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

on automatic threshold selection for polygonal approximations of digital curves polygonal approximation is a very common representation of digital curves a polygonal approximation depends on a parameter e which is the error value in this paper we present a method for an automatic selection of the error value e let e be a polygonal approximation of the original curve with an error value e we define a set of function ns e s s s such that for a given value of s ns e is the number of edges that contain at least s vertices in e the time complexity for computing the set of functions ns e s ss s is almost linear in n the number of vertices in in this paper we analyse the ns graph and show that for adequate values of s a wide plateau is expected to appear at the top of the graph this plateau corresponds to a stable state in the multi scale representation of e we show that the functions ns s s are a statistical representation of some kind of scale space image

review of shape representation and description techniques more and more images have been generated in digital form around the world there is a growing interest in 1nding images in large collections or from remote databases in order to 1nd an image the image has to be described or represented by certain features shape is an important visual feature of an image searching for images using shape features has attracted much attention there are many shape representation anddescription techniques in the literature in this paper we classify and review these important techniques we examine implementation procedures for each technique and discuss its advantages and disadvantages some recent research results are also included and discussed in this paper finally we identify some promising techniques for image retrieval according to standard principles

accurate junction detection and characterization in natural images accurate junction detection and characterization are of primary importance for several aspects of scene analysis including depth recovery and motion analysis in this work we introduce a generic junction analysis scheme the first asset of the proposed procedure is an automatic criterion for the detection of junctions permitting to deal with textured parts in which no detection is expected second the method yields a characterization of l y and x junctions including a precise computation of their type localization and scale contrary to classical approaches scale characterization does not rely on the linear scale space first an a contrario approach is used to compute the meaningfulness of a junction this approach relies on a statistical modeling of suitably normalized gray level gradients then exclusion principles between junctions permit their precise characterization we give implementation details for this procedure and evaluate its efficiency through various experiments

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

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

an approach to the parameterization of structure for fast categorization a decomposition is described which parameterizes the geometry and appearance of contours and regions of gray scale images with the goal of fast categorization to express the contour geometry a contour is transformed into a local global space from which parameters are derived classifying its global geometry arc inflexion or alternating and describing its local aspects degree of curvature edginess symmetry regions are parameterized based on their symmetric axes which are evolved with a wave propagation process enabling to generate the distance map for fragmented contour images the methodology is evaluated on three image sets the caltech 101 set and two sets drawn from the corel collection the performance nearly reaches the one of other categorization systems for unsupervised learning

representative patterns for model based matching abstract a single intensity image of a three dimensional 3d object obtained under perspective projection can be reduced to a two dimensional 2d line drawing which contains patterns characteristic of the object and of its pose these patterns can provide clues to narrow down the search for possible objects and poses when comparing 2d view class models of 3d objects against images this paper describes a general way of representing 2d line patterns and of using the patterns to consistently label 2d models of 3d objects the representation is based on groups of three line segments that are likely to be found in most images containing man made objects but are unlikely to occur by accident experimental results using representative patterns to match 2d view class models of 3d objects against real images of the objects are included

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

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

perceptually relevant and piecewise linear matching of silhouettes in this paper a novel alignment method for silhouettes is proposed this method is based on the establishment of correspondences between landmarks on their boundaries and in turn on the establishment of correspondences of the boundary pieces in between these landmarks the method yields more correct correspondences than conventional methods that scale the arc length descriptions of silhouettes to align them the selection of landmarks is investigated as to the robustness of their localization and their perceptual relevance matching of silhouettes is then achieved by quantifying the dissimilarity of a pair of silhouette boundaries based on a novel dissimilarity metric the matching procedure is evaluated based on retrieval experiments and it is concluded that the precision of the results is higher than that obtained by conventional pointwise comparison methods

quantitative error measures for edge detection how to measure the performance of an edge detection method is an open problem many performance measures have been presented in the literature however there seems to be no agreement on the best option and there exist no practical studies of the features of some of the most recent measures in this work we make a comprehensive overview of the different proposals followed by a practical comparison of the most representative measures on synthetic as well as natural edge images

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

robust symbolic representation for shape recognition and retrieval a new method for shape recognition and retrieval is proposed here the suggested algorithm is based on several steps the algorithm analyzes the contour of pairs of shapes their contours are recovered and represented by a pair of n points obtained by linear interpolation given two points p i and q j from the two shapes the cost of their matching is evaluated by using the shape context and by using dynamic programming the best matching between the point sets is obtained dynamic programming not only recovers the best matching but also identifies occlusions i e points in the two shapes which cannot be properly matched given the correspondence between the two point sets the two contours are aligned using procrustes analysis after alignment each contour is transformed into a string of symbols and a modified version of edit distance is used to compute the similarity between strings of symbols finally recognition and retrieval are obtained by a simple nearest neighbor procedure the algorithm has been tested on a large set of shape databases kimia mpeg 7 natural silhouette database gesture database marine database swedish leaf database diatom database eth 80 3d object database providing performances for both in recognition and in retrieval superior to most of previously proposed approaches

