backward perturbation analysis of certain characteristic subspaces this paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemax x the generalized eigenvalue problem ax bx and the singular value decomposition of a matrixa this paper also gives residual bounds for certain eigenvalues generalized eigenvalues and singular values

reasoning about edges in scale space explores the role of reasoning in early vision processing in particular the problem of detecting edges is addressed the authors do not try to develop another edge detector but rather they study an edge detector rigorously to understand its behavior well enough to formulate a reasoning process that allow appliance of the detector judiciously to recover useful information they present a multiscale reasoning algorithm for edge recovery reasoning about edges in scale space ress the knowledge in ress is acquired from the theory of edge behavior in scale space and represented by a number of procedures ress recovers desired edge curves through a number of reasoning processes on zero crossing images at various scales the knowledge of edge behavior in scale space enables ress to select proper scale parameters recover missing edges eliminate noise or false edges and correct the locations of edges a brief evaluation of ress is performed by comparing it with two well known multistage edge detection algorithms

image processing for numerical approximations of conservation laws nonlinear anisotropic artificial dissipation we employ a nonlinear anisotropic diffusion operator like the ones used as a means of filtering and edge enhancement in image processing and in numerical methods for conservation laws it turns out that algorithms currently used in image processing are very well suited for the design of nonlinear higher order dissipative terms in particular we stabilize the well known lax wendroff formula by means of a nonlinear diffusion term

structured mapping problems for linearly structured matrices abstract given an appropriate class of structured matrices s we characterize matrices x and b for which there exists a matrix a s such that a x b and determine all matrices in s mapping x to b we also determine all matrices in s mapping x to b and having the smallest norm we use these results to investigate structured backward errors of approximate eigenpairs and approximate invariant subspaces and structured pseudospectra of structured matrices

non parametric dominant point detection abstract a new method for detecting dominant points is presented it does not require any input parameter and the dominant points obtained by this method remain relatively the same even when the object curve is scaled or rotated in this method for each boundary point a support region is assigned to the point based on its local properties each point is then smoothed by a gaussian filter with a width proportional to its determined support region a significance measure for each point is then computed dominant points are finally obtained through nonmaximum suppression unlike other dominant point detection algorithms which are sensitive to scaling and rotation of the object curve the new method will overcome this difficulty furthermore it is robust in the presence of noise the proposed new method is compared to the teh chin algorithm see c teh and r t chin ieee trans pattern anal mach intell 11 859 872 1989 in terms of the computational complexity the approximation errors and the number of detected dominant points of an object curve

robust contour decomposition using a constant curvature criterion the problem of decomposing an extended boundary or contour into simple primitives is addressed with particular emphasis on laplacian of gaussian zero crossing contours a technique is introduced for partitioning such contours into constant curvature segments a nonlinear blip filter matched to the impairment signature of the curvature computation process an overlapped voting scheme and a sequential contiguous segment extraction mechanism are used this technique is insensitive to reasonable changes in algorithm parameters and robust to noise and minor viewpoint induced distortions in the contour shape such as those encountered between stereo image pairs the results vary smoothly with the data and local perturbations induce only local changes in the result robustness and insensitivity are experimentally verified

recursive regularization filters design properties and applications least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed for each problem there is a family of discrete regularization filters r filters which allow an efficient determination of the solutions these operators are stable symmetric lowpass filters with an adjustable scale factor two decomposition theorems for the z transform of such systems are presented one facilitates the determination of their impulse response while the other allows an efficient implementation through successive causal and anticausal recursive filtering a case of special interest is the design of r filters for the first and second order difference operators these results are extended for two dimensional signals and for illustration purposes are applied to the problem of edge detection this leads to a very efficient implementation 8 multiplies 10 adds per pixel of the optimal canny edge detector based on the use of a separable second order r filter

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

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

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

reducing aliasing in images a pde based diffusion revisited in this paper we introduce a new diffusion algorithm that can be used for reducing aliasing on both step edges and lines it derives from the diffusion model of perona and malik and works as an adaptive level curve method in which diffusion is carried out in the normal direction of the gradient for step edges while the eigenvalues of the hessian matrix are used for lines to get sharp images we use high pass filters to preserve as much as possible the high frequency content while diffusing experimental tests using grayscale and colour images show that our algorithm efficiently reduces aliasing

scaling theorems for zero crossings of bandlimited signals scale space filtering is the only known method which provides a hierarchic signal description method by extracting features across a continuum of scales one of its important characteristics is that it demands the filtering involved does not create generic features as the scale increases it has been shown that the gaussian filter is unique in holding this remarkable property this is in essence the so called scaling theorem in this paper we propose two scaling theorems for band limited signals they are applicable to a broader class of signals and a bigger family of filtering kernels than in babaud et al 1986 yuille et al 1986 and wu xie 1990 an in depth discussion of our theorems and the previously published ones is also given

