describing 1 d intensity transitions with gaussian derivatives at the resolutions matching the transition widths a full description of image edges requires a complete characterization of their local intensity transitions the spatial structure of those transitions and a description of adjacent image regions the authors propose as a step toward this end a 1 d algorithm for describing local intensity transitions by their gaussian derivatives at a resolution where the support of the gaussian smoothing matches their widths blur the algorithm estimates the transition width from the second derivative of 1 d gaussian response zero crossing slope and leads to a characterization of the transition with its first three derivatives at the resolution matching the width the authors describe how this algorithm can be applied to images and give an example

finite element methods for active contour models and balloons for 2 d and 3 d images the use of energy minimizing curves known as snakes to extract features of interest in images has been introduced by kass witkin and terzopoulos 1987 a balloon model was introduced by cohen 1991 as a way to generalize and solve some of the problems encountered with the original method a 3 d generalization of the balloon model as a 3 d deformable surface which evolves in 3 d images is presented it is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential we also show properties of energy minimizing surfaces concerning their relationship with 3 d edge points to solve the minimization problem for a surface two simplified approaches are shown first defining a 3 d surface as a series of 2 d planar curves then after comparing finite element method and finite difference method in the 2 d problem we solve the 3 d model using the finite element method yielding greater stability and faster convergence this model is applied for segmenting magnetic resonance images

a generalized depth estimation algorithm with a single image a depth estimation algorithm proposed by a p pentland 1987 is generalized in the proposed algorithm the raw image data in the vicinity of the edge is used to estimate the depth from defocus since no differentiation operation on the image data is required before the optimization process the method is less sensitive to the noise disturbance of measurements furthermore the edge orientation that was critical in pentland s approach will not be required in the case this algorithm is then applied to synthetic images containing various amounts of noise to test its performance experimental results indicate that the depth estimation errors are kept within 5 of true values on the average when it is applied to real images

the asymptotic optimal frequency domain filter for edge detection in an earlier paper by shanmugam dickey and green an edge detection filter was derived which maximized the energy within a specified interval about an edge feature the initial expression of this filter involved a prolate spheroidal wave function however a careful analysis of the application of an asymptotic approximation to this function uncovered a major dimensional error the corrected derivation of the asymptotic optimal filter forms the subject of this paper to verify the results the filter found is compared to a similar filter developed independently by marr and hildreth

inequalities in metric spaces with applications abstract analogues of the parallelogram identity and the cn inequality of bruhat and tits in uniformly convex metric spaces are established as an application of the new inequalities we prove two fixed point results for single valued and multi valued lipschitzian mappings

on the minimality and global consistency of row convex constraint networks constraint networks have been shown to be useful in formulating such diverse problems as scene labeling natural language parsing and temporal reasoning given a constraint network we often wish to i find a solution that satisfies the constraints and ii find the corresponding minimal network where the constraints are as explicit as possible both tasks are known to be np complete in the general case task 1 is usually solved using a backtracking algorithm and task ii is often solved only approximately by enforcing various levels of local consistency in this paper we identify a property of binary constraint called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent globally consistent networks have the property that a solution can be found without backtracking we show that one can test for the row convexity property efficiently and we show by examining applications of constraint networks discussed in the literature that our results are useful in practice thus we identify a class of binary constraint networks for which we can solve both tasks i and ii efficiently finally we generalize the results for binary constraint networks to networks with nonbinary constraints

tractable constraints on ordered domains abstract finding solutions to a constraint satisfaction problem is known to be an np complete problem in general but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted in this paper we identify a restricted set of contraints which gives rise to a class of tractable problems this class generalizes the notion of a horn formula in propositional logic to larger domain sizes we give a polynomial time algorithm for solving such problems and prove that the class of problems generated by any larger set of constraints is np complete

