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

the richer representation the better registration in this paper the registration problem is formulated as a point to model distance minimization unlike most of the existing works which are based on minimizing a point wise correspondence term this formulation avoids the correspondence search that is time consuming in the first stage the target set is described through an implicit function by employing a linear least squares fitting this function can be either an implicit polynomial or an implicit b spline from a coarse to fine representation in the second stage we show how the obtained implicit representation is used as an interface to convert point to point registration into point to implicit problem furthermore we show that this registration distance is smooth and can be minimized through the levengberg marquardt algorithm all the formulations presented for both stages are compact and easy to implement in addition we show that our registration method can be handled using any implicit representation though some are coarse and others provide finer representations hence a tradeoff between speed and accuracy can be set by employing the right implicit function experimental results and comparisons in 2d and 3d show the robustness and the speed of convergence of the proposed approach

on recovering hyperquadrics from range data this paper discusses the applications of hyperquadric models in computer vision and focuses on their recovery from range data hyperquadrics are volumetric shape models that include superquadrics as a special case a hyperquadric model can be composed of any number of terms and its geometric bound is an arbitrary convex polytope thus hyperquadrics can model more complex shapes than superquadrics hyperquadrics also possess many other advantageous properties compactness semilocal control and intuitive meaning our proposed algorithm starts with a rough fit using only six terms in 3d four in 2d and adds additional terms as necessary to improve fitting suitable constraints are used to ensure proper convergence experimental results with real 2d and 3d data are presented

implicit polynomials orthogonal distance regression and the closest point on a curve implicit polynomials i e multinomials have a number of properties that make them attractive for modeling curves and surfaces in computer vision the paper considers the problem of finding the best fitting implicit polynomial or algebraic curve to a collection of points in the plane using an orthogonal distance metric approximate methods for orthogonal distance regression have been shown by others to be prone to the problem of cusps in the solution and this is confirmed here consequently this work focuses on exact methods for orthogonal distance regression the most difficult and costly part of exact methods is computing the closest point on the algebraic curve to an arbitrary point in the plane the paper considers three methods for achieving this in detail the first is the standard newton s method the second is based on resultants which are making a resurgence in computer graphics and the third is a novel technique based on successive circular approximations to the curve it is shown that newton s method is the quickest but that it can fail sometimes even with a good initial guess the successive circular approximation algorithm is not as fast but is robust the resultant method is the slowest of the three but does not require an initial guess the driving application of this work was the fitting of implicit quartics in two variables to thinned oblique ionogram traces

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

very fast best fit circular and elliptical boundaries by chord data many machine vision tasks require objects to be delineated during image segmentation that have shapes that are well approximated by circles or ellipses due to their computational efficiency least squares algebraic methods are a popular choice for fitting an elliptic primitive to noisy image data when real time processing is required these methods however suffer from biased estimates and sensitivity to outlier data in this paper a real time least squares method is proposed that provides an indirect geometric fit based on the quadratic polynomial form of parallel chord lengths the algorithm is shown to be more computationally efficient and more easily made robust to outlier data than algebraic methods experimental results also suggest that it provides estimates that suffer less from bias error

segmentation through variable order surface fitting the solution of the segmentation problem requires a mechanism for partitioning the image array into low level entities based on a model of the underlying image structure a piecewise smooth surface model for image data that possesses surface coherence properties is used to develop an algorithm that simultaneously segments a large class of images into regions of arbitrary shape and approximates image data with bivariate functions so that it is possible to compute a complete noiseless image reconstruction based on the extracted functions and regions surface curvature sign labeling provides an initial coarse image segmentation which is refined by an iterative region growing method based on variable order surface fitting experimental results show the algorithm s performance on six range images and three intensity images

a paraboloid fitting technique for calculating curvature from piecewise linear interface reconstructions on 3d unstructured meshes abstract we present a novel method for calculating interface curvature on 3d unstructured meshes from piecewise linear interface reconstructions typically generated in the volume of fluid method interface curvature is a necessary quantity to calculate in order to model surface tension driven flow curvature needs only to be computed in cells containing an interface the approach requires a stencil containing only neighbors sharing a node with a target cell and calculates curvature from a least squares paraboloid fit to the interface reconstructions this involves solving a 6 6 symmetric linear system in each mixed cell we present verification tests where we calculate the curvature of a sphere an ellipsoid and a sinusoid in a 3d domain on regular cartesian meshes distorted hex meshes and tetrahedral meshes for both regular and unstructured meshes we find in all cases the paraboloid fitting method for curvature to converge between first and second order with grid refinement