generalization to novel views universal class based andmodel based processing a major problem in object recognition is that a novel image of a given object can be different from all previously seen images images can vary considerably due to changes in viewing conditions such as viewing position and illumination in this paper we distinguish between three types of recognition schemes by the level at which generalization to novel images takes place universal class and model based the first is applicable equally to all objects the second to a class of objects and the third uses known properties of individual objects we derive theoretical limitations on each of the three generalization levels for the universal level previous results have shown that no invariance can be obtained here we show that this limitation holds even when the assumptions made on the objects and the recognition functions are relaxed we also extend the results to changes of illumination direction for the class level previous studies presented specific examples of classes of objects for which functions invariant to viewpoint exist here we distinguish between classes that admit such invariance and classes that do not we demonstrate that there is a tradeoff between the set of objects that can be discriminated by a given recognition function and the set of images from which the recognition function can recognize these objects furthermore we demonstrate that although functions that are invariant to illumination direction do not exist at the universal level when the objects are restricted to belong to a given class an invariant function to illumination direction can be defined a general conclusion of this study is that class based processing that has not been used extensively in the past is often advantageous for dealing with variations due to viewpoint and illuminant changes

adaptive determination of filter scales for edge detection the authors suggest a regularization method for determining scales for edge detection adaptively for each site in the image plane specifically they extend the optimal filter concept of t poggio et al 1984 and the scale space concept of a witkin 1983 to an adaptive scale parameter to avoid an ill posed feature synthesis problem the scheme automatically finds optimal scales adaptively for each pixel before detecting final edge maps the authors introduce an energy function defined as a functional over continuous scale space natural constraints for edge detection are incorporated into the energy function to obtain a set of optimal scales that can minimize the energy function a parallel relaxation algorithm is introduced experiments for synthetic and natural scenes show the advantages of the algorithm in particular it is shown that this system can detect both step and diffuse edges while drastically filtering out the random noise

on detecting dominant points abstract detecting dominant points is a crucial preprocessing step for shape recognition and point based motion estimation polygonal approximation has been a commonly used approach in detecting dominant points this paper presents two alternatives which detect stable dominant points in the first method we find a set of positive maximum and negative minimum curvature points along the gaussian smoothed boundary followed by a split and merge polygonal approximation algorithm the resulting break points vertices of the approximated polygon are the dominant points experimental results show that dominant points obtained by this method are less sensitive to the orientation of the boundary than other polygonal approximation algorithms in the sense that the number and the location of the dominant points along the contour remain relatively unchanged in the second method we smooth a boundary by a gaussian filter using various widths until the extreme curvature points remain relatively unchanged for a range of filter widths the resulting extreme curvature points which are stable to orientation and a reasonable range of scaling are the dominant points

on the edge location error for local maximum and zero crossing edge detectors examines the localization criterion for edge detection and determine the probability density function describing the edge location error canny 1986 defines the measure of localization as the reciprocal of the root mean square edge location error and formulates an expression of this measure for local maximum detectors however tagare and defigueiredo 1990 point out that an incorrect assumption is made in the calculation the same procedure is used by sarkar and boyer 1991 for their localization measure for zero crossing detectors we modify the analysis and obtain a closed form solution of the probability density function of the edge location error examination of the density function indicates the variance of the edge location error does not exist and hence cannot be used directly as a measure of localization

the geometry of differential operators with application to image processing a family of second order nonlinear differential operators that are useful in building simple edge detectors is treated the authors provide a uniform definition for these operators and describe their internal geometry based on their geometric analysis they associate symbolic descriptors with the abstract edges defined by the operators one of them q is essentially the same operator that j f canny ibid vol pami 8 p 552 7 1986 used to define his two dimensional simple edge detector the authors give a parse tree for q which provides a complete analysis of its local geometry the analysis shows that q has a fondness for edges that lift to certain types of asymptotic curves these include select straight line segments lying on the surface inflections of mean curvature and paths with constant rate of ascent the methods are drawn largely from vector calculus and differential geometry the authors illustrate their theoretical work with results obtained when q was applied to image data that were interpolated with a b spline

regularized laplacian zero crossings as optimal edge integrators we view the fundamental edge integration problem for object segmentation in a geometric variational framework first we show that the classical zero crossings of the image laplacian edge detector as suggested by marr and hildreth inherently provides optimal edge integration with regard to a very natural geometric functional this functional accumulates the inner product between the normal to the edge and the gray level image gradient along the edge we use this observation to derive new and highly accurate active contours based on this functional and regularized by previously proposed geodesic active contour geometric variational models we also incorporate a 2d geometric variational explanation to the haralick edge detector into the geometric active contour framework

edge and line feature extraction based on covariance models image segmentation based on contour extraction usually involves three stages of image operations feature extraction edge detection and edge linking this paper is devoted to the first stage a method to design feature extractors used to detect edges from noisy and or blurred images the method relies on a model that describes the existence of image discontinuities e g edges in terms of covariance functions the feature extractor transforms the input image into a log likelihood ratio image such an image is a good starting point of the edge detection stage since it represents a balanced trade off between signal to noise ratio and the ability to resolve detailed structures for 1 d signals the performance of the edge detector based on this feature extractor is quantitatively assessed by the so called average risk measure the results are compared with the performances of 1 d edge detectors known from literature generalizations to 2 d operators are given applications on real world images are presented showing the capability of the covariance model to build edge and line feature extractors finally it is shown that the covariance model can be coupled to a mrf model of edge configurations so as to arrive at a maximum a posteriori estimate of the edges or lines in the image

