practical reliable bayesian recognition of 2d and 3d objects using implicit polynomials and algebraic invariants we treat the use of more complex higher degree polynomial curves and surfaces of degree higher than 2 which have many desirable properties for object recognition and position estimation and attack the instability problem arising in their use with partial and noisy data the scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function stored in the database then given partial data from one of the objects we want to recognize the object or the object class or collect more data in order to get better parameter estimates for more reliable recognition two problems arising in this scenario are discussed 1 the problem of recognizing these polynomials by comparing them in terms of their coefficients and 2 the problem of where to collect data so as to improve the parameter estimates as quickly as possible we use an asymptotic bayesian approximation for solving the two problems the intrinsic dimensionality of polynomials and the use of the mahalanobis distance are discussed

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

on the sequential accumulation of evidence in this paper we introduce a method for sequentially accumulating evidence as it pertains to an active observer seeking to identify an object in a known environment we develop a probabilistic framework based on a generalized inverse theory where assertions are represented by conditional probability density functions this leads to a sequential recognition strategy in which evidence is accumulated over successive viewpoints using bayesian chaining until a definitive assertion can be made to illustrate the theory we show how the characteristics of belief distributions can be exploited in a model based recognition problem where the task is to identify an unknown model from a database of known objects on the basis of parameter estimates we illustrate the robustness of the algorithm through recognition experiments in two very different contexts 1 a highly structured recognition context where 3 d parametric models can be estimated directly from range data 2 a complex environment where the relationship between the data and the model is learned through an appearance based strategy specifically the flow fields computed through the object s motion are used as structural signatures for recognition

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

the determination of implicit polynomial canonical curves a new method is presented for identifying and comparing closed bounded free form curves that are defined by even implicit polynomial ip equations in the x y cartesian coordinates the method provides a new expression for an ip involving a product of conic factors with unique conic factor centers the critical points for an ip curve are also defined the conic factor centers and the critical points are shown to be useful related points that directly map to one another under affine transformations in particular the explicit determination of such points implies both a canonical form for the curves and the transformation matrix which relates affine equivalent curves

quantum field theory grassmannians and algebraic curves this paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite grassmannians and in part to pointing out the existence of a close analogy between conformal field theory on riemann surfaces and the modern theory of automorphic representations along the way we develop a multiplicative analog of the usual additive ward identities of current algebra we also reformulate the additive ward identities in a way which may be useful in terms of the residues of operator valued differential forms a concluding section is devoted to some remarks on string field theory in an appendix we attempt to clarify the recent construction by beilinson manin and schechtman of what might be called global virasoro algebras

efficient algorithms for obtaining algebraic invariants from higher degree implicit polynomials for recognition of curved objects implicit polynomials can be used for describing a large variety of curved shapes in most of the earlier works the emphasis had been on the use of second degree implicit polynomials to describe more complex curves and surfaces it is necessary that higher degree implicit polynomials be used in this paper we have proposed a tensor based approach for obtaining affine and euclidean invariants from the coefficients of higher degree implicit polynomials our approach is more general and computationally efficient than the matrix based approaches for obtaining invariants for the euclidean case the algorithm can be used for both recognition and pose estimation in this paper we have demonstrated our approach for obtaining invariants of 2 d shapes using third and fourth degree implicit polynomials

generalized affine invariant image normalization we provide a generalized image normalization technique which basically solves all problems in image normalization the orientation of any image can be uniquely defined by at most three non zero generalized complex moments the correctness of our method is demonstrated theoretically as well as in practice by applying them to a number of degenerate images which have failed other previously reported techniques for image normalization

fast computation of tchebichef moments for binary and grayscale images discrete orthogonal moments have been recently introduced in the field of image analysis it was shown that they have better image representation capability than the continuous orthogonal moments one problem concerning the use of moments as feature descriptors is the high computational cost which may limit their application to the problems where the online computation is required in this paper we present a new approach for fast computation of the 2 d tchebichef moments by deriving some properties of tchebichef polynomials and using the image block representation for binary images and intensity slice representation for grayscale images a fast algorithm is proposed for computing the moments of binary and grayscale images the theoretical analysis shows that the computational complexity of the proposed method depends upon the number of blocks of the image thus it can speed up the computational efficiency as far as the number of blocks is smaller than the image size

an adaptive and stable method for fitting implicit polynomial curves and surfaces representing 2d and 3d data sets with implicit polynomials ips has been attractive because of its applicability to various computer vision issues therefore many ip fitting methods have already been proposed however the existing fitting methods can be and need to be improved with respect to computational cost for deciding on the appropriate degree of the ip representation and to fitting accuracy while still maintaining the stability of the fit we propose a stable method for accurate fitting that automatically determines the moderate degree required our method increases the degree of ip until a satisfactory fitting result is obtained the incrementability of qr decomposition with gram schmidt orthogonalization gives our method computational efficiency furthermore since the decomposition detects the instability element precisely our method can selectively apply ridge regression based constraints to that element only as a result our method achieves computational stability while maintaining fitting accuracy experimental results demonstrate the effectiveness of our method compared with prior methods