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

a nonoscillation theorem for half linear differential equations with periodic coefficients abstract the half linear differential equation p x a t p x b t p x 0 is considered under the assumption that the coefficient a t and an indefinite integral b t of b t are periodic functions with period t 0 it is proved that p 1 p b t a t b t 0 0 t t is sufficient for all nontrivial solutions to be nonoscillatory here p 1 and q y y q 2 y for q p or q p p p 1 the proof is given by means of riccati technique the condition is shown to be sharp sufficient conditions are also presented for all nontrivial solutions are oscillatory in the linear case p 2 some examples and simulations are included to illustrate our results

using dominant points and variants for profile extraction from chromosomes abstract the most accurate methods for automatic analysis of chromosomes under a light microscope today extract numerical features from band pattern profiles along their longitudinal axes the construction of a reliable axis is a crucial step in this process we propose a new way based on the dominant points of the contour and cubic splines the dominant points serve as candidates for the tips of the chromosome ambiguities are dissolved by the recently developed method of variants for object identification a voronoi diagram decomposes the chromosome in slices for profile extraction the method improves the currently best classification results significantly yielding a test set error rate of 0 61

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

generalized distributed order diffusion equations with composite time fractional derivative abstract in this paper we investigate the solution of generalized distributed order diffusion equations with composite time fractional derivative by using the fourier laplace transform method we represent solutions in terms of infinite series in fox h functions the fractional and second moments are derived by using mittag leffler functions we observe decelerating anomalous subdiffusion in case of two composite time fractional derivatives generalized uniformly distributed order diffusion equation as a model for strong anomalous behavior is analyzed by using tauberian theorem some previously obtained results are special cases of those presented in this paper

epipolar geometry and linear subspace methods a new approach to weak calibration this paper addresses the problem of estimating the epipolar geometry from point correspondences between two images taken by uncalibrated perspective cameras it is shown that jepson s and heeger s linear subspace technique for infinitesimal motion estimation can be generalized to the finite motion case by choosing an appropriate basis for projective space this yields a linear method for weak calibration the proposed algorithm has been implemented and tested on both real and synthetic images and it is compared to other linear and non linear approaches to weak calibration

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

edge junction and corner detection using color distributions for over 30 years 1970 2000 researchers in computer vision have been proposing new methods for performing low level vision tasks such as detecting edges and corners one key element shared by most methods is that they represent local image neighborhoods as constant in color or intensity with deviations modeled as noise due to computational considerations that encourage the use of small neighborhoods where this assumption holds these methods remain popular the research presented models a neighborhood as a distribution of colors the goal is to show that the increase in accuracy of this representation translates into higher quality results for low level vision tasks on difficult natural images especially as neighborhood size increases we emphasize large neighborhoods because small ones often do not contain enough information we emphasize color because it subsumes gray scale as an image range and because it is the dominant form of human perception we discuss distributions in the context of detecting edges corners and junctions and we show results for each

on kamenev type oscillation theorems for second order differential equations with damping abstract oscillation criteria for the nonlinear second order ordinary differential equation with damping x p t x f x 0 are given the results are extensions of integral averaging technique of kamenev and include earlier results of yeh yan and chen

scale based description and recognition of planar curves and two dimensional shapes the problem of finding a description at varying levels of detail for planar curves and matching two such descriptions is posed and solved in this paper a number of necessary criteria are imposed on any candidate solution method path based gaussian smoothing techniques are applied to the curve to find zeros of curvature at varying levels of detail the result is the generalized scale space image of a planar curve which is invariant under rotation uniform scaling and translation of the curve these properties make the scale space image suitable for matching the matching algorithm is a modification of the uniform cost algorithm and finds the lowest cost match of contours in the scale space images it is argued that this is preferable to matching in a so called stable scale of the curve because no such scale may exist for a given curve this technique is applied to register a landsat satellite image of the strait of georgia b c manually corrected for skew to a map containing the shorelines of an overlapping area

