on the estimation of optical flow relations between different approaches and some new results abstract a common basis is suggested for the optical flow estimation approaches of nagel 1983 haralick and lee 1983 and tretiak and pastor 1984 based on a discussion of these approaches an exact solution for the system of partial differential equations proposed by horn and schunck 1981 is given at gray value corners and extrema the insight gained by this solution results in a modification of the oriented smoothness approach of nagel 1983 which thereby becomes considerably simpler in addition the optical flow estimation approach of hildreth 1983 1984 can be shown to represent a kind of special case of this modified oriented smoothness approach in a more direct manner than discussed in nagel 1984

computing optical flow via variational techniques defined as the apparent motion in a sequence of images the optical flow is very important in the computer vision community where its accurate estimation is necessary for many applications it is o

robust optical flow estimation based on a sparse motion trajectory set this paper presents an approach to the problem of estimating a dense optical flow field the approach is based on a multiframe irregularly spaced motion trajectory set where each trajectory describes the motion of a given point as a function of time from this motion trajectory set a dense flow field is estimated using a process of interpolation a set of localized motion models are estimated with each pixel labeled as belonging to one of the motion models a markov random field framework is adopted allowing the incorporation of contextual constraints to encourage region like structures the approach is compared with a number of conventional optical flow estimation algorithms taken over a number of real and synthetic sequences results indicate that the method produces more accurate results for sequences with known ground truth flow also applying the method to real sequences with unknown flow results in lower dfd for all of the sequences tested

comutations underlying the measuremnt of visual motion abstract the organization of movement in a changing image provides a valuable source of information for analyzing the environment in terms of objects their motion in space and their three dimensional structure a description of this movement is not provided to our visual system directly however it must be inferred from the pattern of changing intensity that reaches the eye this paper examines the problem of motion measurement which we formulate as the computation of an instantaneous two dimensional velocity field initial motion measurements occur at the location of significant intensity changes as suggested by marr and ullman 1 these measurements provide only one component of velocity and must be integrated to compute the two dimensional velocity field a fundamental problem for this integration is that motion is not determined uniquely from the changing image an additional constraint of smoothness of the velocity field is formulated based on the physical assumption that surfaces are generally smooth allowing the computation of a unique solution a theoretical analysis of the conditions under which this computation yields the correct velocity field suggests that the solution is physically plausible empirical studies show the results to be consistent with human motion perception

a fast scalable algorithm for discontinuous optical flow estimation multiple moving objects partially occluded objects or even a single object moving against the background gives rise to discontinuities in the optical flow field in corresponding image sequences while uniform global regularization based moderately fast techniques cannot provide accurate estimates of the discontinuous flow field statistical optimization based accurate techniques suffer from excessive solution time a weighted anisotropic smoothness based numerically robust algorithm is proposed that can generate discontinuous optical flow field with high speed and linear computational complexity weighted sum of the first order spatial derivatives of the flow field is used for regularization less regularization is performed where strong gradient information is available the flow field at any point is interpolated more from those at neighboring points along the weaker intensity gradient component such intensity gradient weighted regularization leads to euler lagrange equations with strong anisotropies coupled with discontinuities in their coefficients a robust multilevel iterative technique that recursively generates coarse level problems based on intensity gradient weighted smoothing weights is employed to estimate discontinuous optical flow field experimental results are presented to demonstrate the efficacy of the proposed technique

a morphological affine and galilean invariant scale space for movies we study a model of multiscale analysis or scale space applied to movies this model comes from a thorough formalization that has been done in the theory of scale space of static image this formulation has led one to associate with each multiscale analysis a partial differential equation pde we examine the case of movies and insist on the motion aspects more precisely it has been proved by alvarez guichard lions and morel 1993 that there exists a unique affine and morphological and galilean invariant scale space for movies the amg model this model is described by a partial differential equation we focus on terms appearing in that equation we show that this model provides a reliable definition of an optical multiscale acceleration at the practical level scale is interpreted as a way of characterizing reliable trajectories as we prove by experiments the amg model is a riddle for decimating spurious trajectories due to any kinds of nonadditive impurities and noise simple discrete formulae are given to implement the model