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

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

identification of an algebraic domain in two dimensions from a finite number of its generalized polarization tensors this paper aims at studying how finitely many generalized polarization tensors of an algebraic domain can be used to determine its shape precisely given a planar set with real algebraic boundary it is shown that the minimal polynomial with real coefficients vanishing on the boundary can be identified as the generator of a one dimensional kernel of a matrix whose entries are obtained from a finite number of generalized polarization tensors the size of the matrix depends polynomially on the degree of the boundary of the algebraic domain the density with respect to hausdorff distance of algebraic domains among all bounded domains invites to extend via approximation our reconstruction procedure beyond its natural context based on this a new algorithm for shape recognition classification is proposed with some strong hints about its efficiency

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

seshadri constants and periods of polarized abelian varieties the purpose of this paper is to study the seshadri constants of abelian varieties consider a polarized abelian variety a l of dimension g over the field of complex numbers one can associate to a l a real number e a l its seshadri constant which in effect measures how much of the positivity of l can be concentrated at any given point of a the number e a l can be defined as the rate of growth in k of the number of jets that one can specify in the linear series oa kl alternatively one considers the blow up f x blx x x of x at a point x with exceptional divisor e x over x and defines e a l def sup e r f l ee is nef

generalized bc systems based on hermitian vector bundles a generalized bc system associated to a hermitian vector bundle over a riemann surface is introduced in close analogy to the usual rank one case some of the geometric analogies to the well known case are studied in particular if there are no zero modes the non abelian theta divisor appears in the general case where zero modes exist it seems to be more difficult to find a natural description it is discussed in detail why no such system exists in low genus that is on the riemann sphere and on elliptic curves

nonparametric segmentation of curves into various representations this paper describes and demonstrates the operation and performance of an algorithm for segmenting connected points into a combination of representations such as lines circular elliptical and superelliptical arcs and polynomials the algorithm has a number of interesting properties including being scale invariant nonparametric general purpose and efficient

second order implicit polynomials for segmentation of range images abstract in this paper we present an algorithm for segmentation and feature extraction for noisy range images we use the edge and region based approach for obtaining reliable segmentation maps for edge detection regions having much higher values of planar fit error are assumed to contain edges these edge regions are explored for detection of crease edges which can be located at the maxima of the curvature however due to noise the maximas are generated at a number of points other than the actual edges these extraneous curvature maximas have a substantially smaller value of curvature compared with the curvature of the actual edges the true edges are detected using an approach based on local thresholding within edge regions the detected edges segment the range image into smooth patches on each of the smooth patches a planar quadric fit is obtained using a novel non iterative procedure to obtain reliable fits we had to restrict the permissible class of quadrics to surfaces of revolution the quadric fit obtained gives a very reliable estimate of the position and orientation of the axis for curved surfaces the approach has been demonstrated on real range data and has been found to give good results

fitting curves and surfaces with constrained implicit polynomials a problem which often arises while fitting implicit polynomials to 2d and 3d data sets is the following although the data set is simple the fit exhibits undesired phenomena such as loops holes extraneous components etc previous work tackled these problems by optimizing heuristic cost functions which penalize some of these topological problems in the fit the paper suggests a different approach to design parameterized families of polynomials whose zero sets are guaranteed to satisfy certain topological properties namely we construct families of polynomials with star shaped zero sets as well as polynomials whose zero sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse this is more rigorous than using heuristics which may fail and result in pathological zero sets the ability to parameterize these families depends heavily on the ability to parameterize positive polynomials to achieve this we use some powerful results from real algebraic geometry

