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

symmetry detection by generalized complex gc moments a close form solution this paper presents a unified method for detecting both reflection symmetry and rotation symmetry of 2d images based on an identical set of features i e the first three nonzero generalized complex gc moments this method is theoretically guaranteed to detect all the axes of symmetries of every 2d image if more nonzero gc moments are used in the feature set furthermore we establish the relationship between reflectional symmetry and rotational symmetry in an image which can be used to check the correctness of symmetry detection this method has been demonstrated experimentally using more than 200 images

a survey of the theory of source coding with a fidelity criterion the purpose of this paper is threefold 1 to acquaint the reader with the types of problems that have been considered in the area of source coding with a fidelity criterion 2 to survey results that have been obtained on these problems and 3 to outline future research trends in the area

detection of partial symmetry using correlation with rotated reflected images abstract symmetry is one of the important structural properties of a figure both in visual psychology and in computer vision symmetrical properties of planar two dimensional figures are considered and a new method for the extraction of rotational symmetry and reflectional symmetry is presented a figure is said to have symmetry if it is invariant with some congruent transformation which consists of translation rotation and reflection the approach is to obtain the congruent transformations which make the transformed image a good match with the original image the directional correlation of edge features is used to evaluate how well the original and transformed images match this method does not require any segmentation nor knowledge of the centroid position and the symmetrical properties of partially symmetric figures can also be detected the approach is described and this method is applied to synthetic and real images

normalization of rotationally symmetric shapes for pattern recognition abstract image normalization is useful for image recognition by the matching method recently a normalization algorithm was developed to normalize images under the distortion of translation rotation scaling and skew however the method fails when the image is rotationally symmetric a method is introduced called the modified fourier descriptor to normalize rotationally symmetric shapes this method is simple and fast and can be combined with the preceding normalization algorithm to make it more general and realizable

epsilon entropy and critical distortion of random fields the link between epsilon entropy for binary random fields and the lee yang theorem in statistical mechanics makes it possible to prove the existence of positive critical distortion for most binary random fields with pair interactions and some with many body interactions lower bounds for critical distortion of random fields on two dimensional 2 d lattices have been obtained in particular the new lower bounds for some 2 d ising models improve upon previously known bounds

contribution to the determination of vanishing points using hough transform we propose a method to locate three vanishing points on an image corresponding to three orthogonal directions of the scene this method is based on two cascaded hough transforms we show that even in the case of synthetic images of high quality a naive approach may fail essentially because of the errors due to the limitation of the image size we take into account these errors as well as errors due to detection inaccuracy of the image segments and provide a method efficient even in the case of real complex scenes

frequency domain classification of cyclic and dihedral symmetries of finite 2 d patterns this paper presents a very simple and effective algorithm for classifying cyclic and dihedral symmetries from images of finite two dimensional patterns it consists in an extension of a frequency domain algorithm proposed by the author for estimating two dimensional rotation of rigid objects within this framework the estimation of rotation is conveniently turned into the problem of detecting two orthogonal lines within the locus of the zero crossings of a certain function represented in orthogonal cartesian coordinates the main contribution of this paper is the use and interpretation of the ambiguities that arise in such a locus in the form of additional pairs of orthogonal lines when applied to symmetric patterns and the development of other algorithmic steps which allow a simple and fast discrimination between cyclic and dihedral symmetries unlike any other available method this algorithm does not require any conversion from cartesian to polar image representations besides classification reliability and consistency observed through several dozen testing experiments the nice features of this new method also include robustness to noise and ease of implementation with fast fourier transform algorithms which makes it amenable to almost real time pattern classifications applications several examples representative of the performance of the algorithm through extensive testing are reported and discussed in the paper

a decomposition theorem for binary markov random fields consider a binary markov random field whose neighbor structure is specified by a countable graph with nodes of uniformly bounded degree under a minimal assumption we prove a decomposition theorem to the effect that such a markov random field can be represented as the nodewise modulo 2 sum of two independent binary random fields one of which is white binary noise of positive weight said decomposition provides the information theorist with an exact expression for the per site rate distortion function of the random field over an interval of distortions not exceeding this weight we mention possible implications for communication theory probability theory and statistical physics

