multiple window parallel adaptive boundary finding in computer vision the problem considered in this paper is the estimation of highly variable object boundaries in noisy images boundaries may be those of a tank in an ir image a spinal canal in a cat scan a cloud in a visible light image etc or they may be internal to an object such as the boundary between a spherical surface and a cylindrical surface in a manufactured object the focus of the paper is on parallel multiple window boundary estimation algorithms here the image field is parti tioned into an array of rectangular windows and boundary finders are run simultaneously within the windows the boundary segments found within the windows are then seamed together to obtain meaningful global boundaries the entire procedure is treated within a maximum likelihood estimation framework that we have developed for boundary finding although our multiple window estimation approach can be used with a number of local boundary finding algorithms we concen trate on one which is based on dynamic programming and will produce the true maximum likelihood boundary some theoretical considera tions for boundary model design and boundary finding runtime are covered included is the use of a low computational cost f test for test ing whether a window contains a boundary and an analytical treatment which shows that use of coarse pixels with a chi square test or an f test improves the probability of correctly recognizing whether a boundary is present in a window

persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

blind deconvolution using a variational approach to parameter image and blur estimation following the hierarchical bayesian framework for blind deconvolution problems in this paper we propose the use of simultaneous autoregressions as prior distributions for both the image and blur and gamma distributions for the unknown parameters hyperparameters of the priors and the image formation noise we show how the gamma distributions on the unknown hyperparameters can be used to prevent the proposed blind deconvolution method from converging to undesirable image and blur estimates and also how these distributions can be inferred in realistic situations we apply variational methods to approximate the posterior probability of the unknown image blur and hyperparameters and propose two different approximations of the posterior distribution one of these approximations coincides with a classical blind deconvolution method the proposed algorithms are tested experimentally and compared with existing blind deconvolution methods

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

a new verified optimization technique for the packing circles in a unit square problems this paper presents a new verified optimization method for the problem of finding the densest packings of nonoverlapping equal circles in a square in order to provide reliable numerical results the developed algorithm is based on interval analysis as one of the most efficient parts of the algorithm an interval based version of a previous elimination procedure is introduced this method represents the remaining areas still of interest as polygons fully calculated in a reliable way currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems still as a result of the highly increasing number of subproblems earlier computer aided methods were not able to solve problem instances where the number of circles was greater than 27 the present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together as a demonstration of the capabilities of the new algorithm the problems of packing 28 29 and 30 circles were solved within very tight tolerance values our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases

variational bayesian image restoration with a product of spatially weighted total variation image priors in this paper a new image prior is introduced and used in image restoration this prior is based on products of spatially weighted total variations tv these spatial weights provide this prior with the flexibility to better capture local image features than previous tv based priors bayesian inference is used for image restoration with this prior via the variational approximation the proposed restoration algorithm is fully automatic in the sense that all necessary parameters are estimated from the data and is faster than previous similar algorithms numerical experiments are shown which demonstrate that image restoration based on this prior compares favorably with previous state of the art restoration algorithms

digital image restoration using spatial interaction models this paper is concerned with developing fast nonrecursive algorithms for the minimum mean squared error restoration of degraded images the degradation is assumed to be due to a space invariant periodic nonseparable known point spread function and additive white noise our basic approach is to represent the images by a class of spatial interaction models namely the simultaneous autoregressve models and the conditional markov models defined on toroidal lattices and develop minimum mean squared error restoration algorithms using these models the restoration algorithms are optimal if the parameters characterizing the interaction models are exactly known however in practice the parameters are estimated from the images by using spatial interaction models we develop restoration algorithms that do not require the availability of the original image or its prototype the specific structure of the underlying lattice enables the implementation of the filters using fast fourier transform fft computations several restoration examples are given

alternating kinetics of annihilating random walks near a free interface the kinetics of annihilating random walks in one dimension with the half line x 0 initially filled is investigated the survival probability of the nth particle from the interface exhibits power law decay with for n 1 and all odd values of n for all n even a faster decay with is observed from consideration of the eventual survival probability in a finite cluster of particles the rigorous bound is derived while a heuristic argument gives numerically this latter value appears to be a lower bound for the average position of the first particle moves to the right approximately as with a relatively sharp and asymmetric probability distribution