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

embedding gestalt laws in markov random fields the goal of this paper is to study a mathematical framework of 2d object shape modeling and learning for middle level vision problems such as image segmentation and perceptual organization for this purpose we pursue generic shape models which characterize the most common features of 2d object shapes in this paper shape models are learned from observed natural shapes based on a minimax entropy learning theory the learned shape models are gibbs distributions defined on markov random fields mrfs the neighborhood structures of these mrfs correspond to gestalt laws colinearity cocircularity proximity parallelism and symmetry thus both contour based and region based features are accounted for stochastic markov chain monte carlo mcmc algorithms are proposed for learning and model verification furthermore this paper provides a quantitative measure for the so called nonaccidental statistics and thus justifies some empirical observations of gestalt psychology by information theory our experiments also demonstrate that global shape properties can arise from interactions of local features

the 2 blocks of the covering groups of the symmetric groups abstract let n be a double cover of the finite symmetric group s n of degree n i e n has a central involution z such that n z s n an irreducible character of n is called ordinary or spin according to whether it has z in its kernel or not the purpose of this paper is to determine the distribution of the spin characters of n into 2 blocks the methods applied here are essentially different from those applied to previous questions of this type we also discuss some consequences of our main result for the decomposition numbers an analogue of james well known result for the decomposition numbers of the symmetric groups is proved providing also a generalization of a theorem of benson ben theorem 1 2 in section 1 we present the background for our results and give some preliminaries in section 2 we give an explicit formula for the number of spin characters in a 2 block we also prove a result about the weight of a block containing a given non self associate spin character which will be important for the proof of our theorem on the 2 block distribution of spin characters section 3 presents some fundamental combinatorial concepts used in sections 4 and 5 the theorem concerning the spin characters in a given 2 block is proved in section 4 and in section 5 we present our results on the decomposition numbers

torus n point functions for mathbb r graded vertex operator superalgebras and continuous fermion orbifolds we consider genus one n point functions for a vertex operator superalgebra with a real grading we compute all n point functions for rank one and rank two fermion vertex operator superalgebras in the rank two fermion case we obtain all orbifold n point functions for a twisted module associated with a continuous automorphism generated by a heisenberg bosonic state the modular properties of these orbifold n point functions are given and we describe a generalization of fay s trisecant identity for elliptic functions

the szego kernel on a sewn riemann surface we describe the szeg kernel on a higher genus riemann surface in terms of szeg kernel data coming from lower genus surfaces via two explicit sewing procedures where either two riemann surfaces are sewn together or a handle is sewn to a riemann surface we consider in detail the examples of the szeg kernel on a genus two riemann surface formed by either sewing together two punctured tori or by sewing a twice punctured torus to itself we also consider the modular properties of the szeg kernel in these cases

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

genus two partition and correlation functions for fermionic vertex operator superalgebras i we define the partition and n point correlation functions for a vertex operator superalgebra on a genus two riemann surface formed by sewing two tori together for the free fermion vertex operator superalgebra we obtain a closed formula for the genus two continuous orbifold partition function in terms of an infinite dimensional determinant with entries arising from torus szeg kernels we prove that the partition function is holomorphic in the sewing parameters on a given suitable domain and describe its modular properties using the bosonized formalism a new genus two jacobi product identity is described for the riemann theta series we compute and discuss the modular properties of the generating function for all n point functions in terms of a genus two szeg kernel determinant we also show that the virasoro vector one point function satisfies a genus two ward identity

skewed rotation symmetry group detection we present a novel and effective algorithm for affinely skewed rotation symmetry group detection from real world images we define a complete skewed rotation symmetry detection problem as discovering five independent properties of a skewed rotation symmetry group 1 the center of rotation 2 the affine deformation 3 the type of the symmetry group 4 the cardinality of the symmetry group and 5 the supporting region of the symmetry group in the image we propose a frieze expansion fe method that transforms rotation symmetry group detection into a simple 1d translation symmetry detection problem we define and construct a pair of rotational symmetry saliency maps complemented by a local feature method frequency analysis using discrete fourier transform dft is applied to the frieze expansion patterns feps to uncover the types cyclic dihedral and o 2 the cardinalities and the corresponding supporting regions concentric or otherwise of multiple rotation symmetry groups in an image the phase information of the fep is used to rectify affinely skewed rotation symmetry groups our result advances the state of the art in symmetry detection by offering a unique combination of region based feature based and frequency based approaches experimental results on 170 synthetic and natural images demonstrate superior performance of our rotation symmetry detection algorithm over existing methods

