locating perceptually salient points on planar curves this paper describes the underlying ideas and algorithmic details of a computer program that performs at a human level of competence for a significant subset of the curve partitioning task it extends and rounds out the technique and philosophical approach originally presented by fischler and bolles 1986 in particular it provides a unified strategy for selecting and dealing with interactions between salient points even when these points are salient at different scales of resolution experimental results are presented involving on the order of 1000 real and synthetically generated images

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

three dimensional shape description using the symmetric axis transform i theory blum s two dimensional shape description method based on the symmetric axis transform sat is generalized to three dimensions the method uniquely decomposes an object into a collection of sub objects each drawn from three separate but not completely independent primitive sets defined in the paper width primitives based on radius function properties axis primitives based on symmetric axis curvatures and boundary primitives based on boundary surface curvatures width primitives are themselves comprised of two components slope districts and curvature districts visualizing the radius function as if it were the height function of some mountainous terrain each slope district corresponds to a mountain face together with the valley below it curvature districts further partition each slope district into regions that are locally convex concave or saddle like similarly axis boundary primitives are regions of the symmetric surface where the symmetric surface boundary surfaces are locally convex concave or saddle like relations among the primitive sets are discussed

a discrete geometry approach for dominant point detection we propose two fast methods for dominant point detection and polygonal representation of noisy and possibly disconnected curves based on a study of the decomposition of the curve into the sequence of maximal blurred segments 2 starting from results of discrete geometry 3 4 the notion of maximal blurred segment of width n 2 has been proposed well adapted to possibly noisy curves the first method uses a fixed parameter that is the width of considered maximal blurred segments the second method is deduced from the first one based on a multi width approach to obtain a non parametric method that uses no threshold for working with noisy curves comparisons with other methods in the literature prove the efficiency of our approach thanks to a recent result 5 concerning the construction of the sequence of maximal blurred segments the complexity of the proposed methods is o nlogn an application of vectorization is also given in this paper

optimal polygonal approximation of digital curves abstract an algorithm for optimal polygonal approximation is presented given a value for the maximal allowed distance between the approximation and the curve the algorithm finds an approximation with the minimal number of vertices the city block metric is used to measure the distance between the approximation and the curve the algorithm worst case complexity is o n 2 where n is the number of points in the curve an efficient and optimal solution for the case of closed curves where no initial point is given is also presented

a novel method for splitting clumps of convex objects incorporating image intensity and using rectangular window based concavity point pair search a novel nonparametric concavity point analysis based method for splitting clumps of convex objects in binary images is presented the method is based on finding concavity point pairs by using a variable size rectangular window the concavity point pairs can be either connected with a straight split line or with a line that follows a path of minimum or maximum intensity on an accompanying grayscale image using straight lines can result in non smooth contours therefore post processing steps that remove acute angles between split lines are proposed results obtained with images that have clumps of biological cells show that the method gives accurate results

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

low lying zeros of l functions and random matrix theory by looking at the average behavior n level density of the low lying zeros of certain families of l functions we find evidence as predicted by function field analogs in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups

optimized polygonal approximation by dominant point deletion an algorithm for polygonal approximation based on dominant point dp deletion is presented in this paper the algorithm selects an initial set of dps and starts eliminating them one by one depending upon the error associated with each dp the associated error value is based on global measure a local optimization of few neighboring points is performed after each deletion although the algorithm does not guarantee an optimal solution the combination of local and global optimization is expected to produce optimal results the algorithm is extensively tested on various shapes with varying number of dps and error threshold in general optimal results were observed for about 96 of the times a good comparative study is also presented in this paper