on the fitting of surfaces to data with covariances we consider the problem of estimating parameters of a model described by an equation of special form specific models arise in the analysis of a wide class of computer vision problems including conic fitting and estimation of the fundamental matrix we assume that noisy data are accompanied by known covariance matrices characterizing the uncertainty of the measurements a cost function is first obtained by considering a maximum likelihood formulation and applying certain necessary approximations that render the problem tractable a newton like iterative scheme is then generated for determining a minimizer of the cost function unlike alternative approaches such as sampson s method or the renormalization technique the new scheme has as its theoretical limit the minimizer of the cost function furthermore the scheme is simply expressed efficient and unsurpassed as a general technique in our testing an important feature of the method is that it can serve as a basis for conducting theoretical comparison of various estimation approaches

direct least square fitting of hyperellipsoids this paper presents two new computationally efficient direct methods for fitting n dimensional ellipsoids to noisy data they conduct the fitting by minimizing the algebraic distance in subject to suitable quadratic constraints the hyperellipsoid specific hes method is an elaboration of existing ellipse and 3d ellipsoid specific fitting methods it is shown that hes is ellipsoid specific in n dimensional space a limitation of hes is that it may provide biased fitting results with data originating from an ellipsoid with a large ratio between the longest and shortest main axis the sum of discriminants sod method does not have such a limitation the constraint used by sod rejects a subset of non ellipsoidal quadrics which enables a high tendency to produce ellipsoidal solutions moreover a regularization technique is presented to force the solutions towards ellipsoids with sod the regularization technique is compatible also with several existing 2d and 3d fitting methods the new methods are compared through extensive numerical experiments with n dimensional variants of three commonly used direct fitting approaches for quadratic surfaces the results of the experiments imply that in addition to the superior capability to create ellipsoidal solutions the estimation accuracy of the new methods is better or equal to that of the reference approaches

point to ellipse and point to ellipsoid distance equation analysis for the problem of distance evaluation from a point x0 to an ellipse in r2 and to an ellipsoid in r3 given by algebraic equation g x 0 we investigate the properties of the distance equation i e an algebraic equation whose zeros coincide with the critical values of the squared distance function we detail the structure of this equation and an algorithm for finding the point in the quadric nearest to x0 we also find analytical formulas for distance approximations using the expansion of the zero of distance equation into power series j 1jgj x0 exact values for the approximation error bounds are obtained via construction of an analogue of the distance equations for the curve to curve distance problem

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

estimation of subspace arrangements with applications in modeling and segmenting mixed data recently many scientific and engineering applications have involved the challenging task of analyzing large amounts of unsorted high dimensional data that have very complicated structures from both geometric and statistical points of view such unsorted data are considered mixed as different parts of the data have significantly different structures which cannot be described by a single model in this paper we propose to use subspace arrangements a union of multiple subspaces for modeling mixed data each subspace in the arrangement is used to model just a homogeneous subset of the data thus multiple subspaces together can capture the heterogeneous structures within the data set in this paper we give a comprehensive introduction to a new approach for the estimation of subspace arrangements this is known as generalized principal component analysis gpca in particular we provide a comprehensive summary of important algebraic properties and statistical facts that are crucial for making the inference of subspace arrangements both efficient and robust even when the given data are corrupted by noise or contaminated with outliers this new method in many ways improves and generalizes extant methods for modeling or clustering mixed data there have been successful applications of this new method to many real world problems in computer vision image processing and system identification in this paper we will examine several of those representative applications this paper is intended to be expository in nature however in order that this may serve as a more complete reference for both theoreticians and practitioners we take the liberty of filling in several gaps between the theory and the practice in the existing literature

algorithms for projecting points onto conics we study the problem of projecting 2d points onto quadratic curves ellipses hyperbolas parabolas we investigate various projection algorithms focusing on those that are mathematically proven to produce or converge to correct results in all cases our tests demonstrate that those may be still unfit for practical use due to large computational errors we present two new algorithms that are not only theoretically proven to converge but achieve nearly perfect accuracy

the multiple window parameter transform the multiwindow transform an extension of parameter transform techniques that increase performance and scope by exploiting the long range correlated information contained in multiple portions of an image is presented multiple window transforms allow the extraction of high dimensional features with improvement in accuracy over conventional techniques while keeping linear to low order polynomial computational and space requirements with respect to image size and dimensionality of the features using correlated information provides a direct link between extracted features and supporting regions in the image this coupled with evidence integration techniques is used to suppress noisy or nonexistent feature hypotheses parameter spaces are implemented as constraint satisfaction networks where feature hypotheses with overlapping support in the image compete after an iterative relaxation phase surviving hypotheses have disjoint support forming a segmentation of the image examples show the performance and provide insight about the behavior