automatic symmetry determination and normalization for rotationally symmetric 2d shapes and 3d solid objects abstract determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision application however the usual normalization methods are inapplicable when the shape is n fold rotationally symmetric n 3 in this paper an efficient and reliable new method using modified fourier transform is introduced to normalize such a 2d rotationally symmetric shape furthermore we extend this new method to the normalization of a n fold axial rotationally symmetric 3d object there are two obvious advantages when using this new method 1 it prevents the high computation load and the difficulty in normalizing the shape s orientation which occurred in the old methods using full matching or correlation functions and 2 the weakness introduced by the old methods using data reduction from 2d data to 1d data is remedied experiments will show the validity of our method

on three dimensional surface reconstruction methods a survey is presented of some of the surface reconstruction methods that can be found in the literature the focus is on a small recent and important subset of the published reconstruction techniques the techniques are classified based on the surface representation used implicit versus explicit functions a study is made of the important aspects of the surface reconstruction techniques one aspect is the viewpoint invariance of the methods this is an important property if object recognition is the ultimate objective the robustness of the various methods is examined it is determined whether the parameter estimates are biased and the sensitivity to obscuration is addressed the latter two aspects are particularly important for fitting functions in the implicit form a detailed description is given of a parametric reconstruction method for three dimensional object surfaces that involves numeric grid generation techniques and variational principle formulations this technique is invariant to rigid motion in dimensional space

linear programming fitting of implicit polynomials a new implicit polynomial ip fitting method is presented it provides a different way of viewing the ip fitting problem from those of the nonlinear optimization approaches it requires less computation and can be done automatically or interactively linear programming is used to do the fitting the approach can incorporate a variety of distance measures and global geometric constraints

three dimensional object representation and invariant recognition using continuous distance transform neural networks 3d object recognition under partial object viewing is a difficult pattern recognition task in this paper we introduce a neural network solution that is robust to partial viewing of objects and noise corruption this method directly utilizes the acquired 3d data and requires no feature extraction the object is first parametrically represented by a continuous distance transform neural network cdtnn trained by the surface points of the exemplar object the cdtnn maps any 3d coordinate into a value that corresponds to the distance between the point and the nearest surface point of the object therefore a mismatch between the exemplar object and an unknown object can be easily computed when encountered with deformed objects this mismatch information can be backpropagated through the cdtnn to iteratively determine the deformation in terms of affine transform application to 3d heart contour delineation and invariant recognition of 3d rigid body objects is presented

on the correlation functions of the vector bundle generalization of the bc system it is shown that the determinants of the correlation functions of the generalized bc system introduced recently are given as pullbacks of the non abelian theta divisor

invariant matching and identification of curves using b splines curve representation there have been many techniques for curve shape representation and analysis ranging from fourier descriptors to moments to implicit polynomials to differential geometry features to time series models to b splines etc the b splines stand as one of the most efficient curve surface representations and possess very attractive properties such as spatial uniqueness boundedness and continuity local shape controllability and invariance to affine transformations these properties made them very attractive for curve representation and consequently they have been extensively used in computer aided design and computer graphics very little work however has been devoted to them for recognition purposes one possible reason might be due to the fact that the b spline curve is not uniquely described by a single set of parameters control points which made the curve matching recognition process difficult when comparing the respective parameters of the curves to be matched this paper is an attempt to find matching solutions despite this limitation and as such it deals the problem of using b splines for shape recognition and identification from curves with an emphasis on the following applications affine invariant matching and classification of 2 d curves with applications in identification of aircraft types based on image silhouettes and writer identification based on handwritten text

on recognizing and positioning curved 3 d objects from image contours an approach for explicitly relating the shape of image contours to models of curved three dimensional objects is presented this relationship is used for object recognition and positioning object models consist of collections of parametric surface patches and their intersection curves this includes nearly all representations used in computer aided geometric design and computer vision the image contours considered are the projections of surface discontinuities and occluding contours elimination theory provides a method for constructing the implicit equation of these contours for an object observed under orthographic or perspective projection this equation is parameterized by the object s position and orientation with respect to the observer determining these parameters is reduced to a fitting problem between the theoretical contour and the observed data points the proposed approach readily extends to parameterized models it has been implemented for a simple world composed of various surfaces of revolution and tested on several real images

bayesian recognition of local 3 d shape by approximating image intensity functions with quadric polynomials the recognition in image data of viewed patches of spheres cylinders and planes in the 3 d world is discussed as a first step to complex object recognition or complex object location and orientation estimation accordingly an image is partitioned into small square windows each of which is a view of a piece of a sphere or of a cylinder or of a plane windows are processed in parallel for recognition of content new concepts and techniques include approximations of the image within a window by 2 d quadric polynomials where each approximation is constrained by one of the hypotheses that the 3 d surface shape seen is either planar cylindrical or spherical a recognizer based upon these approximations to determine whether the object patch viewed is a piece of a sphere or a piece of a cylinder or a piece of a plane lowpass filtering of the image by the approximation the shape recognition is computationally simple and for large windows is approximately bayesian minimum probability of error recognition these classifications are useful for many purposes one such purpose is to enable a following processor to use an appropriate estimator to estimate shape and orientation and location parameters for the 3 d surface seen within a window