symmetry maps of free form curve segments via wave propagation this paper presents an approach for computing the symmetries skeletons of an edge map consisting of a collection of curve segments this approach is a combination of analytic computations in the style of computational geometry and discrete propagations on a grid in the style of the numerical solutions of pde s specifically waves from each of the initial curve segments are initialized and propagated as a discrete wavefront along discrete directions in addition to avoid error built up due to the discrete nature of propagation shockwaves are detected and explicitly propagated along a secondary dynamic grid the propagation of shockwaves integrated with the propagation of the wavefront along discrete directions leads to an exact simulation of propagation by the eikonal equation the resulting symmetries are simply the collection of shockwaves formed in this process which can be manipulated locally exactly and efficiently under local changes in an edge map gap completion removal of spurious elements etc the ability to express grouping operations in the language of symmetry maps makes it an appropriate intermediate representation between low level edge maps and high level object hypotheses

oscillation criteria for second order nonlinear delay dynamic equations on time scales by employing the generalized riccati transformation we establish some new oscillation criteria for second order delay dynamic equations on time scales the obtained results essentially improve the well know oscillation results for half linear dynamic equations such as kamenev type and philos type oscillation criteria some interesting examples are given to illustrate the versatility of our results

scale affine invariant interest point detectors in this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations our scale and affine invariant detectors are based on the following recent results 1 interest points extracted with the harris detector can be adapted to affine transformations and give repeatable results geometrically stable 2 the characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives the laplacian 3 the affine shape of a point neighborhood is estimated based on the second moment matrix our scale invariant detector computes a multi scale representation for the harris interest point detector and then selects points at which a local measure the laplacian is maximal over scales this provides a set of distinctive points which are invariant to scale rotation and translation as well as robust to illumination changes and limited changes of viewpoint the characteristic scale determines a scale invariant region for each point we extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood an iterative algorithm modifies location scale and neighborhood of each point and converges to affine invariant points this method can deal with significant affine transformations including large scale changes the characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point we present a comparative evaluation of different detectors and show that our approach provides better results than existing methods the performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points

generalized feature extraction using expansion matching a novel generalized feature extraction method based on the expansion matching exm method and on the karhunen loeve transform klt is presented the method provides an efficient way to locate complex features of interest like corners and junctions with reduced number of filtering operations the exm method is used to design optimal detectors for a set of model elementary features the kl representation of these model exm detectors is used to filter the image and detect candidate interest points from the energy peaks of the eigen coefficients the kl coefficients at these candidate points are then used to efficiently reconstruct the response and differentiate real junctions and corners from arbitrary features in the image the method is robust to additive noise and is able to successfully extract classify and find the myriad compositions of corner and junction features formed by combinations of two or more edges or lines this method differs from previous works in several aspects first it treats the features not as distinct entities but as combinations of elementary features second it employs an optimal set of elementary feature detectors based on the em approach third the method incorporates a significant reduction in computational complexity by representing a large set of exm filters by a relatively small number of eigen filters derived by the kl transform of the basic exm filter set this is a novel application of the kl transform which is usually employed to represent signals and not impulse responses as in our present work

piecewise approximation of contours through scale space selection of dominant points this paper describes a method of approximating a shape contour with a polygon the polygon vertices are extracted from the curvature extremes through a scale space description of the contour via linear diffusion these vertices are located on the contour points where the sharper changes of the contour directions occur using a proper strategy a set of extremes that result in a given approximation level is chosen by adding new vertices the approximation level can be improved and a scalable representation of the contour is identified this method results in an approximation that discriminates local from global geometric features and provides a good visual representation of the original contour this polygonal approximation method is used for scalable encoding of the shape contours in this regard an encoding technique suitable for scalable polygonal approximation has been developed we show that encoding the approximated polygons result in a good relation between the distortion and the bitrate finally we show that in addition to coding this method can be efficiently used for shape comparison and shape retrieval

fractional calculus with an integral operator containing a generalized mittag leffler function in the kernel in this paper we introduce and investigate a fractional calculus with an integral operator which contains the following family of generalized mittag leffler functions e a b c k z email protected n 0 c k email protected email protected z nn z b email protected c r a max 0 r k 1 r k 0 in its kernel l n being the familiar pochhammer symbol a number of corollaries and consequences of the main results presented here are also considered

a multiscanning approach based on morphological filtering it is argued that the mathematical morphology method seems to be more reasonable and powerful in studying certain multiscaling vision problems than the approach that uses derivatives of gaussian shaped filters of different sizes to show the validity of this method the authors concentrated on an application that involves forming scale space image of a 2 d shape using morphological opening filtering a proof is given to show that morphological opening filtering has a property of not introducing additional zero crossings as one moves to a coarser scale this is a different result from the conclusion by a l yuille and t a poggio ibid vol pami 8 jan 1986 that the gaussian filter is the only filter with this property in addition opening filtering is computationaly simpler than the gaussian filter

representing curves at their natural scales abstract the shapes of curves generally contain several structures at different scales an expressive curve representation should explicitly describe each structure at its appropriate scale this greatly simplifies any reasoning processes e g matching applied to the curve rather than exhaustively represent the curve over a full range of scales just the most significant ones should be retained for a compact and useful representation a technique is presented that identifies the significant or natural scales in a curve