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

bayesian clustering for unsupervised estimation of surface and texture models a method of calculating the maximum likelihood clustering for the unsupervised estimation of polynomial models for the data in images of smooth surfaces or for range data for such surfaces is presented an image or a depth map of a region of smooth 3 d surface is modeled as a polynomial plus white noise a region of physically meaningful textured image such as the image of foliage grass or road in outdoor scenes or conductor or lintburn on a thick film substrate is modeled as a colored gaussian markov random field mrf with a polynomial mean value function unsupervised model parameter estimation is accomplished by determining the segmentation and model parameter values that maximize the likelihood of the data or a more general bayesian performance functional agglomerative clustering is used for this purpose

implicit polynomial representation through a fast fitting error estimation this paper presents a simple distance estimation for implicit polynomial fitting it is computed as the height of a simplex built between the point and the surface i e a triangle in 2 d or a tetrahedron in 3 d which is used as a coarse but reliable estimation of the orthogonal distance the proposed distance can be described as a function of the coefficients of the implicit polynomial moreover it is differentiable and has a smooth behavior hence it can be used in any gradient based optimization in this paper its use in a levenberg marquardt framework is shown which is particularly devoted for nonlinear least squares problems the proposed estimation is a generalization of the gradient based distance estimation which is widely used in the literature experimental results both in 2 d and 3 d data sets are provided comparisons with state of the art techniques are presented showing the advantages of the proposed approach

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

robust parameter estimation in computer vision estimation techniques in computer vision applications must estimate accurate model parameters despite small scale noise in the data occasional large scale measurement errors outliers and measurements from multiple populations in the same data set increasingly robust estimation techniques some borrowed from the statistics literature and others described in the computer vision literature have been used in solving these parameter estimation problems ideally these techniques should effectively ignore the outliers and measurements from other populations treating them as outliers when estimating the parameters of a single population two frequently used techniques are least median of squares lms p j rousseeuw j amer statist assoc 79 1984 pp 871 880 and m estimators robust statistics the approach based on influence functions f r hampel et al john wiley 1986 robust statistics p j huber john wiley 1981 lms handles large fractions of outliers up to the theoretical limit of 50 for estimators invariant to affine changes to the data but has low statistical efficiency m estimators have higher statistical efficiency but tolerate much lower percentages of outliers unless properly initialized while robust estimators have been used in a variety of computer vision applications three are considered here in analysis of range images images containing depth or x y z measurements at each pixel instead of intensity measurements robust estimators have been used successfully to estimate surface model parameters in small image regions in stereo and motion analysis they have been used to estimate parameters of what is called the fundamental matrix which characterizes the relative imaging geometry of two cameras imaging the same scene recently robust estimators have been applied to estimating a quadratic image to image transformation model necessary to create a composite mosaic image from a series of images of the human retina in each case a straightforward application of standard robust estimators is insufficient and carefully developed extensions are used to solve the problem

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

least squares based fitting of paraboloids a technique for reconstructing a class of quadric surfaces from 3d data is presented the technique is driven by a linear least squares based fitting mechanism previously such fitting was restricted to recovery of central quadrics here extension of that basic mechanism to allow recovery of one commonly occurring class of non central quadric the elliptic paraboloids is described the extension uses an indirect solution approach that involves introducing a variable to the basic mechanism that is a function of a quadric surface invariant results from fitting real and synthetic data are also exhibited