superquadric segmentation in range images via fusion of region and boundary information the high potential of superquadrics as modeling elements for image segmentation tasks has been pointed out for years in the computer vision community in this work we employ superquadrics as modeling elements for multiple object segmentation in range images segmentation is executed in two stages first a hypothesis about the values of the segmentation parameters is generated second the hypothesis is refined locally in both stages object boundary and region information are considered boundary information is derived via model based edge detection in the input range image hypothesis generation uses boundary information to isolate image regions that can be accurately described by superquadrics within hypothesis refinement a game theoretic framework is used to fuse the two information sources by associating an objective function to each information source iterative optimization of the two objective functions in succession outputs a precise description of all image objects we demonstrate experimentally that this approach substantially improves the most established method in superquadric segmentation in terms of accuracy and computational efficiency we demonstrate the applicability of our segmentation framework in real world applications by constructing a novel robotic system for automatic unloading of jumbled box like objects from platforms

invariant descriptors for 3d object recognition and pose invariant descriptors are shape descriptors that are unaffected by object pose by perspective projection or by the intrinsic parameters of the camera these descriptors can be constructed using the methods of invariant theory which are briefly surveyed a range of applications of invariant descriptors in 3d model based vision is demonstrated first a model based vision system that recognizes curved plane objects irrespective of their pose is demonstrated curves are not reduced to polyhedral approximations but are handled as objects in their own right models are generated directly from image data once objects have been recognized their pose can be computed invariant descriptors for 3d objects with plane faces are described all these ideas are demonstrated using images of real scenes the stability of a range of invariant descriptors to measurement error is treated in detail

an algebraic geometry study of the b c system with arbitrary twist fields and arbitrary statistics we present an analysis of the generalb c system including the system on a compact riemann surface of arbitrary genusg 0 by postulating that its correlation functions should only have the singularities imposed by the operator product expansion ope of the system studying a very in fact optimally general form of theb c system we prove rigorously that the standard practice of eliminating zero modes and even the standard lagrangian follow from the analyticity structure dictated by the ope alone we extend the analysis to consider the most general case of the presence of twist e g spin fields we then determine all the possible correlation functions of theb c system with statistics unspecified compatible with the ope on imposing fermi and bose statistics we obtain the correlation functions of the fermionicb c and systems respectively

a technique for finding the symmetry axes of implicit polynomial curves under perspective projection this paper presents an algebraic technique for detecting the symmetry axes of a perspectively projected plane curve the procedure applies to implicit polynomials which has been fitted to 2d image data acquired by means of a photo camera or a tv set the effectiveness of the proposed method has been demonstrated experimentally by using both synthetic and real images

the bc system of higher rank revisited the bc system of higher rank introduced recently is examined further it is shown that the correlation functions are connected with certain non abelian functions and it is discussed how quasi determinants arise

on cartan matrices and lower defect groups for covering groups of symmetric groups we determine the elementary divisors of the cartan matrices of spin p blocks of the covering groups of the symmetric groups when p is an odd prime as a consequence we also compute the determinants of these cartan matrices and in particular we confirm a conjecture by brundan and kleshchev that these determinants depend only on the weight but not on the sign of the block

a unifying approach to moment based shape orientation and symmetry classification in this paper the problem of moment based shape orientation and symmetry classification is jointly considered a generalization and modification of current state of the art geometric moment based functions is introduced the properties of these functions are investigated thoroughly using fourier series analysis and several observations and closed form solutions are derived we demonstrate the connection between the results presented in this work and symmetry detection principles suggested from previous complex moment based formulations the proposed analysis offers a unifying framework for shape orientation symmetry detection in the context of symmetry classification and matching the second part of this work presents a frequency domain method aiming at computing a robust moment based feature set based on a true polar fourier representation of image complex gradients and a novel periodicity detection scheme using subspace analysis the proposed approach removes the requirement for accurate shape centroid estimation which is the main limitation of moment based methods operating in the image spatial domain the proposed framework demonstrated improved performance compared to state of the art methods