a multi scale edge detector abstract a multi scale edge detector with subpixel accuracy is described a subpixel laplacian edge detector recursively implemented is run at different scales and the recovered edge information is combined the multi scale edge detection is based on the behavior of edges in scale space and takes into account their physical phenomena with this purpose in mind four step edge models are considered the ideal the blurred the pulse and the staircase it is emphasized that the use of two scales the larger and the smaller is sufficient for good edge detection furthermore a set of rules is derived for combining edge information obtained from a laplacian detector which has some special properties however this type of edge detector gives at least two classes of false edges one of which cannot be eliminated by the usual thresholding methods an appropriate thresholding algorithm is given taking into account the origin of the false edges and their behavior in scale space

the hermite transform theory the author introduces a scheme for the local processing of visual information called the hermite transform the problem is addressed from the point of view of image coding and therefore the scheme is presented as an analysis resynthesis system the objectives of the present work however are not restricted to coding the analysis part is designed so that it can also serve applications in the area of computer vision indeed derivatives of gaussians which have found widespread application in feature detection over the past few years play a central role in the hermite analysis it is also argued that the proposed processing scheme is in close agreement with current insight into the image processing that is carried out by the human visual system in particular it is demonstrated that the hermite transform is in better agreement with human visual modeling than gabor expansions

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

face recognition the problem of compensating for changes in illumination direction a face recognition system must recognize a face from a novel image despite the variations between images of the same face a common approach to overcoming image variations because of changes in the illumination conditions is to use image representations that are relatively insensitive to these variations examples of such representations are edge maps image intensity derivatives and images convolved with 2d gabor like filters here we present an empirical study that evaluates the sensitivity of these representations to changes in illumination as well as viewpoint and facial expression our findings indicated that none of the representations considered is sufficient by itself to overcome image variations because of a change in the direction of illumination similar results were obtained for changes due to viewpoint and expression image representations that emphasized the horizontal features were found to be less sensitive to changes in the direction of illumination however systems based only on such representations failed to recognize up to 20 percent of the faces in our database humans performed considerably better under the same conditions we discuss possible reasons for this superiority and alternative methods for overcoming illumination effects in recognition

on optimal infinite impulse response edge detection filters the authors outline the design of an optimal computationally efficient infinite impulse response edge detection filter the optimal filter is computed based on canny s high signal to noise ratio good localization criteria and a criterion on the spurious response of the filter to noise an expression for the width of the filter which is appropriate for infinite length filters is incorporated directly in the expression for spurious responses the three criteria are maximized using the variational method and nonlinear constrained optimization the optimal filter parameters are tabulated for various values of the filter performance criteria a complete methodology for implementing the optimal filter using approximating recursive digital filtering is presented the approximating recursive digital filter is separable into two linear filters operating in two orthogonal directions the implementation is very simple and computationally efficient has a constant time of execution for different sizes of the operator and is readily amenable to real time hardware implementation

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

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

motion and structure from point correspondences with error estimation planar surfaces the determination of motion and structure of a planar scene from monocular image sequences is studied a new and simpler linear algorithm that gives closed form solutions for motion and structure parameters using point correspondences between two images assuming that the coplanar points undergo a rigid motion in 3 d space is presented a series of analytical results is established from two perspective views of a planar scene which is undergoing a rigid motion there are generally two normalized interpretations for motion parameters and the positions of the object plane these two interpretations one vertical and the other illusive are both valid in the sense that they render the same pair of images the authors identify all the special cases in which the number of interpretations is not two and derive necessary and sufficient geometrical conditions for those special cases to occur the approach to error estimation is based on the first order perturbation the estimated errors provide quantitative assessment for the accuracy of the solutions they also indicate degenerate or nearly degenerate configurations in the presence of noise

on the impact of anisotropic diffusion on edge detection content aware edge preserving smoothing techniques have gained visibility in recent years however they have had a rather limited impact on the edge detection literature compared to content unaware linear techniques often based on gaussian filters in this work we focus on anisotropic diffusion covering its initial definition by perona and malik and subsequent extensions a visual case study is used to illustrate their features we perform a quantitative evaluation of the performance of the canny method for edge detection when substituting linear gaussian smoothing filters by anisotropic diffusion highlightsreviewing the main trends in anisotropic diffusion identifying the practical scenarios on which anisotropic diffusion fails quantifying the improvement of edge detection when anisitropic diffusion is applied

an axiomatic approach to image interpolation we discuss possible algorithms for interpolating data given in a set of curves and or points in the plane we propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations the absolute minimal lipschitz extension model amle is singled out and studied in more detail we show experiments suggesting a possible application the restoration of images with poor dynamic range

an edge detection technique using genetic algorithm based optimization abstract in this paper we present a genetic algorithm based optimization technique for edge detection the problem of edge detection is formulated as one of choosing a minimum cost edge configuration the edge configurations are viewed as two dimensional chromosomes with fitness values inversely proportional to their costs the design of the crossover and the mutation operators in the context of the two dimensional chromosomal representation is described the knowledge augmented mutation operator which exploits knowledge of the local edge structure is shown to result in rapid convergence the incorporation of meta level operators and strategies such as the elitism strategy the engineered conditioning operator and adaptation of mutation and crossover rates in the context of edge detection are discussed and are shown to improve the convergence rate the genetic algorithm with various combinations of meta level operators is tested on synthetic and natural images the performance of the genetic algorithm based cost minimization technique is compared both qualitatively and quantitatively with local search based and simulated annealing based cost minimization approaches the genetic algorithm based technique is shown to perform very well in terms of robustness to noise rate of convergence and quality of the final edge image