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

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

reduced resolution and scale space for dominant feature detection in contours abstract analysis of contours or curves at various scales or levels of smoothing scale space is an important tool in curve segmentation and feature detection in this paper we propose the application of the hierarchical discrete correlation algorithm for efficiently calculating and creating a scale space of curves in conjunction with this procedure we also investigate the use of the reduced resolution or gaussian pyramid representation of the set of smoothed curves as the basis for initially localizing and detecting features we also create an approximation to the above algorithms that is computationally less expensive finally we propose a new inter scale method for curve segmentation and feature detection based on the motion of a curve through scale space

a robust method for scale independent detection of curvature based criticalities and intersections in line drawings a novel two phase iterative method is proposed to identify curvature based criticalities and intersections in line drawings the method is based on evaluating a field obtained from the image via diffusion using adaptively changing ellipse shaped analysis windows the deviation of the average field strength values within the analysis windows from that of an external reference field is used as a metric to evaluate the criticality of a region the external reference field is computed from an image of a straight line the experimental results depict that the method is effective in detecting curvature based criticalities and intersections even for noisy and disconnected drawings as well as drawings drawn with a variety of brush characteristics such as glass ocean ripple scatter stamp strokes effects our method can be employed as a part of a design grammar interpreter highlightsthe method detects curvature based criticalities and intersections efficiently the method is effective for the noisy and disconnected line drawings the method is effective for line drawings with a variety of brush characteristics the method is utilized to represent shapes in a design grammar interpreter features in varying scales can be detected in digital drawings

a random matrix model for elliptic curve l functions of finite conductor we propose a random matrix model for families of elliptic curve l functions of finite conductor a repulsion of the critical zeros of these l functions away from the centre of the critical strip was observed numerically by miller 2006 exp math 15 257 79 such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros for large conductors this distribution is expected to approach the one level density of eigenvalues of orthogonal matrices after appropriate rescaling our purpose here is to provide a random matrix model for miller s surprising discovery we consider the family of even quadratic twists of a given elliptic curve the main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra we call this sub ensemble of so 2n the excised orthogonal ensemble the sieving off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of l functions implied by the formulae of waldspurger and kohnen zagier the cut off scale appropriate to modelling elliptic curve l functions is exponentially small relative to the matrix size n the one level density of the excised ensemble can be expressed in terms of that of the well known jacobi ensemble enabling the former to be explicitly calculated it exhibits an exponentially small on the scale of the mean spacing hard gap determined by the cut off value followed by soft repulsion on a much larger scale neither of these features is present in the one level density of so 2n when n we recover the limiting orthogonal behaviour our results agree qualitatively with miller s discrepancy choosing the cut off appropriately gives a model in good quantitative agreement with the number theoretical data

finding contour based abstractions of planar patterns abstract an algorithm is described to detect a number of points on the contour of a planar shape which constitute the vertices of a schematic polygonal representation of the shape itself a set of points initially extracted from the chain coded representation of the contour is iteratively examined while removing some points and inserting new ones the number of selected points decreases in size from iteration to iteration and the selection process converges towards an expected perceptually significant set of points the polygon obtained by linking successive points approximates the contour in an intuitive way it is not constrained within a given tolerance and is likely to locally change from a coarse to a more faithful approximating shape in correspondence with contour regions increasing in details

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

acord an adaptive corner detector for planar curves abstract in contrast to the conventional method of smoothing a curve at multiple scales and integrating the information at various scales a technique for smoothing a curve adaptively based on the roughness present in the curve is suggested the procedure does not require smoothing at all levels of detail and it does not require construction of complete scale space map and representation of the map by tree the procedure has been applied on a number of digital curves and the results have been compared with those of the recent work