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on convex relaxations for quadratically constrained quadratic programming we consider convex relaxations for the problem of minimizing a possibly nonconvex quadratic objective subject to linear and possibly nonconvex quadratic constraints let mathcal f denote the feasible region for the linear constraints we first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on mathcal f is dominated by an alternative methodology based on convexifying the range of the quadratic form genfrac 0 0pt 1 x genfrac 0 0pt 1 x t for x in mathcal f we next show that the use of alpha bb underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms finally we show that the use of a large class of d c difference of convex underestimators is dominated by a relaxation that combines semidefiniteness with rlt constraints

brownian motion in wedges last passage time and the second arc sine law we consider a planar brownian motion starting from o at time t 0 and stopped at t 1 and a set f oii i 1 2 n of n semi infinite straight lines emanating from o denoting by g the last time when f is reached by the brownian motion we compute the probability law of g in particular we show that for a symmetric f and even n values this law can be expressed as a sum of arcsin or arcsin 2 functions the original result of levy is recovered as the special case n 2 a relation with the problem of reaction diffusion of a set of three particles in one dimension is discussed

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

solving circle packing problems by global optimization numerical results and industrial applications abstract a general circle packing is an optimized arrangement of n arbitrary sized circles inside a container e g a rectangle or a circle such that no two circles overlap in this paper we present several circle packing problems review their industrial applications and some exact and heuristic strategies for their solution we also present illustrative numerical results using generic global optimization software packages our work highlights the relevance of global optimization in solving circle packing problems and points towards the necessary advancements in both theory and numerical practice

estimating the granularity coefficient of a potts markov random field within a markov chain monte carlo algorithm this paper addresses the problem of estimating the potts parameter jointly with the unknown parameters of a bayesian model within a markov chain monte carlo mcmc algorithm standard mcmc methods cannot be applied to this problem because performing inference on requires computing the intractable normalizing constant of the potts model in the proposed mcmc method the estimation of is conducted using a likelihood free metropolis hastings algorithm experimental results obtained for synthetic data show that estimating jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of on the other hand choosing an incorrect value of can degrade estimation performance significantly to illustrate the interest of this method the proposed algorithm is successfully applied to real bidimensional sar and tridimensional ultrasound images

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

fast sampling of gaussian markov random fields this paper demonstrates how gaussian markov random fields conditional autoregressions can be sampled quickly by using numerical techniques for sparse matrices the algorithm is general and efficient and expands easily to various forms for conditional simulation and evaluation of normalization constants we demonstrate its use by constructing efficient block updates in markov chain monte carlo algorithms for disease mapping

interval methods for verifying structural optimality of circle packing configurations in the unit square the paper is dealing with the problem of finding the densest packings of equal circles in the unit square recently a global optimization method based exclusively on interval arithmetic calculations has been designed for this problem with this method it became possible to solve the previously open problems of packing 28 29 and 30 circles in the numerical sense tight guaranteed enclosures were given for all the optimal solutions and for the optimum value the present paper completes the optimality proofs for these cases by determining all the optimal solutions in the geometric sense namely it is proved that the currently best known packing structures result in optimal packings and moreover apart from symmetric configurations and the movement of well identified free circles these are the only optimal packings the required statements are verified with mathematical rigor using interval arithmetic tools