automatic gradient threshold determination for edge detection we describe a method to automatically find gradient thresholds to separate edge from nonedge pixels a statistical model that is the weighted sum of two gamma densities corresponding to edge and nonedge pixels is used to identify a threshold results closely match human perceptual thresholds even under low signal to noise ratio snr levels

localized measurement of emergent image frequencies by gabor wavelets the authors derive implement and demonstrate a computational approach for the measurement of emergent image frequencies measuring emergent signal frequencies requires spectral measurements accurate in both frequency and time or space conflicting requirements that are shown to be balanced by a generalized uncertainty relationship such spectral measurements can be obtained from the responses of multiple wavelet like channel filters that sample the signal spectrum and that yield a locus of possible solutions for each locally emergent frequency it is shown analytically that this locus of solutions is maximally localized in both space and frequency if the channel filters used are gabor wavelets a constrained solution is obtained by imposing a stabilizing term that develops naturally from the assumptions on the signal the measurement of frequencies is then cast as an ill posed extremum problem regularized by the stabilizing term leading to an iterative constraint propagation algorithm the technique is demonstrated by application to a variety of 2 d textured images

design of iir digital differentiators using constrained optimization a new optimization method for the design of fullband and lowpass iir digital differentiators is proposed in the new method the passband phase response error is minimized under the constraint that the maximum passband amplitude response relative error is below a prescribed level for lowpass iir differentiators an additional constraint is introduced to limit the average squared amplitude response in the stopband so as to minimize any high frequency noise that may be present extensive experimental results are included which show that the differentiators designed using the proposed method have much smaller maximum phase response error for the same passband amplitude response error and stopband constraints when compared with several differentiators designed using state of the art competing methods

robust b spline image modeling with application to image processing we present a new approach to two dimensional 2 d robust spline image smoothing based on the m estimator algorithm unlike in other m estimator based image processing algorithms the new algorithm takes into consideration the spatial relations between picture elements the contribution of the sample to the model depends not only on the current residual of that sample but also on the neighboring residuals a smoothing parameter is estimated separately for each processing window and it adapts to the local structure of the image the proposed algorithm is applied to image filtering the resulting filter preserves details and suppresses additive gaussian and impulsive noise efficiently

gabor filter based edge detection abstract it is common practice to utilize evidence from biological and psychological vision experiments to develop computational models for low level feature extraction the receptive profiles of simple cells in mammalian visual systems have been found to closely resemble gabor filters daugman proved that gabor filters achieve joint minimal joint uncertainty these results led researchers to develop computational models based on gabor filters for several low level vision applications such as edge detection texture classification optical flow estimation and data compression in this paper the performance of a gabor odd filter based edge detector is investigated using the measures proposed by canny based on this performance analysis a design criterion for one dimensional 1d gabor filter based edge detector is derived it is shown that this design criterion also holds good for a two dimensional 2d gabor filter based edge detector experimental results are presented to demonstrate the performance of the gabor filter based edge detector

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

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

extensions of scale space filtering to machine sensing systems major components of scale space theory are gaussian filtering and the use of zero crossing thresholders and laplacian operators properties of scale space filtering may be useful for data analysis in multiresolution machine sensing systems however these systems typically violate the gaussian filter assumption and often the data analyses used e g trend analysis and classification are not consistent with zero crossing thresholders and laplacian operators the authors extend the results of scale space theory to include these more general conditions in particular it is shown that relaxing the requirement of linear scaling allows filters to have non gaussian spatial characteristics and that relaxing of the scale requirements s to 0 of the impulse response allows the use of scale space filters with limited frequency support i e bandlimited filters bandlimited scale space filters represent an important extension of scale space analysis for machine sensing

the multiscale veto model a two stage analog network for edge detection and image reconstruction this article presents the theory behind a model for a two stage analog network for edge detection and image reconstruction to be implemented in analog vlsi edges are detected in the first stage using the multiscale veto rule which states that an edge exists only if it can pass a threshold test for each of a set of smoothing filters of decreasing bandwidth the image is reconstructed in the second stage from the brightness values at the pixels between which edges occur the effect of the multiscale veto rule is that noise is removed with the efficiency of the narrowest band smoothing filter while edges are well localized to feature boundaries without having to identify maxima in the magnitude of the gradient unlike previous analog models for edge detection and reconstruction there are no problems of local minima and for any given set of parameters there is a unique solution the reconstructed images appear natural and are very similar visually to the originals

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

coping with discontinuities in computer vision their detection classification and measurement the general principles of detection classification and measurement of discontinuities are studied the following issues are discussed detecting the location of discontinuities classifying discontinuities by their degrees measuring the size of discontinuities and coping with the random noise and designing optimal discontinuity detectors an algorithm is proposed for discontinuity detection from an input signal s for degree k discontinuity detection and measurement a detector p phi is used where p is the pattern and phi is the corresponding filter if there is a degree k discontinuity at location t sub 0 then in the filter response there is a scaled pattern alpha p at t sub 0 where alpha is the size of the discontinuity this reduces the problem to searching for the scaled pattern in the filter response a statistical method is proposed for the approximate pattern matching to cope with the random noise a study is made of optimal detectors which minimize the effects of noise

