fast algorithms for low level vision a recursive filtering structure is proposed that drastically reduces the computational effort required for smoothing performing the first and second directional derivatives and carrying out the laplacian of an image these operations are done with a fixed number of multiplications and additions per output point independently of the size of the neighborhood considered the key to the approach is first the use of an exponentially based filter family and second the use of the recursive filtering applications to edge detection problems and multiresolution techniques are considered and an edge detector allowing the extraction of zero crossings of an image with only 14 operations per output element at any resolution is proposed various experimental results are shown

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

state maps for linear systems modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differential algebraic equations as such state space models are not a natural starting point for modeling while they have utmost importance in the simulation and control phase the paper addresses the problem of computing state variables for systems of linear differential algebraic equations of various forms the point of view from which the problem is considered is the behavioral one as put forward in j c willems automatica j ifac 22 1986 pp 561 580 dynamics reported 2 1989 pp 171 269 ieee trans automat control 36 1991 pp 259 294

acceleration of the shiftable mbi o 1 algorithm for bilateral filtering and nonlocal means a direct implementation of the bilateral filter requires o s2 operations per pixel where s is the effective width of the spatial kernel a fast implementation of the bilateral filter that required o 1 operations per pixel with respect to s was recently proposed this was done by using trigonometric functions for the range kernel of the bilateral filter and by exploiting their so called shiftability property in particular a fast implementation of the gaussian bilateral filter was realized by approximating the gaussian range kernel using raised cosines later it was demonstrated that this idea could be extended to a larger class of filters including the popular non local means filter as already observed a flip side of this approach was that the run time depended on the width r of the range kernel for an image with dynamic range 0 t the run time scaled as o t2 r2 with r this made it difficult to implement narrow range kernels particularly for images with large dynamic range in this paper we discuss this problem and propose some simple steps to accelerate the implementation in general and for small r in particular we provide some experimental results to demonstrate the acceleration that is achieved using these modifications

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

on the gradient inverse weighted filter image processing d c c wang et al 1981 presented a gradient inverse weighted giw filter the proposed noise smoothing scheme is based on the observation that variations of gray levels inside a region are smaller than those between regions reducing noise without smearing edges is the advantage of this nonlinear filter as an improvement of the giw filter a new algorithm is introduced examples show that it can preserve edges and smooth noise more efficiently

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

two sided heat kernel estimates for censored stable like processes in this paper we study the precise behavior of the transition density func tions of censored resurrected stable like processes in c 1 1 open sets in r d where d 1 and 1 2 we first show that the semigroup of the censored stable like process in any bounded lipschitz open set is intrinsically ultracontractive we then establish sharp two sided estimates for the transition density functions of a large class of censored stable like processes in c 1 1 open sets we further obtain sharp two sided estimates for the green functions of these censored stable like processes in bounded c 1 1 open sets

improvements on fast space variant elliptical filtering using box splines it is well known that box filters can be efficiently computed using pre integration and local finite differences by generalizing this idea and by combining it with a nonstandard variant of the central limit theorem we had earlier proposed a constant time or o 1 algorithm that allowed one to perform space variant filtering using gaussian like kernels the algorithm was based on the observation that both isotropic and anisotropic gaussians could be approximated using certain bivariate splines called box splines the attractive feature of the algorithm was that it allowed one to continuously control the shape and size covariance of the filter and that it had a fixed computational cost per pixel irrespective of the size of the filter the algorithm however offered a limited control on the covariance and accuracy of the gaussian approximation in this paper we propose some improvements of our previous algorithm

trace asymptotics for fractional schrodinger operators this paper proves an analogue of a result of banuelos and sa barreto 5 on the asymptotic expansion for the trace of schrodinger operators on rd when the laplacian which is the generator of the brownian motion is replaced by the non local integral operator 2 0 2 which is the generator of the symmetric stable process of order these results also extend recent results of banuelos and yildirim 6 where the first two coefficients for 2 are computed some extensions to schrodinger operators arising from relativistic stable and mixed stable processes are obtained

