comments on nonparametric segmentation of curves into various representations with reply the author points out the existence of a theoretical difficulty that underlies the curve segmentation problem studied in the work of rosin and west ibid vol 17 no 12 p 1140 53 1995 and present a possible solution to it the problem arises because curves are segmented into low and high order features and the former are a subset of the latter in reply rosin and west discuss the advantages and disadvantages of two approaches to model selection their original heuristic sequential method and the information theoretic method suggested by kanatani and others

application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

a symmetry measure for damage detection with mode shapes abstract this paper introduces a feature for detecting damage or changes in structures the continuous symmetry measure which can quantify the amount of a particular rotational mirror or translational symmetry in a mode shape of a structure many structures in the built environment have geometries that are either symmetric or almost symmetric however damage typically occurs in a local manner causing asymmetric changes in the structure s geometry or material properties and alters its mode shapes the continuous symmetry measure can quantify these changes in symmetry as a novel indicator of damage for data based structural health monitoring approaches this paper describes the concept as a basis for detecting changes in mode shapes and detecting structural damage application of the method is demonstrated in various structures with different symmetrical properties a pipe cross section with a finite element model and experimental study the nasa 8 bay truss model and the simulated iasc asce structural health monitoring benchmark structure the applicability and limitations of the feature in applying it to structures of varying geometries is discussed

a signal processing approach to symmetry detection we present an algorithm that detects rotational and reflectional symmetries of two dimensional objects both symmetry types are effectively detected and analyzed using the angular correlation ac which measures the correlation between images in the angular direction the ac is accurately computed using the pseudopolar fourier transform which rapidly computes the fourier transform of an image on a near polar grid we prove that the ac of symmetric images is a periodic signal whose frequency is related to the order of the symmetry this frequency is recovered via spectrum estimation which is a proven technique in signal processing with a variety of efficient solutions we also provide a novel approach for finding the center of symmetry and demonstrate the applicability of our scheme to the analysis of real images

a modified version of the k means algorithm with a distance based on cluster symmetry we propose a modified version of the k means algorithm to cluster data the proposed algorithm adopts a novel nonmetric distance measure based on the idea of point symmetry this kind of point symmetry distance can be applied in data clustering and human face detection several data sets are used to illustrate its effectiveness

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

line drawing interpretation bilateral symmetry results previously derived by the author are used to investigate the implication of bilateral symmetry in line drawings it is shown that the line drawing of an orthographically protected surface of revolution exhibits bilateral symmetry about the projection of its axis of revolution irrespective of the viewing direction barring one exception a bilaterally symmetric line drawing is necessarily the orthographic projection of a local surface of revolution whenever its symmetry axis continues to be the projection of a fixed line in space under perturbation of viewpoint the axis of revolution is the invariant preimage of the symmetry axis various line drawing causes are detailed which facilitate the deduction of invariant preimages of symmetry axes and consequently of local surfaces of revolution

comments on symmetry as a continuous feature we point out the existence of a theoretical difficulty that underlies the symmetry detection studied by zabrodsky et al 1995 and present a possible solution to it

reflection symmetry integrated image segmentation this paper presents a new symmetry integrated region based image segmentation method the method is developed to obtain improved image segmentation by exploiting image symmetry it is realized by constructing a symmetry token that can be flexibly embedded into segmentation cues interesting points are initially extracted from an image by the sift operator and they are further refined for detecting the global bilateral symmetry a symmetry affinity matrix is then computed using the symmetry axis and it is used explicitly as a constraint in a region growing algorithm in order to refine the symmetry of the segmented regions a multi objective genetic search finds the segmentation result with the highest performance for both segmentation and symmetry which is close to the global optimum the method has been investigated experimentally in challenging natural images and images containing man made objects it is shown that the proposed method outperforms current segmentation methods both with and without exploiting symmetry a thorough experimental analysis indicates that symmetry plays an important role as a segmentation cue in conjunction with other attributes like color and texture

3d symmetry detection using the extended gaussian image symmetry detection is important in the area of computer vision a 3d symmetry detection algorithm is presented in this paper the symmetry detection problem is converted to the correlation of the gaussian image once the gaussian image of the object has been obtained the algorithm is independent of the input format the algorithm can handle different kinds of images or objects simulated and real images have been tested in a variety of formats and the results show that the symmetry can be determined using the gaussian image