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

detecting boundaries in a vector field a vector gradient approach is proposed to detect boundaries in multidimensional data with multiple attributes a vector field it is used to extend a gradient edge detector to color images the statistical effects of noise on the distribution of the amplitudes and directions of the vector gradient are characterized the noise behavior of the l sub 2 norm of the scalar gradients is also characterized for comparison when the attribute components are highly correlated as is often the case in color images use of the vector gradient shows a small gain in signal to noise ratio over that of the l sub 2 norm of the scalar gradients this small gain may or may not be significant depending on other measures an edge detector uses to deal with noise

image edge detection based on local dimension a complex networks approach abstract a novel approach based on local dimension of complex networks is proposed to address edge detecting issue in image processing a new method that the image was modeled to complex networks is proposed our method for mapping an image into complex network is based on the weighted combination of the euclidean distance and gray level similarity indices which differs from the way was done previously in several works the local dimension of node characters the local property of pixel it is observed that edge pixels obviously have lower node dimension than non edge pixels which is used in our proposed edge detection method the proposed method is applied to both synthetic and natural images the results show the efficiency of our proposed method

an edge detection technique using the facet model and parameterized relaxation labeling we present a method for detecting and labeling the edge structures in digital gray scale images in two distinct stages first a variant of the cubic facet model is applied to detect the location orientation and curvature of the putative edge points next a relaxation labeling network is used to reinforce meaningful edge structures and suppress noisy edges each node label of this network is a 3d vector parameterizing the orientation and curvature information of the corresponding edge point a hysteresis step in the relaxation process maximizes connected contours for certain types of images prefiltering by adaptive smoothing improves robustness against noise and spatial blurring

linear regularizing algorithms for positive solutions of linear inverse problems linear regularization methods provide a simple technique for determining stable approximate solutions of linear ill posed problems such as fredholm equations of the first kind cauchy problems for elliptic equations and backward solution of forward parabolic equations in most of these problems the solution must be positive to satisfy physical plausibility in this paper we consider ill posed first kind convolution equations and related problems such as numerical differentiation radon transform and laplace transform inversion we investigate several linear regularization algorithms which provide positive approximate solutions for these problems at least in the absence of errors on the data for noisy data the solution is not necessarily positive because the appearance of negative values can then only be an effect of the noise the negative part of the solution should be negligible with a suitable choice of the regularization parameter a price to pay for ensuring positivity is always however a reduction in resolution

modeling for edge detection problems in blurred noisy images the aim of this paper is to provide a theoretical set up and a mathematical model for the problem of image reconstruction the original image belongs to a family of two dimensional 2 d possibly discontinuous functions but is blurred by a gaussian point spread function introduced by the measurement device in addition the blurred image is corrupted by an additive noise we propose a preprocessing of data which enhances the contribution of the signal discontinuous component over that one of the regular part while damping down the effect of noise in particular we suggest to convolute data with a kernel defined as the second order derivative of a gaussian spread function finally the image reconstruction is embedded in an optimal problem framework now convexity and compactness properties for the admissible set play a fundamental role we provide an instance of a class of admissible sets which is relevant from an application point of view while featuring the desired properties

detection of composite edges the paper presents a new parametric model based approach to high precision composite edge detection using orthogonal zernike moment based operators it deals with two types of composite edges a generalized step and b pulse staircase edges a 2 d generalized step edge is modeled in terms of five parameters two gradients on two sides of the edge the distance from the center of the candidate pixel the orientation of the edge and the step size at the location of the edge a 2 d pulse staircase edge is modeled in terms of two steps located at two positions within the mask and the edge orientation a pulse edge is formed if the steps are of opposite polarities whereas a staircase edge results from two steps having the same polarity two complex and two real zernike moment based masks are designed to determine parameters of both the 2 d edge models for a given edge model estimated parameter values at a point are used to detect the presence or absence of that type of edge extensive noise analysis is performed to demonstrate the robustness of the proposed operators experimental results with intensity and range images are included to demonstrate the efficacy of the proposed edge detection technique as well as to compare its performance with the geometric moment based step edge detection technique and canny s 1986 edge detector

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

a luminance and contrast invariant edge similarity measure a novel similarity measure for edge detection that is robust to varying luminance and contrast is presented it incorporates a regularization term and employs directional fir edge filters with hyperbolic tangent profiles to ensure improved noise performance and edge localization compared to classical methods

toward a symbolic representation of intensity changes in images the symbolic representation of gray value variations is studied with emphasis on the gradient of the image function the goal is to relate the results of this analysis to the structure of the picture which is determined by the physics of the image generation process candidates for contour points are the maximal magnitudes of the gray value gradient for different scales in the direction of the gradient based on the output of such a bank of gradient filters a procedure is proposed to select automatically a suitable scale and with that the size of the right convolution kernel the application of poorly adapted filters which make the exact localization of gray value corners or t x and y junctions more difficult is thus avoided possible gaps at such junctions are discussed for images of real scenes and possibilities for the closure of some of these gaps are demonstrated when the extrema of the magnitudes of the gray value gradients are used

on the kahan parlett jiang theorem a globally optimal backward perturbation error for two sided invariant subspaces abstract a crucial question in large non hermitian eigenproblems is whether or not any residual bounds can be very useful in practical situations for this problem the famous kahan parlett jiang theorem kahan et al 1982 15 seems to be the best compromise between generality and sharpness in essence it presents an a posteriori error bound on approximate solutions to eigenproblems which provides a powerful tool to evaluate quality of the computed two sided invariant subspaces of large scale non hermitian matrices however the perturbation error determined by this theorem is only locally optimal rather than globally optimal in this work we revisit this problem and derive a globally optimal backward perturbation error for given two sided approximate invariant subspaces our theorem enhances the kahan parlett jiang theorem and numerical experiments demonstrate the effectiveness of our theoretical results