scale dependency of image derivatives for feature measurement in curvilinear structures extraction of image features is a crucial step in many image analysis tasks in feature extraction methods gaussian derivative kernels are frequently utilized blurring of the image due to convolution with these kernels gives rise to feature measures different from the intended value in the original image we propose to solve this problem by explicitly modeling the scale dependency of derivatives combined with measurement of derivatives at multiple scales this approach is illustrated in methods for feature measurement in curvilinear structures results in 3d confocal images confirm that modelling of scale behavior of derivatives results in improved methods for center line localization in curved line structures and enables curvature and diameter measurement

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

anomalous diffusion modeling by fractal and fractional derivatives this paper makes an attempt to develop a fractal derivative model of anomalous diffusion we also derive the fundamental solution of the fractal derivative equation for anomalous diffusion which characterizes a clear power law this new model is compared with the corresponding fractional derivative model in terms of computational efficiency diffusion velocity and heavy tail property the merits and distinctions of these two models of anomalous diffusion are then summarized

analytical solution of a fractional diffusion equation by variational iteration method in the present paper the analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful variational iteration method by using an initial value the explicit solutions of the equation for different cases have been derived which accelerate the rapid convergence of the series solution the present method performs extremely well in terms of efficiency and simplicity numerical results for different particular cases of the problem are presented graphically

non parametric planar shape representation based on adaptive curvature functions this paper presents a non parametric method to extract a very short feature vector from the curvature function of a planar shape curvature is adaptively calculated using a new procedure that removes noise from the contour without missing relevant points then its fourier transform is projected onto a set of vectors which have been chosen to be as representative as possible to obtain the similarity between the input object and each vector of the set these similarity values are the elements of the feature vector the proposed method is very fast and classification has proven that the representation is good

comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations in this article the homotopy perturbation method proposed by j h he is adopted for solving linear fractional partial differential equations the fractional derivatives are described in the caputo sense comparison of the results obtained by the homotopy perturbation method with those obtained by the variational iteration method reveals that the present methods are very effective and convenient

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

oscillation problems for hill s equation with periodic damping abstract this paper deals with the second order linear differential equation x a t x b t x 0 where a and b are periodic coefficients the main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory our nonoscillation theorem and oscillation theorem are proved by using the riccati technique in our theorem the composite function of an indefinite integral of b and a suitable multiple valued continuously differentiable function are focused and the composite function of them plays an important role the results obtained here include a result by kwong and wong 15 and a result by sugie and matsumura 26 an application to a equation of whittaker hill type is given to show the usefulness of our results finally simulations are also attached to illustrate that our oscillation criterion is sharp

evaluation of interest point detectors many different low level feature detectors exist and it is widely agreed that the evaluation of detectors is important in this paper we introduce two evaluation criteria for interest points repeatability rate and information content repeatability rate evaluates the geometric stability under different transformations information content measures the distinctiveness of features different interest point detectors are compared using these two criteria we determine which detector gives the best results and show that it satisfies the criteria well

motion estimation of two dimensional objects based on the straight line hough transform a new approach abstract in this paper we propose a novel technique for estimating motion of two dimensional objects the technique involves representing objects uniquely by straight line approximations of the boundary p and estimating motion from shifts in the p space the straight line approximation of the boundary is obtained by detecting dominant points during contour tracing and then approximating the pixels between adjacent dominant points along the contour by a straight line using the hough transform we have developed a computationally efficient way of detecting dominant points in a single pass during contour tracing the proposed object representation enables us to determine the translation and rotation with significantly fewer computations compared with existing methods our method can also detect the motion parameters correctly for partially overlapping objects

2d affine invariant contour matching using b spline model this paper presents a new affine invariant matching algorithm based on b spline modeling which solves the problem of the nonuniqueness of b spline in curve matching this method first smoothes the b spline curve by increasing the degree of the curve it is followed by a reduction of the curve degree using the least square error lse approach to construct the curvature scale space css image css matching is then carried out our method combines the advantages of b spline that are continuous curve representation and the robustness of css matching with respect to noise and affine transformation it avoids the need for other matching algorithms that have to use the resampled points on the curve thus the curve matching error is reduced the proposed algorithm has been tested by matching similar shapes from a prototype database the experimental results showed the robustness and accuracy of the proposed method in b spline curve matching