analyzing skewed symmetries symmetry is pervasive in both man made objects and nature since symmetries project to skew symmetries finding axes of skew symmetry is an important vision task this paper presents a linear time algorithm for finding the axes of skew symmetry where the degree of symmetry is known first we present a review and critique of current methods for finding the axes of skew symmetry next we decompose the problem of finding skew symmetry into the subproblems of solving for the rotational parameter of a shear symmetry and recovering the shear parameter of a reflexive symmetry using this approach the authors derive a direct non heuristic moment based technique for finding the axes of skew symmetry for skew symmetric figures with degree of symmetry less than five we obtain a closed form solution the method does not rely on continuous contours but assumes there is no occlusion and requires knowing the contour s degree of symmetry it is the first algorithm to find the axes of skew symmetry ino n time where n is the number of contour points the method is especially suited to industrial applications where the degree of symmetry is often knowna priori examples of the method are presented for both real and synthetic images and an error analysis of the method is given

chiral coefficient a measure of the amount of structural chirality a new concept of quantitative measurement of chiral structure is introduced and labelled as chiral coefficient chi a distinction is made between geometric and physical chiralities and chi is defined for various physical cases few general geometrical features are discussed and it is shown that for every chiral body there exists a natural z direction depending on the physical property treated possible practical applications of this concept are discussed a few simple examples of the chiral coefficient of mass chi m for chiral molecules are treated

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

geometric information criterion for model selection in building a 3 d model of the environment from image and sensor data one must fit to the data an appropriate class of models which can be regarded as a parametrized manifold or geometric model defined in the data space in this paper we present a statistical framework for detecting degeneracies of a geometric model by evaluating its predictive capability in terms of the expected residual and derive the geometric aic we show that it allows us to detect singularities in a structure from motion analysis without introducing any empirically adjustable thresholds we illustrate our approach by simulation examples we also discuss the application potential of this theory for a wide range of computer vision and robotics problems

oneuclideanalgebraofhermitianoperators on a quaternionic hilbert space abstract we solve the problem of finding the best possible constant of ultraprimeness for the special class of euclidean algebra called algebra of hermitian operators on a quaternionic hilbert space more precisely we prove that for algebra of hermitian operators equipped with spectral norm the best possible constant of ultraprimeness is 1 2

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

spectral symmetry analysis we present a spectral approach for detecting and analyzing rotational and reflectional symmetries in n dimensions our main contribution is the derivation of a symmetry detection and analysis scheme for sets of points irn and its extension to image analysis by way of local features each object is represented by a set of points s irn where the symmetry is manifested by the multiple self alignments of s the alignment problem is formulated as a quadratic binary optimization problem with an efficient solution via spectral relaxation for symmetric objects this results in a multiplicity of eigenvalues whose corresponding eigenvectors allow the detection and analysis of both types of symmetry we improve the scheme s robustness by incorporating geometrical constraints into the spectral analysis our approach is experimentally verified by applying it to 2d and 3d synthetic objects as well as real images

symmetry seeking models and 3d object reconstruction we propose models of 3d shape which may be viewed as deformable bodies composed of simulated elastic material in contrast to traditional purely geometric models of shape deformable models are active their shapes change in response to externally applied forces we develop a deformable model for 3d shape which has a preference for axial symmetry symmetry is represented even though the model does not belong to a parametric shape family such as generalized cylinders rather a symmetry seeking property is designed into internal forces that constrain the deformations of the model we develop a framework for 3d object reconstruction based on symmetry seeking models instances of these models are formed from monocular image data through the action of external forces derived from the data the forces proposed in this paper deform the model in space so that the shape of its projection into the image plane is consistent with the 2d silhouette of an object of interest the effectiveness of our approach is demonstrated using natural images

uniform primeness of the jordan algebra of hermitian quaternion matrices abstract in this note we improve the constant of ultraprimeness for the euclidean algebra of hermitian operators on a quaternionic hilbert space more precisely we shall prove that a constant of ultraprimeness is 2 1

regularized neural networks some convergence rate results in a recent paper poggio and girosi 1990 proposed a class of neural networks obtained from the theory of regularization regularized networks are capable of approximating arbitrarily well any continuous function on a compactum in this paper we consider in detail the learning problem for the one dimensional case we show that in the case of output data observed with noise regularized networks are capable of learning and approximating on compacta elements of certain classes of sobolev spaces known as reproducing kernel hilbert spaces rkhs at a nonparametric rate that optimally exploits the smoothness properties of the unknown mapping in particular we show that the total squared error given by the sum of the squared bias and the variance will approach zero at a rate of n 2m 2m 1 where m denotes the order of differentiability of the true unknown function on the other hand if the unknown mapping is a continuous function but does not belong to an rkhs then there still exists a unique regularized solution but this is no longer guaranteed to converge in mean square to a well defined limit further even if such a solution converges the total squared error is bounded away from zero for all n sufficiently large

full and partial symmetries of non rigid shapes symmetry and self similarity are the cornerstone of nature exhibiting themselves through the shapes of natural creations and ubiquitous laws of physics since many natural objects are symmetric the absence of symmetry can often be an indication of some anomaly or abnormal behavior therefore detection of asymmetries is important in numerous practical applications including crystallography medical imaging and face recognition to mention a few conversely the assumption of underlying shape symmetry can facilitate solutions to many problems in shape reconstruction and analysis traditionally symmetries are described as extrinsic geometric properties of the shape while being adequate for rigid shapes such a description is inappropriate for non rigid ones extrinsic symmetry can be broken as a result of shape deformations while its intrinsic symmetry is preserved in this paper we present a generalization of symmetries for non rigid shapes and a numerical framework for their analysis addressing the problems of full and partial exact and approximate symmetry detection and classification