on detecting edges in speckle imagery methods for detecting sustained intensity changes in images corrupted by speckle are analyzed the problem is complicated by the nature of the speckle which is characterized by a high degree of correlation and approximately multiplicative signal dependence these characteristics make the automated extraction of edges in speckle very difficult for applications requiring the location and identifications of objects e g synthetic aperture radar in particular the laplacian of a gaussian log edge detector for this problem is analyzed while the log is found to be effective for detecting meaningful edges use of the log also gives rise to numerous extraneous edges having no physical correlate to alleviate these effects a ratio of averages edge detector is used in conjunction with the log the combined scheme is found to be much more effective than either of the individual edge detectors actual seasat synthetic aperture radar sar images are used to demonstrate the effectiveness of each technique

a multiscale edge detection algorithm based on wavelet domain vector hidden markov tree model abstract the wavelet analysis is an efficient tool for the detection of image edges based on the wavelet analysis we present an unsupervised learning algorithm to detect image edges in this paper a wavelet domain vector hidden markov tree wd vhmt is employed in our algorithm to model the statistical properties of multiscale and multidirectional subband wavelet coefficients of an image with this model each wavelet coefficient is viewed as an observation of its hidden state and the hidden state indicates if the wavelet coefficient belongs to an edge the wd vhmt model can be learned by an expectation maximization algorithm after the model is learned we employ an extended viterbi algorithm to uncover the hidden state sequences according to the maximum a posterior estimation the experiment results of the edge detection for several images are provided to evaluate our algorithm

hybrid image segmentation using watersheds and fast region merging a hybrid multidimensional image segmentation algorithm is proposed which combines edge and region based techniques through the morphological algorithm of watersheds an edge preserving statistical noise reduction approach is used as a preprocessing stage in order to compute an accurate estimate of the image gradient then an initial partitioning of the image into primitive regions is produced by applying the watershed transform on the image gradient magnitude this initial segmentation is the input to a computationally efficient hierarchical bottom up region merging process that produces the final segmentation the latter process uses the region adjacency graph rag representation of the image regions at each step the most similar pair of regions is determined minimum cost rag edge the regions are merged and the rag is updated traditionally the above is implemented by storing all rag edges in a priority queue we propose a significantly faster algorithm which additionally maintains the so called nearest neighbor graph due to which the priority queue size and processing time are drastically reduced the final segmentation provides due to the rag one pixel wide closed and accurately localized contours surfaces experimental results obtained with two dimensional three dimensional 2 d 3 d magnetic resonance images are presented

orthogonal moment operators for subpixel edge detection abstract a new approach to detect step edges with subpixel accuracy is presented the proposed approach is based on a set of orthogonal complex moments of the image known as zernike moments an ideal two dimensional 2d step edge is modeled in terms of four parameters the background gray level the step size the distance of the edge from the center of the mask and the orientation of the edge discrete zernike moments are used to obtain a total of three complex masks to compute all the edge parameters for subpixel detection for pixel level edge detection only two masks one real and one complex are required the theoretical analysis of the influence of noise on the location and the orientation of an edge is presented this analysis reveals that the accuracy of the proposed approach is virtually unaffected by the additive noise the technique is effective in detecting both the pixel level and subpixel level edges experimental results are presented to demonstrate the efficacy of the proposed technique

a contour based stereo matching algorithm using disparity continuity abstract a contour based stereo matching algorithm is presented using the disparity continuity property along image contours the basic idea is to first match specific high information points along a contour then use the matches found to guide other matches on the same contour by sequentially propagating the disparities along the length of the contour the matching algorithm consists of three stages 1 extraction of high curvature edge points terminations and junctions collectively refered to as extremal points 2 matching of the extremal points and 3 matching other edge points by propagating the measured disparities at the extremal points along contours we present a number of examples illustrating the performance of the algorithm in natural stereo images ranging from aerial images to laboratory images

self adaptive regularization often an image g x y is regularized and even restored by minimizing the mumford shah functional properties of the regularized image u x y depends critically on the numerical value of the two parameters spl alpha and spl beta controlling smoothness and fidelity when spl alpha and spl beta are constant over the image small details are lost when an extensive filtering is used in order to remove noise in this paper it is shown how the two parameters spl alpha and spl beta can be made self adaptive in fact spl alpha and spl beta are not constant but automatically adapt to the local scale and contrast of features in the image in this way edges at all scales are detected and boundaries are well localized and preserved in order to preserve trihedral junctions spl alpha and spl beta become locally small and the regularized image u x y maintains sharp and well defined trihedral junctions images regularized by the proposed procedure are well suited for further processing such as image segmentation and object recognition

a computational approach for corner and vertex detection corners and vertexes are strong and useful features in computer vision for scene analysis stereo matching and motion analysis here we deal with the development of a computational approach to these important features we consider first a corner model and study analytically its behavior once it has been smoothed using the well known gaussian filter this allows us to clarify the behavior of some well knowncornerness measure based approaches used to detect these points of interest most of these classical approaches appear to detect points that do not correspond to the exact position of the corner a new scale space based approach that combines useful properties from the laplacian and beaudet s measure beaudet 1978 is then proposed in order to correct and detect exactly the corner position an extension of this approach is then developed to solve the problem of trihedral vertex characterization and detection in particular it is shown that a trihedral vertex has two elliptic maxima on extremal contrast surfaces if the contrast is sufficient and this allows us to classify trihedral vertexes in 2 classes vertex and vertex as corner the corner detection approach developed is applied to accurately detect trihedral vertexes using an additional test in order to make a distinction between trihedral vertexes and corners many experiments have been carried out using noisy synthetic data and real images containing corners and vertexes most of the promising results obtained are used to illustrate the experimental section of this paper