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

infinitesimal plane based pose estimation estimating the pose of a plane given a set of point correspondences is a core problem in computer vision with many applications including augmented reality ar camera calibration and 3d scene reconstruction and interpretation despite much progress over recent years there is still the need for a more efficient and more accurate solution particularly in mobile applications where the run time budget is critical we present a new analytic solution to the problem which is far faster than current methods based on solving pose from n n points pnp and is in most cases more accurate our approach involves a new way to exploit redundancy in the homography coefficients this uses the fact that when the homography is noisy it will estimate the true transform between the model plane and the image better at some regions on the plane than at others our method is based on locating a point where the transform is best estimated and using only the local transformation at that point to constrain pose this involves solving pose with a local non redundant 1st order pde we call this framework infinitesimal plane based pose estimation ippe because one can think of it as solving pose using the transform about an infinitesimally small region on the surface we show experimentally that ippe leads to very accurate pose estimates because ippe is analytic it is both extremely fast and allows us to fully characterise the method in terms of degeneracies number of returned solutions and the geometric relationship of these solutions this characterisation is not possible with state of the art pnp methods

a data driven intermediate level feature extraction algorithm an algorithm is presented that is based on the regression updating theory for fitting linearly parameterizable curves without prior classification of edge data an initial estimate of a hypothesized curve is first obtained by a statistical windowing technique a search region is determined and iteratively grown edge points in the search region are tested for their goodness of fit to the previous estimate the estimate is iteratively updated with the edge points that have favorable goodness of fit measures the edge points having poor goodness of fit measures are rejected as outliers the algorithm drives the estimate to converge to a final solution by repeatedly applying this procedure to the edge data excluded from the previous fitting all the underlying curves are reconstructed the major feature that distinguishes this approach from that of others is that classifying the edge data prior to fitting is not required advantages of this algorithm are 1 the fitting procedure achieves higher robustness and accuracy by dynamically analyzing the data consistency 2 the computational complexity increases only linearly with the number of edge data 3 the algorithm readily extends to reconstruct surfaces from range data thus the algorithm provides a powerful technique enabling a data driven intermediate level vision module to extract parametric features needed for higher level processing

small sample statistical condition estimation of large scale generalized eigenvalue problems we consider the evaluation of the sensitivity or condition number of generalized eigenvalue problems for a large and sparse real matrix or matrix pair in r n n through some coupled sylvester equation using newton s method the technique of the statistical condition estimation has been adapted to the sensitivity of symmetric matrices as well as general matrices with special structures under some assumptions on various types of perturbations

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

segmenting images corrupted by correlated noise image segmentation is fundamental to many image analysis problems it aims to partition a digital image into a set of nonoverlapping homogeneous regions the main contribution of this paper is the development of a new segmentation procedure which is designed to segment images corrupted by correlated noise this new segmentation procedure is based on rissanen s minimum description length mdl principle and consists of two components 1 an mdl based criterion in which the best segmentation is defined as its minimizer and 2 a merging algorithm which attempts to locate this minimizer the performance of this procedure is illustrated via a simulation study with promising results

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

on optimally combining pieces of information with application to estimating 3 d complex object position from range data new asymptotic methods are introduced that permit computationally simple bayesian recognition and parameter estimation for many large data sets described by a combination of algebraic geometric and probabilistic models the techniques introduced permit controlled decomposition of a large problem into small problems for separate parallel processing where maximum likelihood estimation or bayesian estimation or recognition can be realized locally these results can be combined to arrive at globally optimum estimation or recognition the approach is applied to the maximum likelihood estimation of 3 d complex object position to this end the surface of an object is modeled as a collection of patches of primitive quadrics i e planar cylindrical and spherical patches possibly augmented by boundary segments the primitive surface patch models are specified by geometric parameters reflecting location orientation and dimension information the object position estimation is based on sets of range data points each set associated with an object primitive probability density functions are introduced that model the generation of range measurement points this entails the formulation of a noise mechanism in three space accounting for inaccuracies in the 3 d measurements and possibly for inaccuracies in the 3 d modeling we develop the necessary techniques for optimal local parameter estimation and primitive boundary or surface type recognition for each small patch of data and then optimal combining of these inaccurate locally derived parameter estimates in order to arrive at roughly globally optimum object position estimation

robust estimation for range image segmentation and reconstruction this correspondence presents a segmentation and fitting method using a new robust estimation technique we present a robust estimation method with high breakdown point which can tolerate more than 80 of outliers the method randomly samples appropriate range image points in the current processing region and solves equations determined by these points for parameters of selected primitive type from k samples we choose one set of sample points that determines a best fit equation for the largest homogeneous surface patch in the region this choice is made by measuring a residual consensus resc using a compressed histogram method which is effective at various noise levels after we get the best fit surface parameters the surface patch can be segmented from the region and the process is repeated until no pixel left the method segments the range image into planar and quadratic surfaces the resc method is a substantial improvement over the least median squares method by using histogram approach to inferring residual consensus a genetic algorithm is also incorporated to accelerate the random search

