edge location to subpixel values in digital imagery a new method for locating edges in digital data to subpixel values and which is invariant to additive and multiplicative changes in the data is presented for one dimensional edge patterns an ideal edge is fit to the data by matching moments it is shown that the edge location is related to the so called christoffel numbers also presented is the study of the effect of additive noise on edge location the method is extended to include two dimensional edge patterns where a line equation is derived to locate an edge this in turn is compared with the standard hueckel edge operator an application of the new edge operator as an edge detector is also provided and is compared with sobel and hueckel edge detectors in presence and absence of noise

on the spaces lp x ω and wm p x ω in this paper we present some basic results on the generalized lebesgue spaces l p x omega and generalized lebesgue sobolev spaces w m w p x omega these results provide the necessary framework for the study of variational problems and elliptic equations with non standard p x growth conditions c 2001 academic press

boundary trace embedding theorems for variable exponent sobolev spaces we study boundary trace embedding theorems for variable exponent sobolev space w1 p let be an open bounded or unbounded domain in rn satisfying strong local lipschitz condition under the hypotheses that p l 1 infp x supp x n we prove that there is a continuous boundary trace embedding w1 p lq provided q a measurable function on satisfies condition p x q x n 1 p x n p x for x

superposition rules for nonlinear coupled first order differential equations pairs of coupled nonlinear differential equations of the polynomial type have been studied no higher powers than quadratic were considered lie s theorem provides a superposition rule exists if certain operators generate a finite dimensional lie algebra the lie algebras possible were divided into twenty categories for each case the general form of the coupled differential equations is obtained the coupled differential equations are then separated where possible solutions are obtained expressed in terms of a finite number of particular solutions

elliptic boundary value problems with nonstandard growth conditions abstract we study different boundary value problems to the p x laplacean equation namely of robin and signorini types we prove the existence and uniqueness of weak solutions to the problems under study and also their continuous dependence on the exponent data

existence and multiplicity of solutions for quasilinear nonhomogeneous problems an orlicz sobolev space setting abstract we study the boundary value problem div log 1 u q u p 2 u f u in u 0 on where is a bounded domain in r n with smooth boundary we distinguish the cases where either f u u p 2 u u r 2 u or f u u p 2 u u r 2 u with p q 1 p q min n r and r n p n p n p in the first case we show the existence of infinitely many weak solutions for any 0 in the second case we prove the existence of a nontrivial weak solution if is sufficiently large our approach relies on adequate variational methods in orlicz sobolev spaces

a multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids the goal of this correction is to correct a mistake that appears in the proof of both lemma 3 9 and 3 10 in the article mih ilescu r dulescu 2006 1 the mistake in the proof of lemma 3 9 and 3 10 is mainly owing to the fact that we wrongly computed the expression of function g x t

existence and multiplicity results for a new p x kirchhoff problem abstract in this work we study the existence and multiplicity results for the following nonlocal p x kirchhoff problem 0 1 a b 1 p x u p x d x d i v u p x 2 u u p x 2 u g x u in u 0 on where a b 0 are constants r n is a bounded smooth domain p c with n p x 1 is a real parameter and g is a continuous function the analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties

existence and non existence results for a quasilinear problem with nonlinear boundary condition abstract we study the problem div a x u p 2 u 1 x 1 u q 2 u h x u r 2 u in r n a x u p 2 u n b x u p 2 u g x u on u 0 in where is an unbounded domain with smooth boundary n denotes the unit outward normal vector on and 0 are real parameters we assume throughout that p q r p pn n p 1 p n n 1 q n p p n while a b and h are positive functions we show that there exist an open interval i and 0 such that the problem has no solution if i and 0 furthermore there exist an open interval j i and 0 0 such that for any j the above problem has at least a solution if 0 but it has no solution provided that 0 0 our paper extends previous results obtained by j chabrowski and k pfluger

barankin type lower bound on multiple change point estimation we compute lower bounds on the mean square error of multiple change point estimation in this context the parameters are discrete and the cramer rao bound is not applicable consequently we focus on computing the barankin bound bb the greatest lower bound on the covariance of any unbiased estimator which is still valid for discrete parameters in particular we compute the multi parameter version of the hammersley chapman robbins which is a barankin type lower bound we first give the structure of the so called barankin information matrix bim and derive a simplified form of the bb we show that the particular case of two change points is fundamental to finding the inverse of this matrix several closed form expressions of the elements of bim are given for changes in the parameters of gaussian and poisson distributions the computation of the bb requires finding the supremum of a finite set of positive definite matrices with respect to the loewner partial ordering although each matrix in this set of candidates is a lower bound on the covariance matrix of the estimator the existence of a unique supremum w r t to this set i e the tightest bound might not be guaranteed to overcome this problem we compute a suitable minimal upper bound to this set given by the matrix associated with the loewner john ellipsoid of the set of hyper ellipsoids associated to the set of candidate lower bound matrices finally we present some numerical examples to compare the proposed approximated bb with the performance achieved by the maximum likelihood estimator