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

invariant surface segmentation through energy minimization with discontinuities the computational problems in segmenting range data into surface patches based on the invariant surface properties i e mean curvature h and gaussian curvature k are investigated the goal is to obtain reliable hk surface maps two commonly encountered problems are firstly the noise effect in computing derivative estimates and secondly the smoothing across discontinuities here the segmentation is formulated as finding minimization solutions of energy functionals involving discontinuities a two stage approach to the goal is presented stage 1 from a range image to curvature images and stage 2 from the curvature images to the hk maps in both stages solutions are found through minimizing energy functionals that measure the degree of bias of a solution from two constraints the closeness of the solution to the data and the smoothness of the solution controlled by predetermined discontinuities propagation across discontinuities is prevented during minimization which preserves the original surface shapes experimental results are given for a variety of test images

multi edge detection by isotropical 2 d isef cascade most research in edge detection has been based on the classical mono edge model in the present paper we propose an optimal edge detector for complicated noisy multi edge images and the isotropical 2 d realization of the optimal filter we introduce at first the complicated multi edge model and show that the optimal filter as a preparation for multi edge detection is a cascade of infinite size symmetrical exponential filters isef we then propose a fast realization of the isotropical 2 d optimal filter experimental results are also presented

localization and noise in edge detection two aspects of edge detection are analyzed namely accuracy of localization and sensitivity to noise the detection of corners and trihedral vertices is analyzed for gradient schemes and zero crossing schemes it is shown that neither scheme correctly detects corners of trihedral vertices but that the gradient schemes are less sensitive to noise a simple but important conclusion is that the noise present in digital images of typical indoor scenes is small and the signal to noise ratio is high the noise present in digital images is so small as to make the performances of a variety of filters almost indistinguishable as a consequence small filters can be used and the exact shape of the filter is not critical

singularity theory and phantom edges in scale space the process of detecting edges in a one dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal it is shown that the second derivative of the contrast function at the critical point is related to the classification of the associated edge as being phantom or authentic the contrast of authentic edges decreases with filter scale while the contrast of phantom edges are shown to increase with scale as the filter scale increases an authentic edge must either turn into a phantom edge or join with a phantom edge and vanish the points in the scale space at which these events occur are seen to be singular points of the contrast function using ideas from singularity or catastrophy theory the scale map contours near these singular points are found to be either vertical or parabolic

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

unsupervised image restoration and edge location using compound gauss markov random fields and the mdl principle discontinuity preserving bayesian image restoration typically involves two markov random fields one representing the image intensities gray levels to be recovered and another one signaling discontinuities edges to be preserved the usual strategy is to perform joint maximum a posterori map estimation of the image and its edges which requires the specification of priors for both fields instead of taking an edge prior we interpret discontinuities in fact their locations as deterministic unknown parameters of the compound gauss markov random field cgmrf which is assumed to model the intensities this strategy should allow inferring the discontinuity locations directly from the image with no further assumptions however an additional problem emerges the number of parameters edges is unknown to deal with it we invoke the minimum description length mdl principle according to mdl the best edge configuration is the one that allows the shortest description of the image and its edges taking the other model parameters noise and cgmrf variances also as unknown we propose a new unsupervised discontinuity preserving image restoration criterion implementation is carried out by a continuation type iterative algorithm which provides estimates of the number of discontinuities their locations the noise variance the original image variance and the original image itself restored image experimental results with real and synthetic images are reported

a nonlinear derivative scheme applied to edge detection this paper presents a nonlinear derivative approach to addressing the problem of discrete edge detection this edge detection scheme is based on the nonlinear combination of two polarized derivatives its main property is a favorable signal to noise ratio snr at a very low computation cost and without any regularization a 2d extension of the method is presented and the benefits of the 2d localization are discussed the performance of the localization and snr are compared to that obtained using classical edge detection schemes tests of the regularized versions and a theoretical estimation of the snr improvement complete this work

edges transitions and criticality in this article various notions of edges encountered in digital image processing are reviewed in terms of compact representation or completion we show that critical exponents defined in statistical physics lead to a much more coherent definition of edges consistent across the scales in acquisitions of natural phenomena such as high resolution natural images or turbulent acquisitions edges belong to the multiscale hierarchy of an underlying dynamics they are understood from a statistical perspective well adapted to fit the case of natural images numerical computation methods for the evaluation of critical exponents in the non ergodic case are recalled which apply for the vast majority of natural images we study the framework of reconstructible systems in a microcanonical formulation show how it redefines edge completion and how it can be used to evaluate and assess quantitatively the adequation of edges as candidates for compact representations we study with particular attention the case of turbulent data in which edges in the classical sense are particularly challenged tests are conducted and evaluated on a standard database for natural images we test the newly introduced compact representation as an ideal candidate for evaluating turbulent cascading properties of complex images and we show better reconstruction performance than the classical tested methods highlightsclassical edge detection algorithms are reviewed in terms of completion a new edge detection scheme based on multiscale hierarchical classification assessing the consistency of edges across scales evaluating edge performance through the framework of reconstructible systems comparing the performance with classical edge operators