recursive anisotropic 2 d gaussian filtering based on a triple axis decomposition we describe a recursive algorithm for anisotropic 2 d gaussian filtering based on separating the filter into the cascade of three rather two 1 d filters the filters operate along axes obtained by integer horizontal and or vertical pixel shifts this eliminates interpolation which removes spatial inhomogeneity in the filter and produces more elliptically shaped kernels it also results in a more regular filter structure which facilitates implementation in dsp chips finally it improves matching between filters with the same eccentricity and width but different orientations our analysis and experiments indicate that the computational complexity is similar to an algorithm that operates along two axes 11ms for a 512times512 image using a 3 2 ghz pentium 4 pc on the other hand given a limited set of basis filter axes there is an orientation dependent lower bound on the achievable aspect ratios

spatially variant convolution with scaled b splines we present an efficient algorithm to compute multidimensional spatially variant convolutions or inner products between n dimensional signals and b splines or their derivatives of any order and arbitrary sizes the multidimensional b splines are computed as tensor products of 1 d b splines and the input signal is expressed in a b spline basis the convolution is then computed by using an adequate combination of integration and scaled finite differences as to have for moderate and large scale values a computational complexity that does not depend on the scaling factor to show in practice the benefit of using our spatially variant convolution approach we present an adaptive noise filter that adjusts the kernel size to the local image characteristics and a high sensitivity local ridge detector

l p independence of spectral radius for generalized feynman kac semigroups under mild conditions on measures used in the perturbation we establish the l p independence of spectral radius for generalized feynman kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric markov processes these results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized feynman kac functionals developed by the second and third authors

the kalman yakubovich popov lemma in a behavioural framework the classical kalman yakubovich popov lemma provides a link between dissipativity of a system in state space form and the solution to a linear matrix inequality in this paper we derive the kyp lemma for linear systems described by higher order differential equations the result is an lmi in terms of the original coefficients in which the dissipativity problem is posed subsequently we study the connection between dissipativity and spectral factorization of polynomial matrices this enables us to derive a new algorithm for polynomial spectral factorization in terms of an lmi in the coefficients of a polynomial matrix

on the trace of symmetric stable processes on lipschitz domains it is shown that the second term in the asymptotic expansion as t 0 of the trace of the semigroup of symmetric stable processes fractional powers of the laplacian of order for any 0 2 in lipschitz domains is given by the surface area of the boundary of the domain this brings the asymptotics for the trace of stable processes in domains of euclidean space on par with those of brownian motion the laplacian as far as boundary smoothness is concerned

systems with sector bound nonlinearities a behavioral approach the class of nonlinear dynamical systems obtained by interconnection of a linear time invariant dynamical system and a memoryless time invariant sector bound nonlinearity has been widely studied in literature in this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems systems with slope restricted nonlinearities are also considered

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

trace estimates for stable processes in this paper we study the behaviour in time of the trace the partition function of the heat semigroup associated with symmetric stable processes in domains of rd in particular we show that for domains with the so called r smoothness property the second terms in the asymptotic as t 0 involves the surface area of the domain just as in the case of brownian motion

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

a parametrization for dissipative behaviors the behavioral equivalent of single input single output siso systems are behaviors with two manifest variables passive siso systems can therefore be viewed as j dissipative behaviors with two manifest variables here the special matrix j defines a qdf that captures the passivity property of siso systems in this paper we investigate more general qdfs q s induced by some operator these qdfs define some relation between the input the output and their derivatives of a siso system we characterize all behaviors that are dissipative with respect to the prescribed qdf q in fact we parametrize all the behaviors dissipative with respect to q in terms of those dissipative with respect to the special qdf qj induced by the matrix j similar results can also be given for lossless systems

regularity results for stable like operators for 1 2 we consider operators of the form lf x rd f x h f x 1 h 1 f x h a x h h d dh and for 0 1 we consider the same operator but where the f term is omitted we prove under appropriate conditions on a x h that any solution u to lu f will be in c if f c

heat kernel estimates and parabolic harnack inequalities for symmetric dirichlet forms abstract in this paper we consider the following symmetric dirichlet forms on a metric measure space m d e f g e c f g m m f x f y g x g y j d x d y where e c is a strongly local symmetric bilinear form and j d x d y is a symmetric radon measure on m m under general volume doubling condition on m d and some mild assumptions on scaling functions we establish stability results for upper bounds of heat kernel resp two sided heat kernel estimates in terms of the jumping kernels the cut off sobolev inequalities and the faber krahn inequalities resp the poincare inequalities we also obtain characterizations of parabolic harnack inequalities our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2