optimization of random searches on defective lattice networks we study the general problem of how to search efficiently for targets randomly located on defective lattice networks i e regular lattices which have some fraction of its nodes randomly removed we consider large but finite triangular lattices and assume for the search dynamics that the walker chooses steps lengths lj from the power law distribution p lj approximately lj mu with the exponent mu regulating the strategy of the search process at each step lj the searcher moves in straight lines and constantly looks within a detection radius of vision rv for the targets along the way if there is contact with a defect the movement stops and a new step length is chosen hence the presence of defects decreases the efficiency of the overall process we study numerically how three different aspects of the lattice influence the optimization of the search efficiency i the type of boundary conditions ii the concentration of targets and defects and iii the category or class of search destructive nondestructive or regenerative motivated by the results we develop a type of mean field model for the problem and obtain an analytical approximation for the search efficiency function finally we discuss in the context of searches how defective lattices compare with perfect lattices and with continuous environments

on the strong limit point and dirichlet properties of second order differential expressions second order differential expressions of the form w 1 pf qf are considered at a singular end point some of the known relationships between properties such as dirichlet weak dirichlet and strong limit point are extended to incorporate an arbitrary positive weight function and complexvalued coefficients

motion estimation using higher order statistics the objective of this paper is to introduce a fourth order cost function of the displaced frame difference dfd capable of estimating motion even for small regions or blocks using higher than second order statistics is appropriate in case the image sequence is severely corrupted by additive gaussian noise some results are presented and compared to those obtained from the mean kurtosis and the mean square error of the dfd

levy flights in random searches we review the general search problem of how to find randomly located objects that can only be detected in the limited vicinity of a forager and discuss its quantitative description using the theory of random walks we illustrate levy flight foraging by comparison to brownian random walks and discuss experimental observations of levy flights in biological foraging we review recent findings suggesting that an inverse square probability density distribution p l l 2 of step lengths l can lead to optimal searches finally we survey the explanations put forth to account for these unexpected findings

optimal search behavior and classic foraging theory random walk methods and diffusion theory pervaded ecological sciences as methods to analyze and describe animal movement consequently statistical physics was mostly seen as a toolbox rather than as a conceptual framework that could contribute to theory on evolutionary biology and ecology however the existence of mechanistic relationships and feedbacks between behavioral processes and statistical patterns of movement suggests that beyond movement quantification statistical physics may prove to be an adequate framework to understand animal behavior across scales from an ecological and evolutionary perspective recently developed random search theory has served to critically re evaluate classic ecological questions on animal foraging for instance during the last few years there has been a growing debate on whether search behavior can include traits that improve success by optimizing random stochastic searches here we stress the need to bring together the general encounter problem within foraging theory as a mean for making progress in the biological understanding of random searching by sketching the assumptions of optimal foraging theory oft and by summarizing recent results on random search strategies we pinpoint ways to extend classic oft and integrate the study of search strategies and its main results into the more general theory of optimal foraging

on the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction gradient based approaches to the computation of optical flow often use a minimization technique incorporating a smoothness constraint on the optical flow field the author derives the most general form of such a smoothness constraint that is quadratic in first derivatives of the grey level image intensity function based on three simple assumptions about the smoothness constraint 1 it must be expressed in a form that is independent of the choice of cartesian coordinate system in the image 2 it must be positive definite and 3 it must not couple different component of the optical flow it is shown that there are essentially only four such constraints any smoothness constraint satisfying 1 2 or 3 must be a linear combination of these four possibly multiplied by certain quantities invariant under a change in the cartesian coordinate system beginning with the three assumptions mentioned above the author mathematically demonstrates that all best known smoothness constraints appearing in the literature are special cases of this general form and in particular that the weight matrix introduced by h h nagel is essentially modulo invariant quantities the only physically plausible such constraint

a region based selective optical flow back projection for genuine motion vector estimation motion vector plays one significant feature in moving object segmentation however the motion vector in this application is required to represent the actual motion displacement rather than regions of visually significant similarity in this paper region based selective optical flow back projection rsofb which back projects optical flows in a region to restore the region s motion vector from gradient based optical flows is proposed to obtain genuine motion displacement the back projection is performed based on minimizing the projection mean square errors of the motion vector on gradient directions as optical flows of various magnitudes and directions provide various degrees of reliability in the genuine motion restoration the optical flows to be used in the rsofb are optimally selected based on their sensitivity to noises and their tendency in causing motion estimation errors in this paper a deterministic solution is also derived for performing the minimization and obtaining the genuine motion magnitude and motion direction