point algebras for temporal reasoning algorithms and complexity we investigate the computational complexity of temporal reasoning in different time models such as totally ordered partially ordered and branching time our main result concerns the satisfiability problem for point algebras and point algebras extended with disjunctions for these problems we identify all tractable subclasses we also provide a number of additional results for instance we present a new time model suitable for reasoning about systems with a bounded number of unsynchronized clocks we investigate connections with spatial reasoning and we present improved algorithms for deciding satisfiability of the tractable point algebras

on some lp inequalities in best approximation theory h x jxp p ax p is strictly increasing in the interval p j o thus h x 0 has a unique solution x p note that x l 2 1 x p is strictly increasing in p and x 0 is the unique solution of the equation p 2 xp p l f2 1 in the interval 0 6 x 1 1 1 in 11 we proved the inequality

a sufficient condition for backtrack bounded search backtrack search is often used to solve constraint satisfaction problems a relationship involving the structure of the constraints is described that provides a bound on the backtracking required to advance deeper into the backtrack tree this analysis leads to upper bounds on the effort required for solution of a class of constraint satisfaction problems the solutions involve a combination of relaxation preprocessing and backtrack search the bounds are expressed in terms of the structure of the constraint connections specifically the effort is shown to have a bound exponential in the size of the largest biconnected component of the constraint graph as opposed to the size of the graph as a whole

on edge detection edge detection is the process that attempts to characterize the intensity changes in the image in terms of the physical processes that have originated them a critical intermediate goal of edge detection is the detection and characterization of significant intensity changes this paper discusses this part of the edge detection problem to characterize the types of intensity changes derivatives of different types and possibly different scales are needed thus we consider this part of edge detection as a problem in numerical differentiation we show that numerical differentiation of images is an ill posed problem in the sense of hadamard differentiation needs to be regularized by a regularizing filtering operation before differentiation this shows that this part of edge detection consists of two steps a filtering step and a differentiation step following this perspective the paper discusses in detail the following theoretical aspects of edge detection 1 the properties of different types of filters with minimal uncertainty with a bandpass spectrum and with limited support are derived minimal uncertainty filters optimize a tradeoff between computational efficiency and regularizing properties 2 relationships among several 2 d differential operators are established in particular we characterize the relation between the laplacian and the second directional derivative along the gradient zero crossings of the laplacian are not the only features computed in early vision 3 geometrical and topological properties of the zero crossings of differential operators are studied in terms of transversality and morse theory

on the classification of image regions by colour texture and shape abstract a unified approach to how colour texture and shape can be encoded in a single spatio chromatic feature space is developed in particular it is shown how such features can be extracted using multi scaled filtering and correlation methods which capture the variations of colour over space in ways which encode important image features not extracted by techniques which separate colour texture and shape into separate channels finally it is shown how such spatio chromatic features can be used in difficult classification problems

using canny s criteria to derive a recursively implemented optimal edge detector a highly efficient recursive algorithm for edge detection is presented using canny s design 1 we show that a solution to his precise formulation of detection and localization for an infinite extent filter leads to an optimal operator in one dimension which can be efficiently implemented by two recursive filters moving in opposite directions in addition to the noise truncature immunity which results the recursive nature of the filtering operations leads with sequential machines to a substantial saving in computational effort five multiplications and five additions for one pixel independent of the size of the neighborhood the extension to the two dimensional case is considered and the resulting filtering structures are implemented as two dimensional recursive filters hence the filter size can be varied by simply changing the value of one parameter without affecting the time execution of the algorithm performance measures of this new edge detector are given and compared to canny s filters various experimental results are shown

on the location error of curved edges in low pass filtered 2 d and 3 d images the authors study the location error of curved edges in two and three dimensional images after analog and digital low pass filtering the zero crossing of a second derivative filter is a well known edge localization criterion the second derivative in gradient direction sdgd produces a predictable bias in edge location towards the centers of curvature while the linear laplace filter produces a shift in the opposite direction their sum called plus plus laplace sdgd leads to an edge detector that finds curved edges one order more accurately than its constituents this argument holds irrespective of the dimension the influence of commonly used low pass filters such as the psf originating from diffraction limited optics using incoherent light 2 d the gaussian filter with variable cutoff point d d and the isotropic uniform filter d d is studied