low lying zeros of families of elliptic curves the random matrix model predicts that many statistics associated to zeros of a family of l functions can be modeled or predicted by the distribution of eigenval ues of large random matrices in one of the classical linear groups if the statistics of a family of l functions are modeled by the eigenvalues of the group g then we say that g is the symmetry group or symmetry type associated to the family the statistic of interest to us in this work is the density of zeros near the central point also known as the 1 level density the random matrix model predicts that the distribution of these zeros should be modeled by the eigenvalues nearest 1 for one of the symmetry types g unitary symplectic and orthogonal all of the different groups g have distinct behavior in this regard therefore computing the 1 level density gives a theoretical way to predict the symmetry type of a family the 1 level density has been studied for a wide variety of families of l functions see r ksi ils mil for example it is standard to assume the generalized riemann hypothesis grh to study the 1 level density and we do so throughout this work in particular it is necessary to use grh for the application of obtaining a bound on the average analytic rank from a density theorem in some cases the use of grh improves the range of the density theorem which translates to an improved bound on the average rank besides these crucial applications of grh we have freely assumed grh even when it could be removed with extra work since it simplifies arguments in some non essential places it is especially interesting to investigate the 1 level density for families of l functions attached to elliptic curves over the rationals since zeros at the central point have important arithmetic information by the conjecture of birch and swinnerton dyer these investigations have been the main focus of this work

a min max medial axis transformation blum s medial axis transformation mat of the set s of 1 s in a binary picture can be defined by an iterative shrinking and reexpanding process which detects corners on the contours of constant distance from s and thereby yields a skeleton of s for unsegmented gray level pictures one can use an analogous definition in which local min and max operations play the roles of shrinking and expanding to compute a mmmat value at each point of the picture the set of points having high values defines a good skeleton for the set of high gray level points in the given picture

performance comparisons of contour based corner detectors corner detectors have many applications in computer vision and image identification and retrieval contour based corner detectors directly or indirectly estimate a significance measure e g curvature on the points of a planar curve and select the curvature extrema points as corners while an extensive number of contour based corner detectors have been proposed over the last four decades there is no comparative study of recently proposed detectors this paper is an attempt to fill this gap the general framework of contour based corner detection is presented and two major issues curve smoothing and curvature estimation which have major impacts on the corner detection performance are discussed a number of promising detectors are compared using both automatic and manual evaluation systems on two large datasets it is observed that while the detectors using indirect curvature estimation techniques are more robust the detectors using direct curvature estimation techniques are faster

a new measurement for assessing polygonal approximation of curves this paper presents a novel method for assessing the accuracy of unsupervised polygonal approximation algorithms this measurement relies on a polygonal approximation called the reference approximation the reference approximation is obtained using the method of perez and vidal 11 by an iterative method that optimizes an objective function then the proposed measurement is calculated by comparing the reference approximation with the approximation to be evaluated taking into account the similarity between the polygonal approximation and the original contour and penalizing polygonal approximations with an excessive number of points a comparative experiment by using polygonal approximations obtained with commonly used algorithms showed that the proposed measurement is more efficient than other proposed measurements at comparing polygonal approximations with different number of points

partitioning histopathological images an integrated framework for supervised color texture segmentation and cell splitting for quantitative analysis of histopathological images such as the lymphoma grading systems quantification of features is usually carried out on single cells before categorizing them by classification algorithms to this end we propose an integrated framework consisting of a novel supervised cell image segmentation algorithm and a new touching cell splitting method for the segmentation part we segment the cell regions from the other areas by classifying the image pixels into either cell or extra cellular category instead of using pixel color intensities the color texture extracted at the local neighborhood of each pixel is utilized as the input to our classification algorithm the color texture at each pixel is extracted by local fourier transform lft from a new color space the most discriminant color space mdc the mdc color space is optimized to be a linear combination of the original rgb color space so that the extracted lft texture features in the mdc color space can achieve most discrimination in terms of classification segmentation performance to speed up the texture feature extraction process we develop an efficient lft extraction algorithm based on image shifting and image integral for the splitting part given a connected component of the segmentation map we initially differentiate whether it is a touching cell clump or a single nontouching cell the differentiation is mainly based on the distance between the most likely radial symmetry center and the geometrical center of the connected component the boundaries of touching cell clumps are smoothed out by fourier shape descriptor before carrying out an iterative concave point and radial symmetry based splitting algorithm to test the validity effectiveness and efficiency of the framework it is applied to follicular lymphoma pathological images which exhibit complex background and extracellular texture with nonuniform illumination condition for comparison purposes the results of the proposed segmentation algorithm are evaluated against the outputs of superpixel graph cut mean shift and two state of the art pathological image segmentation methods using ground truth that was established by manual segmentation of cells in the original images our segmentation algorithm achieves better results than the other compared methods the results of splitting are evaluated in terms of under splitting over splitting and encroachment errors by summing up the three types of errors we achieve a total error rate of 5 25 per image