3 d reconstruction using mirror images based on a plane symmetry recovering method three dimensional reconstruction from a perspective 2d image using mirrors is addressed the mirrors are used to form symmetrical relations between the direct image and mirror images by finding correspondences between them the 3d shape can be reconstructed by means of plane symmetry recovering method using the vanishing point two constraints are used in determining the correspondence in the case where only one mirror is used invisible parts both in the direct image and in the mirror image may still remain using multiple mirrors however occluded parts will decrease or disappear and occlusion free object reconstruction becomes possible

faster and more robust point symmetry based k means algorithm based on the recently published point symmetry distance psd measure this paper presents a novel psd measure namely symmetry similarity level ssl operator for k means algorithm our proposed modified point symmetry based k means mpsk algorithm is more robust than the previous psk algorithm by su and chou not only the proposed mpsk algorithm is suitable for the symmetrical intra clusters as the psk algorithm does the proposed mpsk algorithm is also suitable for the symmetrical inter clusters in addition two speedup strategies are presented to reduce the time required in the proposed mpsk algorithm experimental results demonstrate the significant execution time improvement and the extension to the symmetrical inter clusters of the proposed mpsk algorithm when compared to the previous psk algorithm

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a new measure of symmetry and its application to classification of bifurcating structures this paper presents a new measure of symmetry for bifurcating structures which relies not only on topology and ordering but also on quantitative properties e g length of branches this measure is based on a specific biological mechanism and on the concept of minimum energy the effectiveness of the approach is demonstrated in a classification test where leaves taken from plants growing under different stress conditions are classified results show that the proposed measure improves classification performance compared to classification based on other leading measures

approximate reflectional symmetries of fuzzy objects with an application in model based object recognition this paper is devoted to the study of reflectional symmetries of fuzzy objects we introduce a symmetry measure which defines the degree of symmetry of an object with respect to a given plane it is computed by measuring the similarity between the original object and its reflection the choice of an appropriate measure of comparison is based on the desired properties of the symmetry measure then an algorithm for computing the symmetry plane of fuzzy objects is proposed this is done using an optimization technique in the space of plane parameters finally we illustrate our approach with an application where the symmetry measure is used as an attribute in graph matching for model based object recognition

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

identification of a discrete planar symmetric shape from a single noisy view in this paper we propose a method for identifying a discrete planar symmetric shape from an arbitrary viewpoint our algorithm is based on a newly proposed notion of a view s skeleton we show that this concept yields projective invariants which facilitate the identification procedure it is furthermore shown that the proposed method may be extended to the case of noisy data to yield an optimal estimate of a shape in question substantiating examples are provided

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

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

finding axes of symmetry from potential fields this paper addresses the problem of detecting axes of bilateral symmetry in images in order to achieve robustness to variation in illumination only edge gradient information is used to overcome the problem of edge breaks a potential field is developed from the edge map which spreads the information in the image plane pairs of points in the image plane are made to vote for their axes of symmetry with some confidence values to make the method robust to overlapping objects only local features in the form of taylor coefficients are used for quantifying symmetry we define an axis of symmetry histogram which is used to accumulate the weighted votes for all possible axes of symmetry to reduce the computational complexity of voting a hashing scheme is proposed wherein pairs of points whose potential fields are too asymmetric are pruned by not being counted for the vote experimental results indicate that the proposed method is fairly robust to edge breaks and is able to detect symmetries even when only 0 05 of the possible pairs are used for voting

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

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

detection of rotational symmetry in curves represented by the slope chain code abstract we present a new approach based on the slope chain code to determine whether a curve is rotational symmetrical and its order of symmetry the proposed approach works for open and closed perfectly symmetrical or quasi symmetrical 2d curves simple operations on the scc and its invariant properties are central to our methodology to evaluate the proposed methodology we use 1400 curves from a public database for the symmetrical asymmetrical classification task a recall r of 0 86 a balanced accuracy ba of 0 92 and a precision p of 0 87 were obtained for the quasi symmetrical quasi asymmetrical classification task r 0 77 ba 0 83 and p 0 70 were obtained for the order of rotational symmetry detection task the following performance was achieved r 0 97 ba 0 98 and p 0 95 for a symmetrical set of curves and r 0 98 ba 0 98 and p 0 90 for a quasi symmetrical set of curves we conclude our presentation demonstrating the usefulness of our methodology with three practical applications