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

generalized space time fractional diffusion equation with composite fractional time derivative abstract we investigate the solution of space time fractional diffusion equations with a generalized riemann liouville time fractional derivative and riesz feller space fractional derivative the laplace and fourier transform methods are applied to solve the proposed fractional diffusion equation the results are represented by using the mittag leffler functions and the fox h function special cases of the initial and boundary conditions are considered numerical scheme and grunwald letnikov approximation are also used to solve the space time fractional diffusion equation the fractional moments of the fundamental solution of the considered space time fractional diffusion equation are obtained many known results are special cases of those obtained in this paper we investigate also the solution of a space time fractional diffusion equations with a singular term of the form x t 1 0

growth conditions for uniform asymptotic stability of damped oscillators abstract the present paper is devoted to an investigation on the uniform asymptotic stability for the linear differential equation with a damping term x h t x 2 x 0 and its generalization p x h t p x p p x 0 where 0 and p z z p 2 z with p 1 sufficient conditions are obtained for the equilibrium x x 0 0 to be uniformly asymptotically stable under the assumption that the damping coefficient h t is integrally positive the obtained condition for the damped linear differential equation is given by the form of a certain uniform growth condition on h t another representation which is equivalent to this uniform growth condition is also given our results assert that the equilibrium can be uniformly asymptotically stable even if h t is unbounded an example is attached to show this fact in addition easy to use conditions are given to guarantee that the uniform growth condition is satisfied moreover a sufficient condition expressed by an infinite series is presented the relation between the representation of an infinite series and the uniform growth condition is also clarified finally our results are extended to be able to apply to the above mentioned nonlinear differential equation

towards effective planar shape representation with multiscale digital curvature analysis based on signal processing techniques abstract this paper presents and discusses a new approach to multiscale curvature analysis of digital contours that is based on digital signal processing techniques the shape contour is expressed in terms of two one dimensional signals linearized x and y coordinates and the derivative theorem is applied as a means of obtaining an interesting expression relating contour curvature and the spectra of its parametrized x and y signals multiscale curvature analysis is achieved through gaussian lowpass filtering and the shrinkage of the original signals implied by such a process is effectively circumvented by an energy based compensation scheme which has allowed accurate quantitative identification of the curvature value the concept of curvegram is introduced and exemplified and the overall performance of the proposed technique for curvature estimation is formally assessed with respect to a contour defined in terms of b splines application examples to synthetic and real images have been included

tissue boundary refinement in magnetic resonance images using contour based scale space matching an algorithm for computationally focusing the tissue boundaries detected from magnetic resonance images is presented the proposed approach is a novel whole contour based technique for tracing edges selected at a coarse scale into successively finer scales to recover the needed precision the tracing algorithm builds consensus through a fast pixel voting scheme also presented is a rigorous method for determining the appropriate itinerary when traversing scale space beginning from the premise of a maximum pixel migration per unit change in scale parameter this leads to an efficient method of processing images so as to maximize accuracy and minimize noise although the log laplacian of gaussian is used for many of the experiments results using a novel edge detector which is mathematically superior to and faster to compute than the log and for which fewer steps are required to traverse the same effective span in scale space are presented experimental results on real data are presented and other potential applications are discussed

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

a robust boundary based object recognition in occlusion environment by hybrid hopfield neural networks this paper presents a new method of occluded object matching for machine vision applications the current methods for occluded object matching lack robustness and require high computational effort in this paper a new hybrid hopfield neural network hhn algorithm which combines the advantages of both a continuous hopfield network chn and a discrete hopfield network dhn will be described and applied for partially occluded object recognition in a multi context scenery the hhn proposed as a new approach provides great fault tolerance and robustness and requires less computation time also advantages of hhn such as reliability and speed will be discussed

oscillation and nonoscillation of hill s equation with periodic damping we prove new results on the oscillation and nonoscillation of the hill s equation with periodic damping y p t y q t y 0 t 0 where p t and q t are continuous and periodic the results show that the equation y sint y cost y 0 is nonoscillatory whilst the equation y cost y sint y 0 is oscillatory

subpixel edge location in binary images using dithering this paper concerns the problem of obtaining subpixel estimates of the locations of straight edges in binary digital images using dithering by adding uniformly distributed independent random noise it is shown that estimation bias may be removed and that the estimation variance is inversely proportional to the length of the line segment the sensitivity to incorrect dither amplitude is calculated and implementation is discussed

deb definite error bounded tangent estimator for digital curves we propose a simple and fast method for tangent estimation of digital curves this geometric based method uses a small local region for tangent estimation and has a definite upper bound error for continuous as well as digital conics i e circles ellipses parabolas and hyperbolas explicit expressions of the upper bounds for continuous and digitized curves are derived which can also be applied to nonconic curves our approach is benchmarked against 72 contemporary tangent estimation methods and demonstrates good performance for conic nonconic and noisy curves in addition we demonstrate a good multigrid and isotropic performance and low computational complexity of o 1 and better performance than most methods in terms of maximum and average errors in tangent computation for a large variety of digital curves