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

strong szegő asymptotics and zeros of the zeta function assuming the riemann hypothesis we prove the weak convergence of linear statistics of the zeros of the riemann function to a gaussian field with covariance structure corresponding to the h1 2 norm of the test functions for this purpose we obtain an approximate form of the explicit formula relying on selberg s smoothed expression for and the helffer sjostrand functional calculus our main result is an analogue of the strong szeg theorem known for toeplitz operators and random matrix theory 2014 wiley periodicals inc

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

polygonal approximation of digital planar curves through break point suppression this paper presents a new algorithm that detects a set of dominant points on the boundary of an eight connected shape to obtain a polygonal approximation of the shape itself the set of dominant points is obtained from the original break points of the initial boundary where the integral square is zero for this goal most of the original break points are deleted by suppressing those whose perpendicular distance to an approximating straight line is lower than a variable threshold value the proposed algorithm iteratively deletes redundant break points until the required approximation which relies on a decrease in the length of the contour and the highest error is achieved a comparative experiment with another commonly used algorithm showed that the proposed method produced efficient and effective polygonal approximations for digital planar curves with features of several sizes

min max operators in texture analysis a signature is generated for a given picture by operating on it with different masks the operations are gray level generalizations of shrink and expand for binary pictures using serra s morphological methods 10 the signature is a set of numbers each corresponding to an application of an operator at a certain scale and direction and can be used to analyze the discriminate textures it is shown that this family of signatures includes as special cases several currently used texture descriptors

unsupervised segmentation and approximation of digital curves with rate distortion curve modeling this paper considers the problem of unsupervised segmentation and approximation of digital curves and trajectories with a set of geometrical primitives model functions an algorithm is proposed based on a parameterized model of the rate distortion curve the multiplicative cost function is then derived from the model by analyzing the minimum of the cost function a solution is defined that produces the best possible balance between the number of segments and the approximation error the proposed algorithm was tested for polygonal approximation and multi model approximation circular arcs and line segments for digital curves and polynomials for trajectory the algorithm demonstrated its efficiency in comparisons with known methods with a heuristic cost function the proposed method can additionally be used for segmentation and approximation of signals and time series a new algorithm for unsupervised segmentation of digital curves is introduced this method gives solutions with the best balance between error and description length a multiplicative criterion for evaluation of solutions is introduced

low lying zeros of l functions with orthogonal symmetry we investigate the moments of a smooth counting function of the zeros near the central point of l functions of weight k cuspidal newforms of prime level n we split by the sign of the functional equations and show that for test functions whose fourier transform is supported in 1 n 1 n as n the first n centered moments are gaussian by extending the support to 1 n 1 1 n 1 we see non gaussian behavior in particular the odd centered moments are nonzero for such test functions if we do not split by sign we obtain gaussian behavior for support in 2 n 2 n if 2k n the nth centered moments agree with random matrix theory in this extended range providing additional support for the katz sarnak conjectures the proof requires calculating multidimensional integrals of the nondiagonal terms in the bessel kloosterman expansion of the petersson formula we convert these multidimensional integrals to one dimensional integrals already considered in the work of iwaniec luo and sarnak ils and derive a new and more tractable expression for the nth centered moments for such test functions this new formula facilitates comparisons between number theory and random matrix theory for test functions supported in 1 n 1 1 n 1 by simplifying the combinatorial arguments as an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point

deformable shape detection and description via model based region grouping a method for deformable shape detection and recognition is described deformable shape templates are used to partition the image into a globally consistent interpretation determined in part by the minimum description length principle statistical shape models enforce the prior probabilities on global parametric deformations for each object class once trained the system autonomously segments deformed shapes from the background while not merging them with adjacent objects or shadows the formulation can be used to group image regions obtained via any region segmentation algorithm e g texture color or motion the recovered shape models can be used directly in object recognition experiments with color imagery are reported