implicit b spline surface reconstruction this paper presents a fast and flexible curve and surface reconstruction technique based on implicit b spline this representation does not require any parameterization and it is locally supported this fact has been exploited in this paper to propose a reconstruction technique through solving a sparse system of equations this method is further accelerated to reduce the dimension to the active control lattice moreover the surface smoothness and user interaction are allowed for controlling the surface finally a novel weighting technique has been introduced in order to blend small patches and smooth them in the overlapping regions the whole framework is very fast and efficient and can handle large cloud of points with very low computational cost the experimental results show the flexibility and accuracy of the proposed algorithm to describe objects with complex topologies comparisons with other fitting methods highlight the superiority of the proposed approach in the presence of noise and missing data

determining pose of 3d objects with curved surfaces a method is presented for computing the pose of rigid 3d objects with arbitrary curved surfaces given an input image and a candidate object model and aspect the method will verify whether or not the object is present and if so report pose parameters the curvature method of bash and ullman is used to model points on the object rim while stereo matching is used for internal edge points the model allows an object edge map to be predicted from pose parameters pose is computed via an iterative search for the best pose parameters heuristics are used so that matching can succeed in the presence of occlusion and artifact and without resetting to use of corresponding salient feature points bench tests and simulations show that the method almost always converges to ground truth pose parameters for a variety of objects and for a broad set of starting parameters in the same aspect

sampling and reconstruction of shapes with algebraic boundaries we present a sampling theory for a class of binary images with finite rate of innovation fri every image in our model is the restriction of mathbb 1 p leq 0 to the image plane where mathbb 1 denotes the indicator function and p is some real bivariate polynomial this particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial p we show that the image parameters i e the polynomial coefficients satisfy a set of linear annihilation equations with the coefficients being the image moments the inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels as a remedy to these problems we replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel the benefits are threefold 1 it relaxes the requirements on the sampling kernels 2 produces annihilation equations that are robust at numerical precision and 3 extends the results to images with unbounded boundaries we further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points and sequentially enforcing measurement consistency we consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images including low to moderate noise levels and a range of realistic sampling kernels

parameterized families of polynomials for bounded algebraic curve and surface fitting interest in algebraic curves and surfaces of high degree as geometric models or shape descriptors for different model based computer vision tasks has increased in recent years and although their properties make them a natural choice for object recognition and positioning applications algebraic curve and surface fitting algorithms often suffer from instability problems one of the main reasons for these problems is that while the data sets are always bounded the resulting algebraic curves or surfaces are in most cases unbounded in this paper the authors propose to constrain the polynomials to a family with bounded zero sets and use only members of this family in the fitting process for every even number d the authors introduce a new parameterized family of polynomials of degree d whose level sets are always bounded in particular its zero sets this family has the same number of degrees of freedom as a general polynomial of the same degree three methods for fitting members of this polynomial family to measured data points are introduced experimental results of fitting curves to sets of points in r sup 2 and surfaces to sets of points in r sup 3 are presented

a moment based approach for deskewing rotationally symmetric shapes the covariance matrix of a pattern is composed by its second order central moments for a rotationally symmetric shape its covariance matrix is a scalar identity matrix in this work we apply this property to restore the skewed shape of rotational symmetry the relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived by computing the covariance matrix of the skewed shape and letting the covariance matrix of the original shape be a scalar identity matrix the skew transformation matrix can be solved then the rotationally symmetric shape can be recovered by multiplying the inverse transformation matrix with the skewed shape the method does not rely on continuous contours and is robust to noise because only the second order moments of the input shape are required experimental results are also presented

detection of generalized principal axes is rotationally symmetric shapes abstract automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper the inapplicability of a well known equation which is used to compute the direction of the principal axis of a given shape using moment functions to a class of so called degenerate shapes is first pointed out rotationally symmetric shapes often encountered in real applications are shown to belong to this class this problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions analytic equations for computing the direction of high order principal axes are derived they include the well known equation for computing the direction of the second order principal axis as a special case some experimental results are included finally to show the effectiveness of the derived analytic equations

local invariants for recognition geometric invariants are shape descriptors that remain unchanged under geometric transformations such as projection or changing the viewpoint a new method of obtaining local projective and affine invariants is developed and implemented for real images being local the invariants are much less sensitive to occlusion than global invariants the invariants computation is based on a canonical method this consists of defining a canonical coordinate system by the intrinsic properties of the shape independently of the given coordinate system since this canonical system is independent of the original one it is invariant and all quantities defined in it are invariant the method was applied without the use of a curve parameter this was achieved by fitting an implicit polynomial to an arbitrary curve in a vicinity of each curve point several configurations are treated a general curve without any correspondence and curves with known correspondences of one or two feature points or lines experimental results for different 2d objects in 3d space are presented