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

affine morphological multiscale analysis of corners andmultiple junctions in this paper we study the application of the affine morphological scale space amss to the analysis of singularities corners or multiple junctions of the shapes present in a 2 d image we introduce a new family of travelling wave solutions of amss which determines the evolution of the initial shapes given by conics we characterize the evolution of corners accross the scales according to their angle we develop a numerical algorithm to compute amss accross the scales and we present some experimental results about corners and multiple junction detection

global surface reconstruction by purposive control of observer motion abstract what viewpoint control strategies are important for performing global visual exploration tasks such as searching for specific surface markings building a global model of an arbitrary object or recognizing an object in this paper we consider the task of purposefully controlling the motion of an active monocular observer in order to recover a global description of a smooth arbitrarily shaped object we formulate global surface reconstruction as the task of controlling the motion of the observer so that the visible rim slides over the maximal connected reconstructible surface regions intersecting the visible rim at the initial viewpoint we show that these regions are bounded by a subset of the visual event curves defined on the surface by studying the epipolar parameterization we develop two basic strategies that allow reconstruction of a surface region around any point in a reconstructible surface region these strategies control viewpoint to achieve and maintain a well defined geometric relationship with the object s surface rely only on information extracted directly from images e g tangents to the occluding contour and are simple enough to be performed in real time we then show how global surface reconstruction can be provably achieved by 1 appropriately integrating these strategies to iteratively grow the reconstructed regions and 2 obeying four simple rules

multimodality image registration by maximization of quantitative qualitative measure of mutual information this paper presents a novel image similarity measure referred to as quantitative qualitative measure of mutual information q mi for multimodality image registration conventional information measures e g shannon s entropy and mutual information mi reflect quantitative aspects of information because they only consider probabilities of events in fact each event has its own utility to the fulfillment of the underlying goal which can be independent of its probability of occurrence thus it is important to consider both quantitative i e probability and qualitative i e utility measures of information in order to fully capture the characteristics of events accordingly in multimodality image registration q mi should be used to integrate the information obtained from both the image intensity distributions and the utilities of voxels in the images different voxels can have different utilities for example in brain images two voxels can have the same intensity value but their utilities can be different e g a white matter wm voxel near the cortex can have higher utility than a wm voxel inside a large uniform wm region in q mi the utility of each voxel in an image can be determined according to the regional saliency value calculated from the scale space map of this image since the voxels with higher utility values or saliency values contribute more in measuring q mi of the two images the q mi based registration method is much more robust compared to conventional mi based registration methods also the q mi based registration method can provide a smoother registration function with a relatively larger capture range in this paper the proposed q mi has been validated and applied to the rigid registrations of clinical brain images such as mr ct and pet images

fractional diffusion equation with a generalized riemann liouville time fractional derivative in this paper the solution of a fractional diffusion equation with a hilfer generalized riemann liouville time fractional derivative is obtained in terms of mittag leffler type functions and fox s h function the considered equation represents a quite general extension of the classical diffusion heat conduction equation the methods of separation of variables laplace transform and analysis of the sturm liouville problem are used to solve the fractional diffusion equation defined in a bounded domain by using the fourier laplace transform method it is shown that the fundamental solution of the fractional diffusion equation with a generalized riemann liouville time fractional derivative defined in the infinite domain can be expressed via fox s h function it is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the caputo and riemann liouville sense are special cases of those diffusion equations with the hilfer generalized riemann liouville time fractional derivative the asymptotic behaviour of the solutions are found for large values of the spatial variable the fractional moments of the fundamental solution of the fractional diffusion equation are obtained the obtained results are relevant in the context of glass relaxation and aquifer problems

local reproducible smoothing without shrinkage a simple local smoothing filter is defined for curves or surfaces combining the advantages of gaussian smoothing and fourier curve description unlike gaussian filters the filter described has no shrinkage problem repeated application of the filter does not yield a curve or surface smaller than the original but simply reproduces the approximate result that would have been obtained by a single application at the largest scale unlike fourier description the filter is local in space the wavelet transform of y meyer 1989 is also shown to have these properties

generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

edge detection with embedded confidence computing the weighted average of the pixel values in a window is a basic module in many computer vision operators the process is reformulated in a linear vector space and the role of the different subspaces is emphasized within this framework wellknown artifacts of the gradient based edge detectors such as large spurious responses can be explained quantitatively it is also shown that template matching with a template derived from the input data is meaningful since it provides an independent measure of confidence in the presence of the employed edge model the widely used three step edge detection procedure gradient estimation non maxima suppression hysteresis thresholding is generalized to include the information provided by the confidence measure the additional amount of computation is minimal and experiments with several standard test images show the ability of the new procedure to detect weak edges