segmentation registration and measurement of shape variation via image object shape a model of object shape by nets of medial and boundary primitives is justified as richly capturing multiple aspects of shape and yet requiring representation space and image analysis work proportional to the number of primitives metrics are described that compute an object representation s prior probability of local geometry by reflecting variabilities in the net s node and link parameter values and that compute a likelihood function measuring the degree of match of an image to that object representation a paradigm for image analysis of deforming such a model to optimize a posteriori probability is described and this paradigm is shown to be usable as a uniform approach for object definition object based registration between images of the same or different imaging modalities and measurement of shape variation of an abnormal anatomical object compared with a normal anatomical object examples of applications of these methods in radiotherapy surgery and psychiatry are given

curve fitting approach for tangent angle and curvature measurements abstract the measures of tangent angle and curvature of a digital curve play an important role in image analysis such as corner detection 1d shape representation and shape signature in the generalized hough transform instead of using the discrete measurement approach the least squares method is employed to fit known digital points to two cubic polynomial functions one with y f x that minimizes the sum of the vertical distances and the other with x g y that minimizes the sum of the horizontal distances from the known points to the approximated curve the tangent angle and curvature are therefore directly computable from the first and second order derivatives of the continuous functions hybrid procedures are also proposed to select the better curve from f and g for accurate evaluation of tangent angle and curvature experimented results on both analytic curves and real world images are included

techniques for assessing polygonal approximations of curves given the enormous number of available methods for finding polygonal approximations to curves techniques are required to assess different algorithms some of the standard approaches are shown to be unsuitable if the approximations contain varying numbers of lines instead we suggest assessing an algorithm s results relative to an optimal polygon and describe a measure which combines the relative fidelity and efficiency of a curve segmentation we use this measure to compare the application of 23 algorithms to a curve first used by teh and chin 1989 their integral square errors ises are assessed relative to the optimal ise in addition using an example of pose estimation it is shown how goal directed evaluation can be used to select an appropriate assessment criterion

on the detection of dominant points on digital curves a parallel algorithm is presented for detecting dominant points on a digital closed curve the procedure requires no input parameter and remains reliable even when features of multiple sizes are present on the digital curve the procedure first determines the region of support for each point based on its local properties then computes measures of relative significance e g curvature of each point and finally detects dominant points by a process of nonmaximum suppression this procedure leads to the observation that the performance of dominant points detection depends not only on the accuracy of the measure of significance but also on the precise determination of the region of support this solves the fundamental problem of scale factor selection encountered in various dominant point detection algorithms the inherent nature of scale space filtering in the procedure is addressed and the performance of the procedure is compared to those of several other dominant point detection algorithms using a number of examples

nonlinear integral equations for electromagnetic inverse problems the scaling properties of maxwell s equations allow the existence of simple yet general nonlinear integral equations for electrical conductivity these equations were developed in an attempt to reduce the generality of linearization to the exclusive scope of electromagnetic problems the reduction is achieved when the principle of similitude for quasi static fields is imposed on linearized forms of the field equations the combination leads to exact integral relations which represent a unifying framework for the general electromagnetic inverse problem the equations are of the same form in both time and frequency domains and hold for all observations that scale as electric and magnetic fields do direct current resistivity and magnetometric resistivity methods are considered as special cases the kernel functions of the integral equations are closely related through a normalization factor to the frechet kernels of the conventional equations obtained by linearization accordingly the sensitivity functio