application of elliptic fourier descriptors to symmetry detection under parallel projection in this paper the method of elliptic fourier descriptors using arc length parameterization is applied to tackle the problem of detection and recovery of symmetry under parallel projection a simple and fast iteration algorithm together with the invariants of symmetry provides sufficient information for the detection and recovery of symmetry under parallel projection the proposed method has been extensively tested using symmetric figures under different parallel projections simulation results of the algorithm are presented the extension of this method for planar object recognition under parallel projection is also addressed

from uncertainty to visual exploration the authors attempt to determine what can be inferred from ambiguity in processes of visual interpretation they discuss this question in a specific context the interpretation of scene geometry in the form of parametrized volumetric models ambiguity is described as a local probabilistic property of the misfit error surface in the parameter space of superellipsoid models namely as an ellipsoid of confidence in which there is a given probability that the true parameters can be found the authors show how to project the ellipsoid of confidence back into 3d space to obtain the shell in which the true 3d surface most probably lies and introduce what they call the uncertainty as a local property of the fitted model s surface they propose a technique that can use this information to plan a new direction of view that minimizes the ambiguity of subsequent interpretation

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

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

affine permutation invariance of 2 d shapes shapes provide a rich set of clues on the identity and topological properties of an object in many imaging environments however the same object appears to have different shapes due to distortions such as translation rotation reflection scaling or skewing further the order by which the object s feature points are scanned changes i e the order of the pixels may be permuted relating two dimensional shapes of the same object distorted by different affine and permutation transformations is a challenge we introduce a shape invariant that we refer to as the intrinsic shape of an object and describe an algorithm blaiser to recover it the intrinsic shape is invariant to affine permutation distortions it is a uniquely defined representative of the equivalence class of all affine permutation distortions of the same object blaiser computes the intrinsic shape from any arbitrarily affine permutation distorted image of the object without prior knowledge regarding the distortions or the undistorted shape of the object the critical step of blaiser is the determination of the shape orientation and we provide a detailed discussion on this topic the operations of blaiser are based on low order moments of the input shape and thus robust to error and noise examples illustrate the performance of the algorithm

detection and localisation of reflectional and rotational symmetry under weak perspective projection in this paper a novel method is proposed for detecting and recovering the pose of a reflectional and rotational symmetry from a single weak perspective image in our paradigm an effective and novel hough based approach is proposed for extracting skew symmetry axes from edge contours a refinement algorithm is developed for verifying and pruning away the implausible and spurious hypothesised symmetry axes for each of the detected skew symmetry axes and its corresponding skew angle a mathematical formulation is devised for evaluating the geometric relationships between an orthogonal junction formed by a symmetry axis and its perpendicular axis in an original plane and its corresponding skew angle in an image plane the derived geometric formulation is integrated into an effective voting scheme for estimating the projective parameters of a reflectionally and rotationally symmetric planar shape using the information of the plausible skew symmetry axes experimental results are presented to verify the computational performance of the proposed paradigm the practical significance and generality of the proposed method are manifold

improving the stability of algebraic curves for applications an algebraic curve is defined as the zero set of a polynomial in two variables algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics the main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data usually arguments against using algebraic curves involve references to mathematicians wilkinson and runge the first goal of this article is to understand the stability issue of algebraic curve fitting then a fitting method based on ridge regression and restricting the representation to well behaved subsets of polynomials is proposed and its properties are investigated the fitting algorithm is of sufficient stability for very fast position invariant shape recognition position estimation and shape tracking based on invariants and new representations among appropriate applications are shape based indexing into image databases