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

partition function estimation of gibbs random field images using monte carlo simulations a monte carlo simulation technique for estimating the partition function of a general gibbs random field image is proposed by expressing the partition function as an expectation an importance sampling approach for estimating it using monte carlo simulations is developed as expected the resulting estimators are unbiased and consistent computations can be performed iteratively by using simple monte carlo algorithms with remarkable success as demonstrated by simulations the work concentrates on binary second order gibbs random fields defined on a rectangular lattice however the proposed methods can be easily extended to more general gibbs random fields their potential contribution to optimal parameter estimation and hypothesis testing problems for general gibbs random field images using a likelihood approach is anticipated

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

matching widely separated views based on affine invariant regions invariant regions are self adaptive image patches that automatically deform with changing viewpoint as to keep on covering identical physical parts of a scene such regions can be extracted directly from a single image they are then described by a set of invariant features which makes it relatively easy to match them between views even under wide baseline conditions in this contribution two methods to extract invariant regions are presented the first one starts from corners and uses the nearby edges while the second one is purely intensity based as a matter of fact the goal is to build an opportunistic system that exploits several types of invariant regions as it sees fit this yields more correspondences and a system that can deal with a wider range of images to increase the robustness of the system two semi local constraints on combinations of region correspondences are derived one geometric the other photometric they allow to test the consistency of correspondences and hence to reject falsely matched regions experiments on images of real world scenes taken from substantially different viewpoints demonstrate the feasibility of the approach

detecting approximate reflection symmetry in a point set using optimization on manifold we propose an algorithm to detect approximate reflection symmetry present in a set of volumetrically distributed points belonging to mathbb r d containing a distorted reflection symmetry pattern we pose the problem of detecting approximate reflection symmetry as the problem of establishing correspondences between the points which are reflections of each other and we determine the reflection symmetry transformation we formulate an optimization framework in which the problem of establishing the correspondences amounts to solving a linear assignment problem and the problem of determining the reflection symmetry transformation amounts to solving an optimization problem on a smooth riemannian product manifold the proposed approach estimates the symmetry from the geometry of the points and is descriptor independent we evaluate the performance of the proposed approach on the standard benchmark dataset and achieve the state of the art performance we further show the robustness of our approach by varying the amount of distortion in a perfect reflection symmetry pattern where we perturb each point by a different amount of perturbation we demonstrate the effectiveness of the method by applying it to the problem of 2 d two dimensional and 3 d reflection symmetry detection along with comparisons

point matching under large image deformations and illumination changes to solve the general point correspondence problem in which the underlying transformation between image patches is represented by a homography a solution based on extensive use of first order differential techniques is proposed we integrate in a single robust m estimation framework the traditional optical flow method and matching of local color distributions these distributions are computed with spatially oriented kernels in the 5d joint spatial color space the estimation process is initiated at the third level of a gaussian pyramid uses only local information and the illumination changes between the two images are also taken into account subpixel matching accuracy is achieved under large projective distortions significantly exceeding the performance of any of the two components alone as an application the correspondence algorithm is employed in oriented tracking of objects

robust midsagittal plane extraction from normal and pathological 3 d neuroradiology images this paper focuses on extracting the ideal midsagittal plane imsp from three dimensional 3 d normal and pathological neuroimages the main challenges in this work are the structural asymmetry that may exist in pathological brains and the anisotropic unevenly sampled image data that is common in clinical practice we present an edge based cross correlation approach that decomposes the plane fitting problem into discovery of two dimensional symmetry axes on each slice followed by a robust estimation of plane parameters the algorithm s tolerance to brain asymmetries input image offsets and image noise is quantitatively evaluated we find that the algorithm can extract the imsp from input 3 d images with 1 large asymmetrical lesions 2 arbitrary initial rotation offsets 3 low signal to noise ratio or high bias field the imsp algorithm is compared with an approach based on maximization of mutual information registration and is found to exhibit superior performance under adverse conditions finally no statistically significant difference is found between the midsagittal plane computed by the imsp algorithm and that estimated by two trained neuroradiologists