sato tate theorem for families and low lying zeros of automorphic l functions we consider certain families of automorphic representations over number fields arising from the principle of functoriality of langlands let g be a reductive group over a number field f which admits discrete series representations at infinity let l g widehat g rtimes mathrm gal bar f f be the associated l group and r l g rightarrow mathrm gl d mathbb c a continuous homomorphism which is irreducible and does not factor through mathrm gal bar f f the families under consideration consist of discrete automorphic representations of g mathbb a f of given weight and level and we let either the weight or the level grow to infinity we establish a quantitative plancherel and a quantitative sato tate equidistribution theorem for the satake parameters of these families this generalizes earlier results in the subject notably of sarnak prog math 70 321 331 1987 and serre j am math soc 10 1 75 102 1997 as an application we study the distribution of the low lying zeros of the associated family of l functions l s pi r assuming from the principle of functoriality that these l functions are automorphic we find that the distribution of the 1 level densities coincides with the distribution of the 1 level densities of eigenvalues of one of the unitary symplectic and orthogonal ensembles in accordance with the katz sarnak heuristics we provide a criterion based on the frobenius schur indicator to determine this symmetry type if r is not isomorphic to its dual r vee then the symmetry type is unitary otherwise there is a bilinear form on mathbb c d which realizes the isomorphism between r and r vee if the bilinear form is symmetric resp alternating then r is real resp quaternionic and the symmetry type is symplectic resp orthogonal

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

radial symmetries based decomposition of cell clusters in binary and gray level images the segmentation of structures of complex cytological and histological images is a necessary intermediate step for image analysis that give rise to binary images in many cases these binary images can be rather far away from a subsequent object specific quantification because biological structures digitized by optoelectronic devices may situated close together so that they appear as one fused object in the projective image such fusions of objects may become complex so that large clusters of biological structures emerge to quantify individual objects of a cluster they must be separated the shape size and intensity variation of cells in complex organs like the brain may breed planar configurations that can be splitted only inadequate by common techniques e g watershed separation or basic morphological processing of images considering iteratively object contours suitable features of saliency can be accumulated that give rise to markers of singular objects such significant markers may drive a separation process more effective than common approaches the determination of markers by an iterative method should be scale translation and rotation invariant and robust with regard to noise due to the variability of biological specimen we realize a technique that splits cell clumps consisting of different cell sizes and shapes into meaningful parts the multiscale method applied here is based on the analysis of the contour shape and the object area by iterative voting using oriented kernels these cone shaped kernels vote iteratively for the local center of mass of the components of an aggregation the voting is performed along the gradient of the distance transformation of the binarized image of aggregates iterative voting is initialized by voting along the gradient direction where at each iteration the voting direction and shape of the kernel is refined resp the kernel topography is refined and reoriented iteratively it turned out that the kernel topography is unique because it votes for the most likely set of grid points where the gravity center of an individual cluster component may be located furthermore a new procedure is realized to use the local intensities of aggregations for kernel voting the last voted iteration provides gravitation centers resp centers of mass of the clumped cells these are extracted and used as markers to determine individual cell boundaries by a marker based watershed postprocessing the subject of this paper is to highlight the basic algorithm of iterative kernel voting and expanding it to process intensities within clusters as well as contour information the approach is applied to synthetic images that were modified systematically with regard to object topology natural aggregates of cells at the light microscopic level and cell clusters derived from high resolution flat bed scanning were splitted in addition to these examples images from a benchmark databases were investigated the splittings generated by the iterative voting approach were compared with expected splittings of test persons and with results of the watershed method especially the gray level based iterative voting method provides superior results for cell cluster separation in comparison to the watershed procedure

polygonal representation of digital planar curves through dominant point detection a nonparametric algorithm we describe a new algorithm that detects a set of feature points on the boundary of an 8 connected shape that constitute the vertices of a polygonal approximation of the shape itself the set of feature points nodes is a ranked subset of the original shape points whose connected left and right arm extents cover the entire shape nodes are ranked based on their strength in terms of their importance to other boundary points length of support region and distance from the centroid the polygon obtained by linking the detected nodes approximates the contour in an intuitive way the proposed algorithm does not require an input parameter and works well for shapes with features of multiple sizes