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

a bayesian method for fitting parametric and nonparametric models to noisy data we present a simple paradigm for fitting models parametric and nonparametric to noisy data which resolves some of the problems associated with classical mse algorithms this is done by considering each point on the model as a possible source for each data point the paradigm can be used to solve problems which are ill posed in the classical mse approach such as fitting a segment as opposed to a line it is shown to be nonbiased and to achieve excellent results for general curves even in the presence of strong discontinuities results are shown for a number of fitting problems including lines circles elliptic arcs segments rectangles and general curves contaminated by gaussian and uniform noise

estimation of planar curves surfaces and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation the author addresses the problem of parametric representation and estimation of complex planar curves in 2 d surfaces in 3 d and nonplanar space curves in 3 d curves and surfaces can be defined either parametrically or implicitly with the latter representation used here a planar curve is the set of zeros of a smooth function of two variables x y a surface is the set of zeros of a smooth function of three variables x y z and a space curve is the intersection of two surfaces which are the set of zeros of two linearly independent smooth functions of three variables x y z for example the surface of a complex object in 3 d can be represented as a subset of a single implicit surface with similar results for planar and space curves it is shown how this unified representation can be used for object recognition object position estimation and segmentation of objects into meaningful subobjects that is the detection of interest regions that are more complex than high curvature regions and hence more useful as features for object recognition

cosmos a representation scheme for 3d free form objects we address the problem of representing and recognizing 3d free form objects when 1 the object viewpoint is arbitrary 2 the objects may vary in shape and complexity and 3 no restrictive assumptions are made about the types of surfaces on the object we assume that a range image of a scene is available containing a view of a rigid 3d object without occlusion we propose a new and general surface representation scheme for recognizing objects with free form sculpted surfaces in this scheme an object is described concisely in terms of maximal surface patches of constant shape index the maximal patches that represent the object are mapped onto the unit sphere via their orientations and aggregated via shape spectral functions properties such as surface area curvedness and connectivity which are required to capture local and global information are also built into the representation the scheme yields a meaningful and rich description useful for object recognition a novel concept the shape spectrum of an object is also introduced within the framework of cosmos for object view grouping and matching we demonstrate the generality and the effectiveness of our scheme using real range images of complex objects

detection of object wings in fused range and intensity imagery abstract we present methods for detecting primitives in scenes containing objects whose surfaces are approximately piecewise quadric using a sensor that yields fused range and intensity the primitives detected in a fused image are viewpoint dependent and are called object wings which consist of adjacent fragments of object contour and surface wings are small enough so that some object parts can be sensed despite significant occlusion yet rich enough to be of use in indexing to candidate models in constraining object pose and in aggregation methods which combine primitives into larger structures or objects fitting procedures are defined for specific wing primitives experiments with both synthetic and real data show that classification within the wing types of planar spherical conical cylindrical can be achieved with over 90 accuracy contour information available in fitting fused data improves fitting relative to use of surface data alone moreover it is shown that only a small number of contour and surface samples are needed for extraction of primitives oc 1997 pattern recognition society

automatic rotational symmetry determination for shape analysis abstract the determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications in this paper a simple but effective algorithm to analyse the rotational symmetry of a given closed curve shape s is proposed a circle c with the centroid of s as the circle center and the average radius of s as the circle radius is superimposed on s resulting in the intersection of c and s at a set of points by theoretical analysis the relationship between the order n s of the rotational symmetry of s and the number of intersection points between c and s is established all the possible values for n s are determined and finally n s is determined by evaluating the similarity between s and its rotated versions in the proposed algorithm only simple pixel operations and second order moment function computation are involved several problems caused by the use of discrete image coordinates are analysed and solved for correct decision making in the algorithm good experimental results are also included

describing complicated objects by implicit polynomials this paper introduces and focuses on two problems first is the representation power of closed implicit polynomials of modest degree for curves in 2 d images and surfaces in 3 d range data super quadrics are a small subset of object boundaries that are well fitted by these polynomials the second problem is the stable computationally efficient fitting of noisy data by closed implicit polynomial curves and surfaces the attractive features of these polynomials for vision is discussed

fay s trisecant identity and conformal field theory we study the correlation functions of a system of free chiral fermions on a compact riemann surface using techniques of algebraic geometry fay s trisecant identity arises as a consequence of the proof of the uniqueness of correlation functions

automatic registration for 3d shapes using hybrid dimensionality reduction shape descriptions automatic registration for 3d shapes is an attractive problem in computer vision various registration algorithms based on different surface representations have been developed for this topic however most of the existing algorithms suffer from some limitations mainly related to discriminating similarity metric partially overlapping data and the robustness to resolution noise and occlusion in this research hybrid dimensionality reduction shape descriptions drsd are proposed for pair wise registration which aims to overcome these limitations based on recently emerging angle preserving parameterization techniques such as harmonic maps and abf 3d shapes are described in low dimension space with both local and global considerations therefore searching for correspondences verifying overlapping regions and calculating registration error all can be implemented in low dimension space moreover a large number of experiments using both real and synthetic images have been carried out to show the accuracy efficiency and robustness of the hybrid drsd algorithm