global heat kernel estimates for symmetric jump processes in this paper we study sharp heat kernel estimates for a large class of symmetric jump type processes in r d for all t 0 a prototype of the processes under consideration are symmetric jump processes on r d with jumping intensity

fast anisotropic gauss filtering we derive the decomposition of the anisotropic gaussian in a one dimensional 1 d gauss filter in the x direction followed by a 1 d filter in a nonorthogonal direction spl phi so also the anisotropic gaussian can be decomposed by dimension this appears to be extremely efficient from a computing perspective an implementation scheme for normal convolution and for recursive filtering is proposed also directed derivative filters are demonstrated for the recursive implementation filtering an 512 spl times 512 image is performed within 40 msec on a current state of the art pc gaining over 3 times in performance for a typical filter independent of the standard deviations and orientation of the filter accuracy of the filters is still reasonable when compared to truncation error or recursive approximation error the anisotropic gaussian filtering method allows fast calculation of edge and ridge maps with high spatial and angular accuracy for tracking applications the normal anisotropic convolution scheme is more advantageous with applications in the detection of dashed lines in engineering drawings the recursive implementation is more attractive in feature detection applications for instance in affine invariant edge and ridge detection in computer vision the proposed computational filtering method enables the practical applicability of orientation scale space analysis

sharp heat kernel estimates for relativistic stable processes in open sets in this paper we establish sharp two sided estimates for the transition densities of relativistic stable processes i e for the heat kernels of the operators m m2 2 in c1 1 open sets here m 0 and 0 2 the estimates are uniform in m 0 m for each fixed m 0 letting m 0 we recover the dirichlet heat kernel estimates for 2 2 in c1 1 open sets obtained in 14 sharp two sided estimates are also obtained for green functions of relativistic stable processes in bounded c1 1 open sets

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models

fast space variant elliptical filtering using box splines the efficient realization of linear space variant non convolution filters is a challenging computational problem in image processing in this paper we demonstrate that it is possible to filter an image with a gaussian like elliptic window of varying size elongation and orientation using a fixed number of computations per pixel the associated algorithm which is based upon a family of smooth compactly supported piecewise polynomials the radially uniform box splines is realized using preintegration and local finite differences the radially uniform box splines are constructed through the repeated convolution of a fixed number of box distributions which have been suitably scaled and distributed radially in an uniform fashion the attractive features of these box splines are their asymptotic behavior their simple covariance structure and their quasi separability they converge to gaussians with the increase of their order and are used to approximate anisotropic gaussians of varying covariance simply by controlling the scales of the constituent box distributions based upon the second feature we develop a technique for continuously controlling the size elongation and orientation of these gaussian like functions finally the quasi separable structure along with a certain scaling property of box distributions is used to efficiently realize the associated space variant elliptical filtering which requires o 1 computations per pixel irrespective of the shape and size of the filter

compactness of isospectral potentials the schrodinger operator v of a compact riemannian manifold m has pure point spectrum suppose that v 0 is a smooth reference potential various criteria are given which guarantee the compactness of all v satisfying spec v spec v 0 in particular compactness is proved assuming an a priori bound on the w s 2 m norm of v where s n 2 2 and n dim m this improves earlier work of bruning an example involving singular potentials suggests that the condition s n 2 2 is appropriate compactness is also proved for non negative isospectral potentials in dimensions n 9

fast o 1 bilateral filtering using trigonometric range kernels it is well known that spatial averaging can be realized in space or frequency domain using algorithms whose complexity does not scale with the size or shape of the filter these fast algorithms are generally referred to as constant time or o 1 algorithms in the image processing literature along with the spatial filter the edge preserving bilateral filter involves an additional range kernel this is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest the range kernel operates by acting on the pixel intensities this makes the averaging process nonlinear and computationally intensive particularly when the spatial filter is large in this paper we show how the o 1 averaging algorithms can be leveraged for realizing the bilateral filter in constant time by using trigonometric range kernels this is done by generalizing the idea presented by porikli i e using polynomial kernels the class of trigonometric kernels turns out to be sufficiently rich allowing for the approximation of the standard gaussian bilateral filter the attractive feature of our approach is that for a fixed number of terms the quality of approximation achieved using trigonometric kernels is much superior to that obtained by porikli using polynomials