maximin latin hypercube designs in two dimensions the problem of finding a maximin latin hypercube design in two dimensions can be described as positioning n nonattacking rooks on an n a n chessboard such that the minimal distance between pairs of rooks is maximized maximin latin hypercube designs are important for the approximation and optimization of black box functions in this paper general formulas are derived for maximin latin hypercube designs for general n when the distance measure is l or l1 furthermore for the distance measure l2 we obtain maximin latin hypercube designs for n 70 and approximate maximin latin hypercube designs for other values of n all these maximin latin hypercube designs can be downloaded from the website http www spacefillingdesigns nl we show that the reduction in the maximin distance caused by imposing the latin hypercube design structure is small this justifies the use of maximin latin hypercube designs instead of unrestricted designs

wavelets ridgelets and curvelets for poisson noise removal in order to denoise poisson count data we introduce a variance stabilizing transform vst applied on a filtered discrete poisson process yielding a near gaussian process with asymptotic constant variance this new transform which can be deemed as an extension of the anscombe transform to filtered data is simple fast and efficient in very low count situations we combine this vst with the filter banks of wavelets ridgelets and curvelets leading to multiscale vsts ms vsts and nonlinear decomposition schemes by doing so the noise contaminated coefficients of these ms vst modified transforms are asymptotically normally distributed with known variances a classical hypothesis testing framework is adopted to detect the significant coefficients and a sparsity driven iterative scheme reconstructs properly the final estimate a range of examples show the power of this ms vst approach for recovering important structures of various morphologies in very low count images these results also demonstrate that the ms vst approach is competitive relative to many existing denoising methods

bayesian estimation for optimized structured illumination microscopy structured illumination microscopy is a recent imaging technique that aims at going beyond the classical optical resolution by reconstructing high resolution hr images from low resolution lr images acquired through modulation of the transfer function of the microscope the classical implementation has a number of drawbacks such as requiring a large number of images to be acquired and parameters to be manually set in an ad hoc manner that have until now hampered its wide dissemination here we present a new framework based on a bayesian inverse problem formulation approach that enables the computation of one hr image from a reduced number of lr images and has no specific constraints on the modulation moreover it permits to automatically estimate the optimal reconstruction hyperparameters and to compute an uncertainty bound on the estimated values we demonstrate through numerical evaluations on simulated data and examples on real microscopy data that our approach represents a decisive advance for a wider use of hr microscopy through structured illumination

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

rigid body segmentation and shape description from dense optical flow under weak perspective we present an algorithm for identifying and tracking independently moving rigid objects from optical flow some previous attempts at segmentation via optical flow have focused on finding discontinuities in the flow field while discontinuities do indicate a change in scene depth they do not in general signal a boundary between two separate objects the proposed method uses the fact that each independently moving object has a unique epipolar constraint associated with its motion thus motion discontinuities based on self occlusion can be distinguished from those due to separate objects the use of epipolar geometry allows for the determination of individual motion parameters for each object as well as the recovery of relative depth for each point on the object the algorithm assumes an affine camera where perspective effects are limited to changes in overall scale no camera calibration parameters are required a kalman filter based approach is used for tracking motion parameters with time

exponents appearing in heterogeneous reaction diffusion models in one dimension we study the following 1d two species reaction diffusion model there is a small concentration of b particles with diffusion constant d b in an homogenous background of w particles with diffusion constant d w two w particles of the majority species either coagulate w w longrightarrow w or annihilate w w longrightarrow emptyset with the respective probabilities p c q 2 q 1 and p a 1 q 1 a b particle and a w particle annihilate w b longrightarrow emptyset with probability 1 the exponent theta q lambda d b d w describing the asymptotic time decay of the minority b species concentration can be viewed as a generalization of the exponent of persistent spins in the zero temperature glauber dynamics of the 1d q state potts model starting from a random initial condition the w particles represent domain walls and the exponent theta q lambda characterizes the time decay of the probability that a diffusive spectator does not meet a domain wall up to time t we extend the methods introduced by derrida hakim and pasquier em phys rev lett bf 75 751 1995 saclay preprint t96 013 to appear in em j stat phys 1996 for the problem of persistent spins to compute the exponent theta q lambda in perturbation at first order in q 1 for arbitrary lambda and at first order in lambda for arbitrary q