a return to the hardy littlewood integral inequality this paper is concerned with the extension and generalization of a classic integral inequality first considered by hardy and littlewood in 1932 the integral inequality is big int b a p f 2 q f 2 big 2 leqslant k int b a w f 2 int b a w w 1 pf qf 2 f in delta where k is a positive number the coefficients p q and w are real valued and w 0 on the interval a b infty a b leqslant infty and p 1 q w in l loc a b delta is the maximal linear manifold of functions f for which the two integrals on the right hand side are both finite the hardy littlewood inequality is the special case a 0 b infty p w 1 and q 0 on 0 infty here k 4 and all the cases of equality can be identified in this paper the results of everitt in 1971 connecting the above inequality with the titchmarsh weyl m coefficient of the associated differential equation py qy lambda wy on a b are restated in an improved form and are shown to hold under the above minimal conditions on the coefficients p q and w it is proved that the best possible value for the number k satisfies 1 k leqslant infty and that within these bounds all values of k can be realized in the course of establishing these results new information of independent interest is obtained about the behaviour of the m coefficient near the real axis examples are given to exhibit the wide range of possibilities for the integral inequality recent results on the asymptotic form of the m coefficient and on the transformation theory of the associated differential equation are shown to have applications for the general properties of the inequality all recent contributions to this general integral inequality are listed in the references and their content is surveyed during the course of the paper

an investigation of smoothness constraints for the estimation of displacement vector fields from image sequences a mapping between one frame from an image sequence and the preceding or following frame can be represented as a displacement vector field in most situations the mere gray value variations do not provide sufficient information in order to estimate such a displacement vector field supplementary constraints are necessary for example the postulate that a displacement vector field varies smoothly as a function of the image position taken as a general requirement this creates difficulties at gray value transitions which correspond to occluding contours nagel therefore introduced the oriented smoothness requirement which restricts variations of the displacement vector field only in directions with small or no variation of gray values this contribution reports results of an investigation about how such an oriented smoothness constraint may be formulated and evaluated

the influence of the environment on levy random search efficiency fractality and memory effects an open problem in the field of random searches relates to optimizing the search efficiency in fractal environments here we address this issue through a systematic study of levy searches in landscapes encompassing several degrees of target aggregation and fractality for scarce resources non destructive searches with unrestricted revisits to targets are shown to present universal optimal behavior irrespective of the general scaling properties of the spatial distribution of targets in contrast no such universal behavior occurs in the destructive case with forbidden revisits in which the optimal strategy strongly depends on the degree of target aggregation we also investigate how the presence of memory and learning skills of the searcher affect the search efficiency by considering a limiting model in which the searcher learns through recent experience to recognize food rich areas we find that a statistical memory of previous encounters does not necessarily increase the rate of target findings in random searches instead there is an optimal extent of memory dependent on specific details of the search space and stochastic dynamics which maximizes the search efficiency this finding suggests a more general result namely that in some instances there are actual advantages to ignoring certain pieces of partial information while searching for objects

d dimensional levy flights exact and asymptotic the analytic and asymptotic properties of the spherically symmetric d dimensional levy stable probability density function p d r are discussed in detail these isotropic stable probability density functions pdfs are analogous to the one dimensional symmetric levy stable pdfs previously studied by the present authors j math phys 43 2670 2002 we construct a hypergeometric representation of p d r when is rational and find a number of new representations of p d r in terms of special functions for various values of d and a recursion relation is found between p d r and p d 2 r which in particular implies there exists a simple map between p 1 r and p 3 r as in our previous paper we discuss the properties of p d r for both the cases 2 and 2 we demonstrate the existence of intricate exponentially small series in the large r asymptotics of p d r when is an integer which are dominant when is even we explicitly construct this beyond all orders expansion of p d r for arbitr

optimization of random searches on regular lattices we investigate random searches on isotropic and topologically regular square and triangular lattices with periodic boundary conditions and study the efficiency of search strategies based on a power law distribution p ensuremath ell ensuremath sim ensuremath ell ensuremath ensuremath mu of step lengths ensuremath ell we consider both destructive searches in which a target can be visited only once and nondestructive searches when a target site is always available for future visits we discuss i the dependence of the search efficiency on the choice of the lattice topology ii the relevance of the periodic boundary conditions iii the behavior of the optimal power law exponent ensuremath mu mathrm opt as a function of target site density iv the differences between destructive and nondestructive environments and finally v how the results for the discrete searches differ from the continuous cases previously studied