optimal edge detector design i parameter selection and noise effects this paper comments on the optimality of the laplacian of a gaussian edge detection filter which localizes edges through zero crossings in the filtered image the arguments of both marr and hildreth and dickey and shanmugam are reviewed to establish that the filter is optimal in the sense of maximizing output image energy near edge features this filter s principal advantage over other edge detectors is that its response is user adjustable through selection of a single parameter the gaussian standard deviation however no clear method for the selection of this parameter has been provided the problem is addressed here by applying the filter to two ideal periodic edge models blurred by a gaussian distribution point spread function the observed response to the edge spacing and blur standard deviation is then translated into a filter parameter design procedure the problems of optimum filter performance in the presence of additive gaussian noise are then addressed the problem of selecting the sampled filter s coefficient word size is dealt with in a companion paper

precision edge contrast and orientation estimation the contrast and orientation estimation accuracy of several edge operators that have been proposed in the literature is examined both for the noiseless case and in the presence of additive gaussian noise the test image is an ideal step edge that has been sampled with a square aperture grid the effects of subpixel translations and rotations of the edge on the performance of the operators are studied it is shown that the effect of subpixel translations of an edge can generate more error than moderate noise levels methods with improved results are presented for sobel angle estimates and the nevatia babu operator and theoretical noise performance evaluations are also provided an edge operator based on two dimensional spatial moments is presented all methods are compared according to worst case and rms error in an ideal noiseless situation and rms error under various noise levels

optimal filters for the detection of linear patterns in 2 d and higher dimensional images abstract it is shown that the rotation invariant operators form a set of orthogonal basis functions which is optimal for a wide range of 2 d and 3 d feature detection operations the optimality criterion used is based on the karhunen loeve expansion of the local image data this is a generalization of the work of zucker and hummel who showed that these filters are optimal filters for the detection of step edges in 2 d and 3 d images

color edge extraction using orthogonal polynomials based zero crossings scheme abstract a new color edge extraction scheme based on identifying the zero crossings in the second directional derivatives in terms of orthogonal polynomials has been proposed in this paper the proposed scheme supports a complete set of finite difference operators which are easily configured from orthogonal polynomials a color image region in presence of gaussian noise is represented as a linear combination of these operators towards color edges and noise statistical hypothesis testing procedures are employed to separate out the responses towards noise from those towards color edges the usage of the proposed framework is highlighted in devising a color edge detector based on identifying the zero crossings in the second directional derivatives of the color image region the proposed color edge extraction scheme is also compared with two existing color edge extraction schemes

fixed point theorems for uniformly lipschitzian semigroups in uniformly convex spaces for all x y in c and each n 1 2 goebel and kirk 2 first studied these mappings and they proved that such t has a fixed point if c is a closed bounded convex subset of a uniformly convex banach space x and k k k is the unique solution of the equation 1 a l k k 1 with 6 being the modulus of convexity of x lifschitz s showed that in a hilbert space i a uniformly k lipschitzian mapping t with k c 21j2 has a fixed point lim 6 7 and lim et al 8 extended goebel kirk s theorem in lp spaces recently downing and ray l and ishihara and takahashi 3 verified that lifschitz s theorem is valid for uniformly lipschitzian semigroups in hifbert spaces the purpose of this paper is to prove fixed point theorems for a uniformly k lipschitzian semigroup in uniformly convex banach spaces which is left reversible or for which the space of right uniformly continuous functions on the semigroup has a left invariant mean the results generalize those of downing and ray l lau 4 lim 6 7 and ishihara and takahashi 3