symmetric stable laws and stable like jump diffusions asymptotic expansions are obtained for finite dimensional symmetric stable distributions they are used to prove the existence of continuous transition probability densities of stable and stable like jump diffusions and to construct local multiplicative asymptotics and global two sided estimates for these densities as a consequence the distribution of the first passage times for stable jump diffusions is estimated and the integral test for the limsup behaviour of their sample paths as t to 0 is provided 1991 mathematics subject classification 60e07 60g17 60j35 47d07

off diagonal upper estimates for the heat kernel of the dirichlet forms on metric spaces we give equivalent characterizations for off diagonal upper bounds of the heat kernel of a regular dirichlet form on the metric measure space in two settings for the upper bounds with the polynomial tail typical for jump processes and for the upper bounds with the exponential tail for diffusions our proofs are purely analytic and do not use the associated hunt process

barriers exit time and survival probability for unimodal levy processes we give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal levy processes in smooth domains

heat kernel estimates for subordinate brownian motions in this article we study transition probabilities of a class of subordinate brownian motions under mild assumptions on the laplace exponent of the corresponding subordinator sharp two sided estimates of the transition probability are established this approach in particular covers subordinators with laplace exponents that vary regularly at infinity with index one e g log 1 1 or log 1 2 0 2 that correspond to subordinate brownian motions with scaling order that is not necessarily stricty between 0 and 2 these estimates are applied to estimate green function potential of subordinate brownian motion we also prove the equivalence of the lower scaling condition of the laplace exponent and the near diagonal upper estimate of the transition estimate

dirichlet heat kernel estimates for subordinate brownian motions with gaussian components in this paper we derive explicit sharp two sided estimates for the dirichlet heat kernels in c 1 1 open sets d in r d of a large class of subordinate brownian motions with gaussian components when d is bounded our sharp two sided dirichlet heat kernel estimates hold for all t 0 integrating the heat kernel estimates with respect to the time variable t we obtain sharp two sided estimates for the green functions in bounded c 1 1 open sets of such subordinate brownian motions with gaussian components

bounds on green s functions of second order differential equations we estimate the diagonal part of the green s function for the equation 2 v x t x t 0 t 0 x b where b is a finite region of the euclidean space rd with a regular boundary in the special case v x 0 x b we also obtain bounds for the nondiagonal part of the green s function which are uniform in t

stochastic calculus over symmetric markov processes without time reversal we refine stochastic calculus for symmetric markov processes without using time reverse operators under some conditions on the jump functions of locally square integrable martingale additive functionals we extend nakao s divergence like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi everywhere starting points which are refinements of the previous results under the law for almost everywhere starting points this refinement of stochastic calculus enables us to establish a generalized fukushima decomposition for a certain class of functions locally in the domain of dirichlet form and a generalized ito formula

on reflected dirichlet spaces reflecting diffusion processes on smooth domains in euclidean space are well understood silverstein in 22 and 23 developed two variant procedures for constructing the reflected processes for a general class of symmetric hunt processes from a dirichlet space point of view a direct approach is given in this paper and these two variant procedures are shown to yield the same result only the techniques of martingales and ordinary markov processes are used

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

parabolic harnack inequality and heat kernel estimates for random walks with long range jumps we investigate the relationships between the parabolic harnack inequality heat kernel estimates some geometric conditions and some analytic conditions for random walks with long range jumps unlike the case of diffusion processes the parabolic harnack inequality does not in general imply the corresponding heat kernel estimates

heat content estimates over sets of finite perimeter abstract this paper studies by means of standard analytic tools the small time behavior of the heat content over a bounded lebesgue measurable set of finite perimeter by working with the set covariance function and by imposing conditions on the heat kernels applications concerning the heat kernels of rotational invariant stable processes are given

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance

kalman yakubovich lemma in the behavioural setting in this paper the kalman yakubovich lemma is formulated and proved in the behavioural theoretic setting the behavioural framework is used to generalize the lemma for behaviours with two manifest variables with respect to supply rates that are quadratic differential forms qdfs

convergence of symmetric markov chains on mathbb z d for each n let y n t be a continuous time symmetric markov chain with state space n 1 mathbb z d conditions in terms of the conductances are given for the convergence of the y n t to a symmetric markov process yt on mathbb r d we have weak convergence of y n t t leq t 0 for every t0 and every starting point the limit process y has a continuous part and may also have jumps