the inverse problem of resistivity sounding the electric potential due to a single point electrode at the surface of a layered conducting medium is calculated by means of a linear combination of the potentials associated with a set of two layer systems this new representation is called the bilayer expansion for the green s function it enables the forward problem of resistivity sounding to be solved very efficiently even for complicated profiles also the bilayer expansion facilitates the solution of the resistivity inverse problem the coefficients in the expansion are linearly related to apparent resistivity as it is measured and they are readily mapped into parameters for a model specifically i consider models comprising uniformly conducting layers of equal thickness for a given finite data set a quadratic program can be used to find the best fitting model in this class for any specified thickness as the thickness is reduced models of this kind can approximate arbitrary profiles with unlimited accuracy if there is a model that satisfies

what is the depth of investigation of a resistivity measurement abstractever since the first computation of resistivity sounding curves there has been the impression that somehow they are averages of the vertical resistivity profile this prompted the idea to represent apparent resistivity as an integral over depth and to define depth of investigation using the integrands of the integrals as elementary contributions however elementary contributions for a boundary value problem cannot be uniquely defined and are not physically meaningful many practical applications that have been derived from this approach might be at stake regarding their theoretical basis on the other hand a sensitivity function has a definite physical meaning and it is uniquely defined but it offers a different picture for a layered earth the concept of elementary contributions must then be abandoned as not real as some respected scholars have suggested or it must be put on solid ground if we are going to continue using it our claim is that any definition of elementary contributions must

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

comparison between the morphological skeleton and morphological shape decomposition the morphological skeleton and morphological shape decomposition msd are two popular approaches for morphological shape representation each method represents an object as an algebraic combination of a number of components where each component is given by a locus of points dilated by a specified structuring element homothetic this correspondence develops a theoretical comparison between the two methods combining the theoretical results with several representation cost measures we make a concrete comparison of the efficiency of the two methods the results indicate that for complex objects i e objects requiring a full range of homothetic sizes in the morphological skeleton representation the msd represents objects more efficiently than the morphological skeleton for three of four suggested cost measures

morphological filters part ii their relations to median order statistic and stack filters this paper extends the theory of median order statistic os and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions dilations openings closings and open closings that commute with thresholding the max min representation of os filters is introduced by showing that any median or other os filter is equal to a maximum of erosions moving local minima and also to a minimum of dilations moving local maxima thus os filters can be computed by a closed formula that involves a max min on prespecified sets of numbers and no sorting stack filters are established as the class of filters that are composed exactly of a finite number of max min operations the kernels of median os and stack filters are collections of input signals that uniquely represent these filters due to their translation invariance the max min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal with respect to a signal ordering kernel elements and also to a minimum of dilations by minimal kernel elements of dual filters the representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum of products expressions of boolean functions it is also shown that median filtering and its iterations of any signal by convex 1 d windows is bounded below by openings and above by closings a signal is a root fixed point of the median iff it is a root of both an opening and a closing the open closing and clos opening yield median roots in one pass suppress impulse noise similarly to the median can discriminate between positive and negative noise impulses and are computationally less complex than the median some similar results are obtained for 2 d median filtering

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

curvature and tangential deflection of discrete arcs a theory based on the commutator of scatter matrix pairs and its application to vertex detection in planar shape data this paper introduces a new theory for the tangential deflection and curvature of plane discrete curves our theory applies to discrete data in either rectangular boundary coordinate or chain coded formats its rationale is drawn from the statistical and geometric properties associated with the eigenvalue eigenvector structure of sample covariance matrices specifically we prove that the nonzero entry of the commutator of a piar of scatter matrices constructed from discrete arcs is related to the angle between their eigenspaces and further we show that this entry is in certain limiting cases also proportional to the analytical curvature of the plane curve from which the discrete data are drawn these results lend a sound theoretical basis to the notions of discrete curvature and tangential deflection and moreover they provide a means for computationally efficient implementation of algorithms which use these ideas in various image processing contexts as a concrete example we develop the commutator vertex detection cvd algorithm which identifies the location of vertices in shape data based on excessive cummulative tangential deflection and we compare its performance to several well established corner detectors that utilize the alternative strategy of finding approximate curvature extrema