object based estimation of dense motion fields motion estimation belongs to key techniques in image sequence processing segmentation of the motion fields such that ideally each independently moving object uniquely corresponds to one region is one of the essential elements in object based image processing this paper is concerned with unsupervised simultaneous estimation of dense motion fields and their segmentations it is based on a stochastic model relating image intensities to motion information based on the analysis of natural images a region based model of motion compensated prediction error is proposed in each region the error is modeled by a white stationary generalized gaussian random process the motion field and its segmentation are themselves modeled by a compound gibbs markov random field accounting for statistical bindings in spatial direction and along the direction of motion trajectories the a posteriori distribution of the motion field for a given image sequence is formulated as an objective function such that its maximization results in the map estimate a deterministic multiscale relaxation technique with regular structure is employed for optimization of the objective function simulation results are in a good agreement with human perception for both the motion fields and their segmentations

computation and analysis of image motion a synopsis of current problems and methods the goal of this paper is to offer a structured synopsis of the problems in image motion computation and analysis and of the methods proposed exposing the underlying models and supporting assumptions a sufficient number of pointers to the literature will be given concentrating mostly on recent contributions emphasis will be on the detection measurement and segmentation of image motion tracking and deformable motion issues will be also addressed finally a number of related questions which could require more investigations will be presented

levy flights and random searches in this work we discuss some recent contributions to the random search problem our analysis includes superdiffusive levy processes and correlated random walks in several regimes of target site density mobility and revisitability we present results in the context of mean field like and closed form average calculations as well as numerical simulations we then consider random searches performed in regular lattices and lattices with defects and we discuss a necessary criterion for distinguishing true superdiffusion from correlated random walk processes we invoke energy considerations in relation to critical survival states on the edge of extinction and we analyze the emergence of levy behavior in deterministic search walks finally we comment on the random search problem in the context of biological foraging

domain decomposition for variational optical flow computation we present an approach to parallel variational optical flow computation by using an arbitrary partition of the image plane and iteratively solving related local variational problems associated with each subdomain the approach is particularly suited for implementations on pc clusters because interprocess communication is minimized by restricting the exchange of data to a lower dimensional interface our mathematical formulation supports various generalizations to linear nonlinear convex variational approaches three dimensional image sequences spatiotemporal regularization and unstructured geometries and triangulations results concerning the effects of interface preconditioning as well as runtime and communication volume measurements on a pc cluster are presented our approach provides a major step toward real time two dimensional image processing using off the shelf pc hardware and facilitates the efficient application of variational approaches to large scale image processing problems

improvements in the statistical approach to random levy flight searches recently it has been shown that the most e0cient strategy for searching randomly located objects when the sites are randomly distributed and can be revisited any number of times leads to a power law distribution p of the ights with 2 we show analytically that the incorporation of energy considerations limits the possible range for the l evy exponent however 2 still emerges as the optimal foraging condition furthermore we show that the probability distribution of ight lengths for the short and intermediate ight length regimes depends on the details of the system c 2001 elsevier science b v all rights reserved

multiple constraints to compute optical flow the computation of the optical flow field from an image sequence requires the definition of constraints on the temporal change of image features in this paper we consider the implications of using multiple constraints in the computational schema in the first step it is shown that differential constraints correspond to an implicit feature tracking therefore the best results either in terms of measurement accuracy and speed in the computation are obtained by selecting and applying the constraints which are best tuned to the particular image feature under consideration considering also multiple image points not only allows us to obtain a locally better estimate of the velocity field but also to detect erroneous measurements due to discontinuities in the velocity field moreover by hypothesizing a constant acceleration motion model also the derivatives of the optical flow are computed several experiments are presented from real image sequences

a note on dirichlet type criteria for complex sturm liouville expressions abstract second order differential expressions of the form w 1 pf qf are considered for real valued w 0 on a real interval i with one singular end point criteria are established for the expression to be dirichlet and strong limit point when p and q are permitted to be complex valued these extend criteria for real valued coefficients due to w n everitt h kalf and others

estimate large motions using the reliability based motion estimation algorithm detecting and estimating motions of fast moving objects has many important applications however most existing motion estimation techniques have difficulties in handling large motions in the scene in this paper we extend our recently proposed reliability based stereo vision technique to solving large motion estimation problem compared with our stereo vision approach the new algorithm removes the constant penalty assumption and explicitly enforces the inter scanline consistency constraint the resulting algorithm can handle sequences that contain large motions and can produce optical flows with 100 density over the entire image domain the experimental results indicate that it can generate more accurate optical flows than existing approaches