existence and multiplicity of solutions for a neumann problem involving the p x laplace operator abstract in this paper we study the following nonlinear neumann boundary value problem div u p x 2 u u p x 2 u f x u x t r u v 0 x t r where r n is a bounded domain with smooth boundary u v is the outer unit normal derivative on 0 is a real number p is a continuous function on with inf x p x 1 f r r is a continuous function using the three critical point theorem due to ricceri under the appropriate assumptions on f we establish the existence of at least three solutions of this problem some known results are generalized

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

detection of intensity changes with subpixel accuracy using laplacian gaussian masks we present a system that takes a gray level image as input locates edges with subpixel accuracy and links them into lines edges are detected by finding zero crossings in the convolution of the image with laplacian of gaussian log masks the implementation differs markedly from m i t s as we decompose our masks exactly into a sum of two separable filters instead of the usual approximation by a difference of two gaussians dog subpixel accuracy is obtained through the use of the facet model 1 we also note that the zero crossings obtained from the full resolution image using a space constant for the gaussian and those obtained from the 1 n resolution image with 1 n pixel accuracy and a space constant of n for the gaussian are very similar but the processing times are very different finally these edges are grouped into lines using the technique described in 2

infinitely many solutions to elliptic problems with variable exponent and nonhomogeneous neumann conditions abstract the aim of this paper is to establish the existence of an unbounded sequence of weak solutions for a class of differential equations with p x laplacian and subject to small perturbations of nonhomogeneous neumann conditions the approach is based on variational methods

on achievable accuracy in edge localization edge localization occurs when an edge detector determines the location of an edge in an image the authors use statistical parameter estimation techniques to derive bounds on achievable accuracy in edge localization these bounds known as the cramer rao bounds reveal the effect on localization of factors such as signal to noise ratio snr extent of edge observed scale of smoothing filter and a priori uncertainty about edge intensity by using continuous values for both image coordinates and intensity the authors focus on the effect of these factors prior to sampling and quantization they also analyze the canny algorithm and show that for high snr its mean squared error is only a factor of two higher than the lower limit established by the cramer rao bound although this is very good the authors show that for high snr the maximum likelihood estimator which is also derived virtually achieves the lower bound

game theoretic integration for image segmentation robust segmentation of structures from an image is essential for a variety of image analysis problems however the conventional methods of region based segmentation and gradient based boundary finding are often frustrated by poor image quality here we propose a method to integrate the two approaches using game theory in an effort to form a unified approach that is robust to noise and poor initialization this combines the perceptual notions of complete boundary information using edge data and shape priors with gray level homogeneity using two computational modules the novelty of the method is that this is a bidirectional framework whereby both computational modules improve their results through mutual information sharing a number of experiments were performed both on synthetic datasets and datasets of real images to evaluate the new approach and it is shown that the integrated method typically performs better than conventional gradient based boundary finding

solutions for neumann boundary value problems involving p x laplace operators abstract in this paper we study the nonlinear neumann boundary value problem of the following form p div u p x 2 u u p x 2 u f x u in u p x 2 u g x u on using the variational method under appropriate assumptions on f and g we obtain a number of results on existence and multiplicity of solutions also what is also interesting is that we obtain some results which can be considered as extensions of the classical result named combined effects of concave and convex nonlinearities moreover the positive solution of the problem is considered

further study on the effect of boundary conditions abstract this paper concerns the least energy solutions to a semilinear elliptic equation the role of geometry of the boundary is studied when a parameter in the boundary condition tends to a critical index as well as to infinity

subpixel measurements using a moment based edge operator recent results in precision measurements using computer vision are presented an edge operator based on two dimensional spatial moments is given the operator can be implemented for virtually any size of window and has been shown to locate edges in digitized images to a twentieth of a pixel this accuracy is unaffected by additive or multiplicative changes to the data values the precision is achieved by correcting for many of the deterministic errors caused by nonideal edge profiles using a lookup table to correct the original estimates of edge orientation and location this table is generated using a synthesized edge which is located at various subpixel locations and various orientations the operator is extended to accommodate nonideal edge profiles and rectangularly sampled pixels the technique is applied to the measurement of imaged machined metal parts theoretical and experimental noise analyses show that the operator has relatively small bias in the presence of noise