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

motif based defect detection for patterned fabric this paper proposes a generalized motif based method for detecting defects in 16 out of 17 wallpaper groups in 2d patterned texture it assumes that most patterned texture can be decomposed into lattices and their constituents motifs it then utilizes the symmetry property of motifs to calculate the energy of moving subtraction and its variance among different motifs by learning the distribution of these values over a number of defect free patterns boundary conditions for discerning defective and defect free patterns can be determined this paper presents the theoretical foundation of the method and defines the relations between motifs and lattice from which a new concept called energy of moving subtraction is derived using norm metric measurement between a collection of circular shift matrices of motif and itself it has been shown in this paper that the energy of moving subtraction amplifies the defect information of the defective motif together with its variance an energy variance space is further defined where decision boundaries are drawn for classifying defective and defect free motifs as the 16 wallpaper groups of patterned fabric can be transformed into three major groups the proposed method is evaluated over these three major groups from which 160 defect free lattices samples are used for defining the decision boundaries with 140 defect free and 113 defective samples used for testing an overall detection success rate of 93 32 is achieved for the proposed method no other generalized approach can achieve this success rate has been reported before and hence this result outperforms all other previously published approaches

on symmetry and multiple view geometry structure pose and calibration from a single image in this paper we provide a principled explanation of how knowledge in global 3 d structural invariants typically captured by a group action on a symmetric structure can dramatically facilitate the task of reconstructing a 3 d scene from one or more images more importantly since every symmetric structure admits a canonical coordinate frame with respect to which the group action can be naturally represented the canonical pose between the viewer and this canonical frame can be recovered too which explains why symmetric objects e g buildings provide us overwhelming clues to their orientation and position we give the necessary and sufficient conditions in terms of the symmetry group admitted by a structure under which this pose can be uniquely determined we also characterize when such conditions are not satisfied to what extent this pose can be recovered we show how algorithms from conventional multiple view geometry after properly modified and extended can be directly applied to perform such recovery from all hidden images of one image of the symmetric structure we also apply our results to a wide range of applications in computer vision and image processing such as camera self calibration image segmentation and global orientation large baseline feature matching image rendering and photo editing as well as visual illusions caused by symmetry if incorrectly assumed

convex set symmetry measurement using blaschke addition in this paper we state a new way for measuring symmetry of convex sets the measures introduced are based on the blaschke sum this operation has some important properties and is a natural way for combining convex objects when represented by their extended gaussian image different kinds of symmetry are considered and their properties outlined to form the blaschke sum it is necessary to solve the minkowski reconstruction problem we study the method of reconstruction and give some improvements to make the scheme more efficient

curved glide reflection symmetry detection we generalize the concept of bilateral reflection symmetry to curved glide reflection symmetry in 2d euclidean space such that classic reflection symmetry becomes one of its six special cases we propose a local feature based approach for curved glide reflection symmetry detection from real unsegmented 2d images furthermore we apply curved glide reflection axis detection for curved reflection surface detection in 3d images our method discovers groups and connects statistically dominant local glide reflection axes in an axis parameter space aps without preassumptions on the types of reflection symmetries quantitative evaluations and comparisons against state of the art algorithms on a diverse 64 test image set and 1 125 swedish leaf data images show a promising average detection rate of the proposed algorithm at 80 and 40 percent respectively and superior performance over existing reflection symmetry detection algorithms potential applications in computer vision particularly biomedical imaging include saliency detection from unsegmented images and quantification of deviations from normality we make our 64 test image set publicly available

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

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

focus of attention from local color symmetries in this paper a continuous valued measure for local color symmetry is introduced the new algorithm is an extension of the successful gray value based symmetry map proposed by reisfeld et al the use of color facilitates the detection of focus points fps on objects that are difficult to detect using gray value contrast only the detection of fps is aimed at guiding the attention of an object recognition system therefore fps have to fulfill three major requirements stability distinctiveness and usability the proposed algorithm is evaluated for these criteria and compared with the gray value based symmetry measure and two other methods from the literature stability is tested against noise object rotation and variations of lighting as a measure for the distinctiveness of fps the principal components of fp centered windows are compared with those of windows at randomly chosen points on a large database of natural images finally usability is evaluated in the context of an object recognition task

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

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions

a theory of multiscale curvature based shape representation for planar curves a shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented the method rests on the describing a curve at varying levels of detail using features that are invariant with respect to transformations that do not change the shape of the curve three different ways of computing the representation are described they result in three different representations the curvature scale space image the renormalized curvature scale space image and the resampled curvature scale space image the process of describing a curve at increasing levels of abstraction is referred to as the evolution or arc length evolution of that curve several evolution and arc length evolution properties of planar curves are discussed