an initial study of time complexity in infinite domain constraint satisfaction abstract the constraint satisfaction problem csp is a widely studied problem with numerous applications in computer science and artificial intelligence for infinite domain csps there are many results separating tractable and np hard cases while upper and lower bounds on the time complexity of hard cases are virtually unexplored hence we initiate a study of the worst case time complexity of such csps we analyze backtracking algorithms and determine upper bounds on their time complexity we present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well studied problems in for instance temporal reasoning finally we prove non trivial lower bounds applicable to many interesting csps under the assumption that certain complexity theoretic assumptions hold the gap between upper and lower bounds is in many cases surprisingly small which suggests that our upper bounds cannot be significantly improved

stereo correspondence by surface reconstruction an algorithm that solves the computational stereo correspondence and the surface reconstruction is presented the algorithm integrates the reconstruction process in the correspondence analysis by means of multipass attribute matching and disparity refinement in the matching process the requirement of attribute similarity is relaxed with the pass number while the requirement for agreement between the predicted and the measured disparity is tightened disparity discontinuities and occluded areas are detected by analyzing the partial derivatives of the reconstructed disparity surface results on synthetic and on real stereo image pairs are reported

building tractable disjunctive constraints many combinatorial search problems can be expressed as constraint satisfaction problems this class of problems is known to be np hard in general but a number of restricted constraint classes have been identified which ensure tractability this paper presents the first general results on combining tractable constraint classes to obtain larger more general tractable classes we give examples to show that many known examples of tractable constraint classes from a wide variety of different contexts can be constructed from simpler tractable classes using a general method we also construct several new tractable classes that have not previously been identified

on the complexity of qualitative spatial reasoning a maximal tractable fragment of the region connection calculus the computational properties of qualitative spatial reasoning have been investigated to some degree however the question for the boundary between polynomial and np hard reasoning problems has not been addressed yet in this paper we explore this boundary in the region connection calculus rcc 8 we extend bennett s encoding of rcc 8 in modal logic based on this encoding we prove that reasoning is np complete in general and identify a maximal tractable subset of the relations in rcc 8 that contains all base relations further we show that for this subset path consistency is sufficient for deciding consistency

detection localization and estimation of edges a method to detect locate and estimate edges in a one dimensional signal is presented it is inherently more accurate than all previous schemes as it explicitly models and corrects interaction between nearby edges the method is iterative with initial estimation of edges provided by the zero crossings of the signal convolved with laplacian of gaussian log filter the necessary computations necessitate knowledge of this convolved output only in a neighborhood around each zero crossing and in most cases could be performed locally by independent parallel processors results on one dimensional slices extracted from real images and on images which have been proposed independently in the row and column directions are shown an analysis of the method is provided including issues of complexity and convergence and directions of future research are outlined

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

disjunctions independence refinements an important question in constraint satisfaction is how to restrict the problem to ensure tractability since the general problem is np hard the use of disjunctions has proven to be a useful method for constructing tractable constraint classes from existing classes the well known max closed and ord horn constraints are examples of tractable classes that can be constructed this way three sufficient conditions the guaranteed satisfaction property 1 independence and 2 independence that each ensure the tractability of constraints combined by disjunctions have been proposed in the literature we show that these conditions are both necessary and sufficient for tractability in three different natural classes of disjunctive constraints this suggests that deciding this kind of property is a very important task when dealing with disjunctive constraints we provide a simple automatic method for checking the 1 independence property this method is applicable whenever the consistency of the constraints under consideration can be decided by path consistency our method builds on a connection between independence and refinements which is a way of reducing one constraint satisfaction problem to another