existence of solutions for p x laplacian dirichlet problem this paper presents several sufficient conditions for the existence of solutions for the dirichlet problem of p x laplacian div u p x 2 u f x u x u 0 x especially an existence criterion for infinite many pairs of solutions for the problem is obtained the discussion is based on the theory of the spaces lp x and w01 p x

lower bounds on estimator performance for energy invariant parameters of multidimensional poisson processes using rate distortion theory lower bounds are developed for the mean square error of estimates of a random parameter of an m dimensional inhomogeneous poisson process with respect to which the energy i e the average number of points is invariant the bounds are derived without stringent assumptions on either the form of the intensity or the prior distribution of the parameter and they can handle random nuisance parameters the derivation makes use of a side information averaging principle applied to the distortion rate function and a maximum entropy property of energy constrained poisson processes under the additional assumption of conditional entropy invariance of the point process with respect to the parameter of interest an explicit bound is given which depends on the information discrimination between the inhomogeneous conditionally poisson process and a nearly homogeneous poisson process the application of the explicit bound is illustrated through a treatment of the problems of time shift estimation and relative time shift estimation for poisson streams

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

on detecting edges an edge in an image corresponds to a discontinuity in the intensity surface of the underlying scene it can be approximated by a piecewise straight curve composed of edgels i e short linear edge elements each characterized by a direction and a position the approach to edgel detection here is to fit a series of one dimensional surfaces to each window kernel of the operator and accept the surface description which is adequate in the least squares sense and has the fewest parameters a one dimensional surface is one which is constant along some direction the tanh is an adequate basis for the stepedge and its combinations are adequate for the roofedge and the line edge the proposed method of step edgel detection is robust with respect to noise for step size noise 2 5 it has subpixel position localization position and an angular localization better than 10 further it is designed to be insensitive to smooth shading these results are demonstrated by some simple analysis statistical data and edgelimages also included is a comparison of performance on a real image with a typical operator difference of gaussians the results indicate that the proposed operator is superior with respect to detection localization and resolution

infinitely many positive solutions for a p x kirchhoff type equation this paper is concerned with the existence of infinitely many positive solutions to a class of p x kirchhoff type problem by means of a direct variational approach and the theory of the variable exponent sobolev spaces we establish the existence of infinitely many distinct positive solutions whose w1 p x norms and l norms tend to zero under suitable hypotheses about nonlinearity

analysis of multiscale products for step detection and estimation we analyze discrete wavelet transform dwt multiscale products for detection and estimation of steps here the dwt is an over complete approximation to smoothed gradient estimation with smoothing varied over dyadic scale as developed by mallat and zhong 1992 the multiscale product approach was first proposed by rosenfeld 1970 for edge detection we develop statistics of the multiscale products and characterize the resulting non gaussian heavy tailed densities the results may be applied to edge detection with a false alarm constraint the response to impulses steps and pulses is also characterized to facilitate the analysis we employ a new general closed form expression for the cramer rao bound crb for discrete time step change location estimation the crb can incorporate any underlying continuous and differentiable edge model including an arbitrary number of steps the crb analysis also includes sampling phase offset effects and is valid in both additive correlated gaussian and independent and identically distributed i i d non gaussian noise we consider location estimation using multiscale products and compare results to the appropriate crb

existence results to elliptic systems with nonstandard growth conditions abstract nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics lecture notes in mathematics vol 1748 2000 c r acad sci paris ser i math 329 1999 393 398 math ussr izv 29 1987 33 66 in this work elliptic systems with nonstandard growth conditions are studied existence and multiplicity results under growth conditions on the reaction terms are established

image field categorization and edge corner detection from gradient covariance edges corners and vertices in an image correspond to 1d one dimensional and 2d discontinuities in the intensity surface of the underlying scene ridges and peaks correspond to 1d and 2d extrema in it all of them can be characterized by the distribution of gradients particularly by the dimensionality of it the approach to image field categorization here is to construct a covariance matrix of the gradient vector in each small window and apply the canonical correlation analysis to it schwarz s inequality on the matrix determinant and the related differential equation is the key to this analysis we obtain two operators p sub eg and q sub eg to categorize the image field into a unidirectionally varying region univar an omidirectionally varying region omnivar and a nonvarying region we investigate the conditions under which their absolute maximum response i e p sub eg 1 and q sub eg 1 occurs in the small window and show that they are respectively the desired 1d and 2d discontinuities extrema and omnivar is in many cases a 1d pattern in polar coordinates this leads to an algorithm to obtain further classification and accurate localization of them into edges ridges peaks corners and vertices through detailed analysis in the informative varying axis of them we examined and compared the performance of the operators and the localization algorithm on various types of images and various noise levels the results indicate that the proposed method is superior with respect to stability localization and resolution