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

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

symmetry as a continuous feature symmetry is treated as a continuous feature and a continuous measure of distance from symmetry in shapes is defined the symmetry distance sd of a shape is defined to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape this general definition of a symmetry measure enables a comparison of the amount of symmetry of different shapes and the amount of different symmetries of a single shape this measure is applicable to any type of symmetry in any dimension the symmetry distance gives rise to a method of reconstructing symmetry of occluded shapes the authors extend the method to deal with symmetries of noisy and fuzzy data finally the authors consider grayscale images as 3d shapes and use the symmetry distance to find the orientation of symmetric objects from their images and to find locally symmetric regions in images

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

corner detection and curve segmentation by multiresolution chain code linking in this paper we propose a new method to characterise a curve by means of the hierarchical computation of a multiresolution structure this structure consisting of successive lower resolution versions of the same object is processed using the linked pyramid approach we adapt the multiresolution pixel linking algorithm to the processing of curve contours which are described by their chain code we also introduce a selective class selection process which allows application of the algorithm to segmentation and detection of contour features the resulting framework presents good performance for a wide range of object sizes without the need of any parameter tweaking and allows detection of shape detail at different scales

hierarchical shape description via the multiresolution symmetric axis transform a method is proposed that produces a shape description in the form of a hierarchy by scale of simple symmetric axis sequences an axis segment that is a child of another has smaller scale and is seen as a branch of its parent the scale value and parent child relationship are induced by following the symmetric axis under successive reduction of resolution the result in two or three dimensions is a figure rather than boundary oriented shape description that has natural segments and is insensitive to noise in the object description the general approach of hierarchy production by following a feature through successive resolution reduction will be presented as will methods of resolution reduction and computer implementation also the relation of this figure based shape description to those based on boundary curvature will be briefly discussed

nonlinear multiresolution signal decomposition schemes ii morphological wavelets interest in multiresolution techniques for signal processing and analysis is increasing steadily an important instance of such a technique is the so called pyramid decomposition scheme this paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis the proposed theory is based on the following ingredients 1 the pyramid consists of a finite or infinite number of levels such that the information content decreases toward higher levels and 2 each step toward a higher level is implemented by an information reducing analysis operator whereas each step toward a lower level is implemented by an information preserving synthesis operator one basic assumption is necessary synthesis followed by analysis yields the identity operator meaning that no information is lost by these two consecutive steps several examples of pyramid decomposition schemes are shown to be instances of the proposed theory a particular class of linear pyramids morphological skeleton decompositions the morphological haar pyramid median pyramids etc furthermore the paper makes a distinction between single scale and multiscale decomposition schemes i e schemes without or with sample reduction finally the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks

extraction of dominant points by estimation of the contour fluctuations a new method of dominant points dps extraction is presented for each boundary point the method assigns a support region whose length is individually limited two steps are used to detect dps firstly points having higher curvature value than the estimated fluctuations over the support region are extracted as candidates secondly adjacent candidates are merged and a final elimination step is performed on each region additionally a new error measurement scheme is introduced which takes into account the error due to the polygonalisation and the deviation brought by each dp this error is used to compare our algorithm to other methods

a curve bend function based method to characterize contour shapes in this paper a new curve bend function cbf based method is presented to extract main features of a contour it utilizes two quantities one called a curve bend angle cba and the other known as a type coefficient of curve segment defined on a contour the cba measures the bending degree at each point on the contour while the type coefficient characterizes the properties convexity and concavity of simple curve segments the cbf combines the advantages of the two quantities and provides an overall description of a contour a local maximum or minimum of the cbf corresponds to a critical point of the contour and the peak value is the cosine of the cba hence the problem of locating critical points of a contour is reduced to the problem of finding peak points of the cbf the sign of a peak value indicates the related cba being an inner or outer angle our experimental results and comparisons between the cbf method and rosenfeld johnston s method show that our method is effective and accurate in generating the description of a contour

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