globally optimal grouping for symmetric closed boundaries by combining boundary and region information many natural and man made structures have a boundary that shows a certain level of bilateral symmetry a property that plays an important role in both human and computer vision in this paper we present a new grouping method for detecting closed boundaries with symmetry we first construct a new type of grouping token in the form of symmetric trapezoids by pairing line segments detected from the image a closed boundary can then be achieved by connecting some trapezoids with a sequence of gap filling quadrilaterals for such a closed boundary we define a unified grouping cost function in a ratio form the numerator reflects the boundary information of proximity and symmetry and the denominator reflects the region information of the enclosed area the introduction of the region area information in the denominator is able to avoid a bias toward shorter boundaries we then develop a new graph model to represent the grouping tokens in this new graph model the grouping cost function can be encoded by carefully designed edge weights and the desired optimal boundary corresponds to a special cycle with a minimum ratio form cost we finally show that such a cycle can be found in polynomial time using a previous graph algorithm we implement this symmetry grouping method and test it on a set of synthetic data and real images the performance is compared to two previous grouping methods that do not consider symmetry in their grouping cost functions

similarity and symmetry measures for convex shapes using minkowski addition this paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on minkowski addition and the brunn minkowski inequality this means in particular that these measures are region based in contrast to most of the literature where one considers contour based measures all measures considered in this paper are invariant under translations furthermore they can be chosen to be invariant under rotations multiplications reflections or the class of affine transformations it is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle spl theta is a piecewise concave function of spl theta this and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures various results obtained in this paper are illustrated by experimental data although the paper deals exclusively with the two dimensional case many of the theoretical results carry over almost directly to higher dimensional spaces

a new method to estimate the critical distortion of random fields it was shown by c m newman and g a baker 1991 that the critical distortion d sub c of the ising model on z sup k is related to the radius of convergence r of the classical mayer series by d sub c r 1 r an approach from statistical mechanics to estimate r as well as d sub c by using the matrix representation of mayer series is found

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

on the lengths of jordan chains for eigenvalues on the boundary of numerical range of quadratic operator polynomial abstract we describe a connection between the geometrical structure of the numerical range w l of a selfadjoint quadratic operator polynomial l and the lengths of jordan chains for eigenvalues on the boundary of the numerical range also for each pair of natural numbers m n 1 m 2 n we construct a monic quadratic polynomial l m n 2 c b with selfadjoint n n matrices c b which has an eigenvalue 0 w l m n with a jordan chain of length m

subpixel detection of the center of an object using a spectral phase algorithm on the image this paper presents an algorithm designed for detecting the center of an object with central symmetry in an image it is well known that the spectral phase of such an object is a plane whose orientation depends on the object position in the image our algorithm applies weighting of the moduli to determine the slope and the orientation of this wrapped phase plane in the image spectrum a comparison is made with other conventional methods and shows the advantages of this algorithm subpixel resolution computing speed ease of implementation

unified detection of skewed rotation reflection and translation symmetries from affine invariant contour features symmetry detection is significant for object detection and recognition since symmetries are salient cues for distinguishing geometrical structures from cluttered backgrounds this paper proposes a unified framework to detect rotation reflection and translation symmetries simultaneously on unsegmented real world images the detection is performed based on affine invariant contour features and is applicable under skewed imaging with distortions contours on a natural image are first matched to each other to find affine invariant matching pairs which are then classified into three groups using a sign change criterion corresponding to the three types of symmetries the three groups are used to vote for the corresponding symmetries the voting for rotation utilizes a scheme of short line segments the voting for reflection is based on a parameterization of axis equation and the voting for translation uses a cascade like approach experimental results of real world images are provided with quantitative evaluations validating that the proposed framework achieves desired performance

on the detection of the axes of symmetry of symmetric and almost symmetric planar images a simple algorithm is presented for finding all the axes of symmetry of symmetric and almost symmetric planar images having nonuniform gray level intensity images the technique which can also be used in conjunction with planar curves is based on the identification of the centroids of the given image and other related sets of points followed by a maximization of a specially defined coefficient of symmetry owing to the nature of the required procedures which are strictly based on the computation of mean values the method is not particularly affected by the presence of noise and is especially suited for application in conjunction with digitized figures

on critical distortion for markov sources corresp the authors underbound the critical distortion below which the rate distortion function of a markov chain equals its shannon lower bound the bound is tight in all cases in which the exact answer is currently known