partial shape recognition a landmark based approach a method of recognizing partially occluded objects is presented in which each object is represented by a set of landmarks given a scene consisting of partially occluded objects a model object in the scene is hypothesized by matching the landmarks of the model with those in the scene a measure of similarity between two landmarks is needed to perform this matching a local shape measure sphericity is introduced it is shown that any invariant function under a similarity transformation is a function of the sphericity to match landmarks between the model and the scene a table of compatibility is constructed a technique known as hopping dynamic programming is described to guide the landmark matching through the compatibility table the location of the model in the scene is estimated with a least squares fit among the matched landmarks a heuristic measure is then computed to decide if the model is in the scene

roof edge detection using regularized cubic b spline fitting abstract a scheme employing one dimensional 1 d regularized cubic b spline rcbs fitting g chen and y h yang ieee trans systems man cybernet 25 636 693 task of step edge detection the regularized fitting is transformed into a quadratic energy equation to simplify the computation this scheme however has three major limitations it is non linear has a limited accuracy and is computationally expensive this paper presents a modified scheme which overcomes these limitations the modified scheme employs the 1 d rcbs fitting on the horizontal and vertical orientations of a window of an image to generate two 1 d signals which provide sufficient information about the local property of the sub image for roof edge detection experimental results show that the scheme of roof edge detection is very sensitive to small signals

the accuracy of the computation of optical flow and of the recovery of motion parameters the accuracy and the dependence on parameters of a general scheme for the analysis of time varying image sequences are discussed the approach is able to produce vector fields from which it is possible to recover 3 d motion parameters such as time to collision and angular velocity the numerical stability of the computed optical flow and the dependence of the recovery of 3 d motion parameters on spatial and temporal filtering is investigated by considering optical flows computed on subsampled images or along single scanlines it is also possible to recover 3 d motion parameters from reduced optical flows an adequate estimate of time to collision can be obtained from sequences of images with spatial resolution reduced to 128 128 pixels or from sequences of single scanlines passing near the focus of expansion the use of kalman filtering increases the accuracy and the robustness of the estimation of motion parameters the proposed approach seems to be able to provide not only a theoretical background but also practical tools that are adequate for the analysis of time varying image sequences

structured procrustes problem abstract let s be a class of structured matrices given a pair of matrices x and b we consider the structured procrustes problem spp a arg min g s g x b f and provide a complete solution when s is either a jordan algebra or a lie algebra associated with an orthosymmetric scalar product we characterize and determine all solutions of the structured procrustes problem as well as those solutions which have the smallest norm we show that for the spectral norm there may be infinitely many smallest norm solutions of the structured procrustes problem whereas for the frobenius norm the smallest norm solution is unique

a cost minimization approach to edge detection using simulated annealing the authors cast edge detection as a problem in cost minimization this is achieved by the formulation of a cost function that evaluates the quality of edge configurations the function is a linear sum of weighted cost factors the cost factors capture desirable characteristics of edges such as accuracy in localization thinness and continuity edges are detected by finding the edge configurations that minimize the cost function the authors give a mathematical description of edges and analyze the cost function in terms of the characteristics of the edges in minimum cost configurations through the analysis guidelines are provided on the choice of weights to achieve certain characteristics of the detected edges the cost function is minimized by the simulated annealing method a set of strategies is presented for generating candidate states and to devise a suitable temperature schedule

early jump out corner detectors the authors present two new corner detectors one works by using dissimilarity along the contour direction to detect curves in the image contour and the other estimates image curvature along the contour direction these operators are fast robust to noise and require no subjective thresholding

efficient image segmentation using partial differential equations and morphology the goal of this paper is to investigate segmentation methods that combine fast preprocessing algorithms using partial differential equations pdes with a watershed transformation with region merging we consider two well founded pde methods a nonlinear isotropic diffusion filter that permits edge enhancement and a convex nonquadratic variational image restoration method which gives good denoising for the diffusion filter an efficient algorithm is applied using an additive operator splitting aos that leads to recursive and separable filters for the variational restoration method a novel algorithm is developed that uses aos schemes within a gaussian pyramid decomposition examples demonstrate that preprocessing by these pde techniques significantly improves the watershed segmentation and that the resulting segmentation method gives better results than some traditional techniques the algorithm has linear complexity and it can be used for arbitrary dimensional data sets the typical cpu time for segmenting a 256 image on a modern pc is far below one second

an eigenstructure approach to edge detection a procedure for detecting step edges in noisy signals that does not involve any prefiltering of the data is proposed it locates step edges in 1 d noisy signals as follows first it computes the eigenvectors corresponding to the three smallest eigenvalues of a matrix formed with the discrete fourier transform of the given data next it estimates the edge locations by finding the local minima in the sum of the spectra of the computed eigenvectors the technique computes a point edge map for 2 d images by analyzing each row column and 45 degrees and 135 degrees diagonal in the image the computational complexity of the proposed procedure is determined

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