hilfer fractional advection diffusion equations with power law initial condition a numerical study using variational iteration method abstract we propose a hilfer advection diffusion equation of order 0 1 and type 0 1 and find the power series solution by using variational iteration method power series solutions are expressed in a form that is easy to implement numerically and in some particular cases solutions are expressed in terms of mittag leffler function absolute convergence of power series solutions is proved and the sensitivity of the solutions is discussed with respect to changes in the values of different parameters for power law initial conditions it is shown that the hilfer advection diffusion pde gives the same solutions as the caputo and riemann liouville advection diffusion pde to leading order the fractional solution compared to the non fractional solution increases rapidly with for 0 7 at a given time t but for 0 7 this factor is weakly sensitive to we also show that the truncation errors arising when using the partial sum as approximate solutions decay exponentially fast with the number of terms n used we find that for 0 7 the number of terms needed is weakly sensitive to the accuracy level and to the fractional order n 20 but for 0 7 the required number of terms increases rapidly with the accuracy level and also with the fractional order

wavelet based approach to character skeleton character skeleton plays a significant role in character recognition the strokes of a character may consist of two regions i e singular and regular regions the intersections and junctions of the strokes belong to singular region while the straight and smooth parts of the strokes are categorized to regular region therefore a skeletonization method requires two different processes to treat the skeletons in theses two different regions all traditional skeletonization algorithms are based on the symmetry analysis technique the major problems of these methods are as follows 1 the computation of the primary skeleton in the regular region is indirect so that its implementation is sophisticated and costly 2 the extracted skeleton cannot be exactly located on the central line of the stroke 3 the captured skeleton in the singular region may be distorted by artifacts and branches to overcome these problems a novel scheme of extracting the skeleton of character based on wavelet transform is presented in this paper this scheme consists of two main steps namely a extraction of primary skeleton in the regular region and b amendment processing of the primary skeletons and connection of them in the singular region a direct technique is used in the first step where a new wavelet based symmetry analysis is developed for finding the central line of the stroke directly a novel method called smooth interpolation is designed in the second step where a smooth operation is applied to the primary skeleton and thereafter the interpolation compensation technique is proposed to link the primary skeleton so that the skeleton in the singular region can be produced experiments are conducted and positive results are achieved which show that the proposed skeletonization scheme is applicable to not only binary image but also gray level image and the skeleton is robust against noise and affine transform

analytical solution of a fractional diffusion equation by adomian decomposition method this paper presents an analytical solution of a fractional diffusion equation by adomian decomposition method by using an initial value the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically the present method performs extremely well in terms of efficiency and simplicity

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

boundary based corner detection using neural networks abstract in this paper we present a novel boundary based corner detection approach using artificial neural networks anns two neural networks are proposed one for detecting corner points with high curvature and the other for detecting tangent points and inflection points that generally have low curvature for a given boundary point p i the first ann uses the normalized coordinates of points on the forward arm neighboring points succeeding p i or on the backward arm neighboring points preceding p i of the point p i as the input vector the output feature of the network is the angle of the forward arm or backward arm with respect to the x axis the boundary point with sufficiently small angle between the forward and backward arms is identified as a corner since the feature points of tangency and inflection have relatively low curvature the signs of curvature rather than the magnitude of curvature for points in the neighborhood of p i are used as the input vector to the second ann the curvature sign at each boundary point is derived from the outputs of the first ann the outputs of the second ann only respond to the sign patterns of tangent points and inflection points by using both anns all features of corners tangent points and inflection points can be extracted from the boundary of any arbitrary shape experimental results have shown that the proposed anns have good detection and localization for objects in random orientations and with moderate scale changes

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

global existence and nonexistence of solutions for second order nonlinear differential equations abstract this paper deals with the global existence and nonexistence of solutions of the second order nonlinear differential equation x x 0 satisfying x 0 x 0 and x 0 x 1 where is a positive parameter and with 0 and 0 is strictly increasing odd bijective and continuous on necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic examples are given to illustrate our main result finally a nonexistence result for the equation with a damping term is discussed as an application to our result

oscillation criterion for half linear differential equations with periodic coefficients abstract in this paper we present an oscillation criterion for second order half linear differential equations with periodic coefficients the method is based on the nonexistence of a proper solution of the related modified riccati equation our result can be regarded as an oscillatory counterpart to the nonoscillation criterion by sugie and matsumura 2008 these two theorems provide a complete half linear extension of the oscillation criterion of kwong and wong 2003 dealing with the hill s equation