a computational approach to edge detection this paper describes a computational approach to edge detection the success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points these goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution we define detection and localization criteria for a class of edges and present mathematical forms for these criteria as functionals on the operator impulse response a third criterion is then added to ensure that the detector has only one response to a single edge we use the criteria in numerical optimization to derive detectors for several common image features including step edges on specializing the analysis to step edges we find that there is a natural uncertainty principle between detection and localization performance which are the two main goals with this principle we derive a single operator shape which is optimal at any scale the optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a gaussian smoothed image we extend this simple detector using operators of several widths to cope with different signal to noise ratios in the image we present a general method called feature synthesis for the fine to coarse integration of information from operators at different scales finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge

existence results for the p laplacian with nonlinear boundary conditions in this paper we study the existence of nontrivial solutions for the problem pu u p 2u in a bounded smooth domain rn with a nonlinear boundary condition given by u p 2 u f u on the boundary of the domain the proofs are based on variational and topological arguments

solutions for p x laplacian dirichlet problems with singular coefficients using the critical point theory we investigate the existence and multiplicity of solutions for p x laplacian dirichlet problem with singular coefficients

on existence and multiplicity of solutions of neumann boundary value problems for quasi linear elliptic equations abstract in the present paper the following neumann boundary value problem involving the p laplacian 1 1 p u f x u in u n 0 on is studied and some new existence and multiplicity results of solutions for the problem 1 1 are obtained

overview of differential equations with non standard growth abstract differential equations with non standard growth have been a very active field of investigation in recent years in this survey we present an overview of the field as well as several of the most important results we consider both existence and regularity questions finally we provide a comprehensive list of papers published to date

existence and blow up rate of large solutions of p x laplacian equations with gradient terms abstract in this paper we investigate boundary blow up solutions of the problem p x u f x u k x u m x in u x as d x 0 where p x u div u p x 2 u is called the p x laplacian our results extend the previous work 25 of y liang q h zhang and c s zhao from the radial case to the non radial setting and 46 due to q h zhang and d motreanu from the assumption that k x u x m x is a small perturbation to the case in which k x u m x is a large perturbation we provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of d x and in terms of the growth of the exponents furthermore the comparison principle is no longer applicable in our context since f x is not assumed to be monotone in this paper

compact imbeddings between variable exponent spaces with unbounded underlying domain abstract in this paper we have established some compact imbedding theorems for some subspaces of w 1 p x u when the underlying domain u is unbounded the domain we consider is mainly of type r n n 2 or r l l 2 where r m is a bounded domain with smooth boundary

density of smooth functions in wk p x ω kovacik rakosnik investigated the spaces l p x omega of functions which are integrable with variable power p x and the corresponding counterparts of the sobolev spaces w k p x omega we continue that investigation and describe a class of functions p x for which the set of smooth functions on omega is dense in w k p x omega as a corollary we obtain in terms of the distance function a condition on elements of w k p x omega sufficient to ensure that they belong to w 0 k p x omega

existence of multiple positive solutions for inhomogeneous neumann problem in this paper we study the existence and nonexistence of multiple positive solutions for the inhomogeneous neumann boundary value problem u up u 0 withd u x under some assumptions on the boundary and the function x for x 0 x 0 x c it is shown that there exists a constant 0 such that problem possesses at least two positive solutions if and at least one positive solution if furthermore there are no positive solutions for problem if

the neumann boundary value problem of higher order quasilinear elliptic equation abstract this paper is concerned with the generalized solution to the neumann boundary value problem of higher order quasilinear elliptic equation 1 m 1 d a x m u f x u x b x m u g x u x we show that there exists at least one non trivial solution for the above problem in w m p x

existence of solutions for p x laplacian problems on a bounded domain in this paper we study the following p x laplacian problem div a x vd uv p x 2 d u b x v uv p x 2 u f x u x e u u 0 on p u where 1 p 1 l p x l p 2 n u s r n is a bounded domain and applying the mountain pass theorem we obtain the existence of solutions in w 0 1 p x for the p x laplacian problems in the superlinear and sublinear cases 2004 elsevier inc all rights reserved