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

reflection symmetry detection using locally affine invariant edge correspondence reflection symmetry detection receives increasing attentions in recent years the state of the art algorithms mainly use the matching of intensity based features such as the sift within a single image to find symmetry axes this paper proposes a novel approach by establishing the correspondence of locally affine invariant edge based features which are superior to the intensity based in the aspects that it is insensitive to illumination variations and applicable to textureless objects the locally affine invariance is achieved by simple linear algebra for efficient and robust computations making the algorithm suitable for detections under object distortions like perspective projection commonly used edge detectors and a voting process are respectively used before and after the edge description and matching steps to form a complete reflection detection pipeline experiments are performed using synthetic and real world images with both multiple and single reflection symmetry axis the test results are compared with existing algorithms to validate the proposed method

estimation of phase for noisy linear phase signals it is well known that a discrete time symmetric signal has a linear phase fourier transform this paper describes a procedure for estimating the parameters associated with a linear phase signal from noisy measurements when the data being modeled is composed of a linear phase signal corrupted by additive gaussian noise the approach taken results in maximum likelihood estimates of the linear phase parameters the fisher information matrix determining the achievable accuracy for the estimates is derived the proposed estimation procedure is applied to the image processing problem of determining the inclination of the axis of symmetry of a reflection symmetric object the effectiveness of the procedure is tested on both synthetic and real images

the s kernel a measure of symmetry of objects in this paper we introduce a new symmetry feature named symmetry kernel sk to support a measure of symmetry given any symmetry transform s sk of a pattern p is the maximal included symmetric sub set of p for all directions and shifts we provide a first algorithm to exhibit this kernel where the centre of symmetry is assumed to be the centre of mass then we prove that in any direction the optimal axis corresponds to the maximal correlation of a pattern with its symmetric version that leads to a second algorithm the associated symmetry measure is a modified difference between the respective surfaces of a pattern and its kernel a series of experiments supports the actual algorithm validation

irregular shape symmetry analysis theory and application to quantitative galaxy classification this paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the amount of symmetry of arbitrary shapes the definitions of both bilateral symmetricity and rotational symmetricity give new insight into analyzing the geometrical property of a shape and enable characterizing arbitrary shapes in a new way we developed a set of criteria for quantitative galaxy classification using our proposed irregular shape symmetry measures our study has demonstrated the effectiveness of the proposed method for the characterization of the shape of the celestial bodies the concepts described in the paper are applicable to many fields such as mathematics artificial intelligence digital image processing robotics biomedicine etc

noncombinatorial detection of regular repetitions under perspective skew we present a geometric framework for the efficient detection of regular repetitions of planar but not necessarily coplanar patterns at the heart of our system lie the fixed structures of the transformations that describe these regular configurations the approach detects a number of symmetric configurations that have traditionally been dealt with separately in that all configurations corresponding to planar homologies are detected these include important cases such as periodicities mirror symmetries and reflections about a point the approach can handle perspective distortions it avoids to get trapped in combinatorics through invariant based hashing for pattern matching and through hough transforms for the detection of fixed structures additional efficiency and robustness are obtained from the system s ability to reason about the consistency of multiple homologies the performance of the system is demonstrated with several examples

interpretation of conic motion and its applications the indeterminacy of conic motion is analyzed in terms of lie group theory it is shown that an image motion of a conic is associated with a group ofinvisible motions that do not cause a visible change of the conic all such groups are isomorphic to the group associated with a special conic called thestandard circle for which the group of invisible motions is the three dimensional lorentz group similar results are obtained forinvisible optical flows finally our analysis is extended toconic stereo the 3 d position and orientation of a conic in the scene are computed from two projections this algorithm also works with one camera if a circular pattern is projected from a light source

decomposition of binary random fields and zeros of partition functions on montre qu il existe une relation etroite entre le probleme de decomposition pour des champs eleatoires binaires etudies dans un article recent de hajek et berger 1987 et le probleme de mecanique statistique de determination de regions independantes du zero pour des fonctions de partition