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

applications of vector fields to image processing we use rotational and curvature properties of vector fields to identify critical features of an image using vector analysis and dif ferential geometry we establish the properties needed and then use these properties in three ways first our results make it theoretically possible to identify extremal edges of an intensity function f x y of two variables by considering the gradient vector field v f there is also enough information in f to find regions of high curvature i e high curvature of the level paths of f for color images we use the vector field v i q in application the image is partitioned into a grid of squares on the boundary of each square v v is sampled and these unit vectors are used as the tangents of a curve the rotation number or topological degree and the average curvature f are computed for each square analysis of these numbers yields infor mation on edges and curvature experimental results from both simu lated and real data are described

a unified approach to boundary perception edges textures and illusory contours a model consisting of a multistage system which extracts and groups salient features in the image at different spatial scales or frequencies is used in the first stage a gabor wavelet decomposition provides a representation of the image which is orientation selective and has optimal localization properties in space and frequency this decomposition is useful in detecting significant features such as step and line edges at different scales and orientations in the image following the wavelet transformation local competitive interactions are introduced to reduce the effects of noise and changes in illumination interscale interactions help in localizing the line ends and corners and play a crucial role in boundary perception the final stage groups similar features aiding in boundary completion the different stages can be identified with processing by simple complex and hypercomplex cells in the visual cortex of mammals experimental results demonstrate the performance of this model in detecting boundaries both real and illusory in real and synthetic images

design of steerable filters for feature detection using canny like criteria we propose a general approach for the design of 2d feature detectors from a class of steerable functions based on the optimization of a canny like criterion in contrast with previous computational designs our approach is truly 2d and provides filters that have closed form expressions it also yields operators that have a better orientation selectivity than the classical gradient or hessian based detectors we illustrate the method with the design of operators for edge and ridge detection we present some experimental results that demonstrate the performance improvement of these new feature detectors we propose computationally efficient local optimization algorithms for the estimation of feature orientation we also introduce the notion of shape adaptable feature detection and use it for the detection of image corners

local grayvalue invariants for image retrieval this paper addresses the problem of retrieving images from large image databases the method is based on local grayvalue invariants which are computed at automatically detected interest points a voting algorithm and semilocal constraints make retrieval possible indexing allows for efficient retrieval from a database of more than 1 000 images experimental results show correct retrieval in the case of partial visibility similarity transformations extraneous features and small perspective deformations

some defects in finite difference edge finders this work illustrates and explains various artifacts in the output of five finite difference edge finders those of j f canny 1983 1986 r a boie et al 1986 and r a boie and i j cox 1987 and three variations on that of d marr and e c hildreth 1980 reimplemented with a common output format and method of noise suppression these artifacts include gaps in boundaries spurious boundaries and deformation of region shape

multiple widths yield reliable finite differences computer vision a finite difference edge finder in which the finite difference is computed at a range of widths i e a range of distances between data points is introduced wide operators report low amplitude responses more reliably than narrow operators so if wide operators are used to fill gaps in narrow operator responses each operator can be restricted to report only statistically reliable responses without losing many real features this sharply reduces the noise in the final output theoretical bounds on spurious responses in the finite difference outputs given only weak assumptions about the signal and noise are presented the expected response of the edge finder to an ideal straight step edge is also analyzed these performance measures are compared with those of a standard algorithm based on gaussian smoothing and those of a second algorithm that also considers the spatial structure of noise the algorithms prove equally good at suppressing noise but are better able to detect faint or blurred features these predictions are confirmed by empirical tests on real images

digitized circular arcs characterization and parameter estimation the digitization of a circular arc causes an inherent loss of geometrical information arcs with slightly different local curvature or position may lead to exactly the same digital pattern in this paper the authors give a characterization of all centers and radii of circular arcs yielding the same digitization pattern the radius of the arcs varies over the set however only one curvature or radius estimate can be assigned to the digital pattern the authors derive an optimal estimator and give expressions for the bound on the precision of estimation this bound due to digitization is the deterministic equivalent of the cramer rao bound known from parameter estimation theory consider the estimation of the local curvature and local radius of a smooth object typically such parameters are estimated by moving a window along the digital boundary methods in literature show a poor precision in estimating curvature values relative errors of over 40 are often found from the definition of curvature it follows that locally the curve can be considered a circular arc and hence the method presented in this paper can be applied to the pattern in the window giving estimates with optimal precision and a measure for the remaining error on the practical side the authors present examples of the residual error due to the discrete grid the estimation of the radius or curvature of a circular arc at random position with an estimation window containing 10 points coded with nine freemancodes has a relative deviation exceeding 2 for a full disk the deviation is below 1 when the radius r exceeds four grid units the presented method is particularly useful for problems where some prior knowledge on the distribution of radii is known and where there is a noise free sampling