in detail two examples of such dynamical systems: unimodal maps of the interval at the accumulation of period– doubling and smooth homeomorphisms of the cir ..."
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Abstract. We study the effect of noise on one–dimensional critical dynamical systems (that is, maps with a renormalization theory). We consider in detail two examples of such dynamical systems: unimodal maps of the interval at the accumulation of period– doubling and smooth homeomorphisms of the circle with a critical point and with golden mean rotation number. We show that, if we scale the space and the time, several properties of the noise (the cumulants or Wick–ordered moments) satisfy some scaling relations. A consequence of the scaling relations is that a version of the central limit theorem holds. Irrespective of the shape of the initial noise, if the bare noise is weak enough, the effective noise becomes close to Gaussian in several senses that we can make precise. We notice that the conclusions are false for maps with positive Lyapunov exponents. The method of analysis is close in spirit to the study of scaling limits in renormalization theory. We also perform several numerical experiments that confirm the rigorous results and that suggest several conjectures. 1.
Contents
, 2002
"... Abstract. In this paper a new multifractal stochastic process called Limit of the Integrated Superposition of Diffusion processes with Linear differencial Generator (LISDLG) is presented which realistically characterizes the network traffic multifractality. Several properties of the LISDLG model are ..."
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Abstract. In this paper a new multifractal stochastic process called Limit of the Integrated Superposition of Diffusion processes with Linear differencial Generator (LISDLG) is presented which realistically characterizes the network traffic multifractality. Several properties of the LISDLG model are presented including long range dependence, cumulants, logarithm of the characteristic function, dilative stability, spectrum and bispectrum. The model captures higherorder statistics by the cumulants. The relevance and validation of the proposed model are demonstrated by real data of Internet traffic.