analysis of multichannel narrow band filters for image texture segmentation a model for texture analysis and segmentation using multiple oriented channel filters is analyzed in the general framework several different arguments are applied leading to the conclusion that the two dimensional gabor filters possess strong optimality properties for this task properties of the multiple channel segmentation approach are analyzed in particular perturbations of textures from an ideal model are found to have important effects on the segmentation that can usually be ameliorated by simply preceding the segmentation process by a logarithmic operation and using a low pass postfilter prior to making region assignments the difficult problems of space variant textures and multiple component textures are also considered local spatial frequency estimation approaches are suggested that use the responses as constraints in estimating the locally emergent texture frequencies complex texture aggregates containing multiple shared frequency components can be analyzed if the textures are distinct and few in number

on the localization performance measure and optimal edge detection the localization performance measure of edge detectors is addressed a one dimensional formulation of the problem is considered a linear space invariant filter is used for the detection the locations of local maxima in the thresholded output of the filter are declared to be the edges the limitations of conventional performance measures are shown and a localization performance measure for edge detection is suggested this performance measure is based on the theory of zero crossings of stochastic processes it is shown that the derivative of a gaussian is the optimal edge detector for the measure

on an inequality of bynum and drew an extension of the bynum drew inequality in the abstract lp space 1 p 2 is presented the extended inequality is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly lipschitzian mappings and to prove an estimate for the self jung constants

inequalities in banach spaces with applications well known that many problems in a hilbert space h are solved by applying the following identity ilax 1 a yi12 a412 1 llyl12 a 1 mx yl12 1 1 for all x y in h and 0 5 i 5 1 therefore one natural method to solve problems in a banach space xis to establish equalities and usually inequalities in xanalogous to 1 1 this is in fact shown by some authors recent work see for example reich l poffald and reich 2 lim 3 lim et af 4 prus and smarzewski 5 smarzewski 6 and reich 21 the purpose of this paper is to continue carrying out investigations in this line more precisely in section 3 we establish some inequalities analogous to 1 l in either uniformly convex or uniformly smooth banach spaces as corollaries we obtain new characteristics of p uniform convexity and q uniform smoothness of a banach space x via the functionals i lip and ii llq respectively we give in section 4 applications of the inequalities obtained in section 3 to fixed points of uniformly lipschitzian mappings and to best approximation theory

reasoning about temporal relations a maximal tractable subclass of allen s interval algebra we introduce a new subclass of allen s interval algebra we call ord horn subclass which is a strict superset of the pointisable subclass we prove that reasoning in the ord horn subclass is a polynomial time problem and show that the path consistency method is sufficient for deciding satisfiability further using an extensive machine generated case analysis we show that the ord horn subclass is a maximal tractable subclass of the full algebra assuming p np in fact it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations

the influence of edge direction on the estimation of edge contrast and orientation abstract this paper concerns the influence of edge direction on the estimation of edge contrast and orientation we present an analytical study of general first order detectors we show that the gradient estimated using radial filters is not affected by edge orientation for non radial filters the gradient may be affected by edge orientation for instance we find that the estimated edge orientation using a non radial filter may be biased even if the image is noise free however there are non radial filters for which gradient is unaffected by edge orientation as in the case of radial filters the properties of these functions are given in this paper the results are illustrated by studying the detectors of canny deriche and shen castan we take into account discretization effects such as subpixel and tesselation of the image plane the quantitative results provide insights into the estimation of gradient by these commonly used detectors taken together the theoretical and experimental results give a clear indication of the effect of the rotation invariance property of an edge detector on its response thus providing a more precise meaning for this property in edge detection

a variational approach for the segmentation of the left ventricle in cardiac image analysis in this paper we propose a level set method to segment mr cardiac images our approach is based on a coupled propagation of two cardiac contours and integrates visual information with anatomical constraints the visual information is expressed through a gradient vector flow based boundary component and a region term that aims at best separating the cardiac contours regions according to their global intensity properties in order to deal with misleading visual support an anatomical constraint is considered that couples the propagation of the cardiac contours according to their relative distance the resulting motion equations are implemented using a level set approach and a fast and stable numerical approximation scheme the additive operator splitting encouraging experimental results are provided using real data