the development and comparison of robust methodsfor estimating the fundamental matrix this paper has two goals the first is to develop a variety of robust methods for the computation of the fundamental matrix the calibration free representation of camera motion the methods are drawn from the principal categories of robust estimators viz case deletion diagnostics m estimators and random sampling and the paper develops the theory required to apply them to non linear orthogonal regression problems although a considerable amount of interest has focussed on the application of robust estimation in computer vision the relative merits of the many individual methods are unknown leaving the potential practitioner to guess at their value the second goal is therefore to compare and judge the methods comparative tests are carried out using correspondences generated both synthetically in a statistically controlled fashion and from feature matching in real imagery in contrast with previously reported methods the goodness of fit to the synthetic observations is judged not in terms of the fit to the observations per se but in terms of fit to the ground truth a variety of error measures are examined the experiments allow a statistically satisfying and quasi optimal method to be synthesized which is shown to be stable with up to 50 percent outlier contamination and may still be used if there are more than 50 percent outliers performance bounds are established for the method and a variety of robust methods to estimate the standard deviation of the error and covariance matrix of the parameters are examined the results of the comparison have broad applicability to vision algorithms where the input data are corrupted not only by noise but also by gross outliers

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

explicit three dimensional topology optimization via moving morphable void mmv approach abstract three dimensional 3d topology optimization problems always involve huge numbers of degrees of freedom dofs in finite element analysis fea and design variables in numerical optimization respectively this will inevitably lead to large computational efforts in the solution process in the present paper an efficient and explicit topology optimization approach which can reduce not only the number of design variables but also the number of degrees of freedom in fea is proposed based on the moving morphable voids mmvs solution framework this is achieved by introducing a set of geometry parameters e g control points of b spline surfaces to describe the boundary of a structure explicitly and removing the unnecessary dofs from the fe model at every step of numerical optimization numerical examples demonstrate that the proposed approach does can overcome the bottleneck problems associated with a 3d topology optimization problem in a straightforward way and enhance the solution efficiency significantly

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

heteroscedastic regression in computer vision problems with bilinear constraint we present an algorithm to estimate the parameters of a linear model in the presence of heteroscedastic noise i e each data point having a different covariance matrix the algorithm is motivated by the recovery of bilinear forms one of the fundamental problems in computer vision which appears whenever the epipolar constraint is imposed or a conic is fit to noisy data points we employ the errors in variables eiv model and show why already at moderate noise levels most available methods fail to provide a satisfactory solution the improved behavior of the new algorithm is due to two factors taking into account the heteroscedastic nature of the errors arising from the linearization of the bilinear form and the use of generalized singular value decomposition gsvd in the computations the performance of the algorithm is compared with several methods proposed in the literature for ellipse fitting and estimation of the fundamental matrix it is shown that the algorithm achieves the accuracy of nonlinear optimization techniques at much less computational cost

the enumerative geometry of projective algebraic surfaces and the complexity of aspect graphs the aspect graph is a popular viewer centered representation that enumerates all the topologically distinct views of an object building the aspect graph requires partitioning viewpoint space in view equivalent cells by a certain number of visual event surfaces if the object is piecewise smooth algebraic then all visual event surfaces are either made of lines having specified contacts with the object or made of lines supporting the points of contacts of planes having specified contacts with the object in this paper we present a general framework for studying the enumerative properties of line and plane systems the context is that of enumerative geometry and intersection theory in particular we give exact results for the degrees of all visual event surfaces coming up in the construction of aspect graphs of piecewise smooth algebraic bodies we conclude by giving a bound on the number of topologically distinet views of such objects

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

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

a formal classification of 3d medial axis points and their local geometry this paper proposes a novel hypergraph skeletal representation for 3d shape based on a formal derivation of the generic structure of its medial axis by classifying each skeletal point by its order of contact we show that genetically the medial axis consists of five types of points which are then organized into sheets curves and points 1 sheets manifolds with boundary which are the locus of bitangent spheres with regular tangency a sub 1 sup 2 a sub k sup n notation means n distinct k fold tangencies of the sphere of contact as explained in the text two types of curves 2 the intersection curve of three sheets and the locus of centers of tritangent spheres a sub 1 sup 3 and 3 the boundary of sheets which are the locus of centers of spheres whose radius equals the larger principal curvature i e higher order contact a sub 3 points and two types of points 4 centers of quad tangent spheres a sub 1 sup 4 and 5 centers of spheres with one regular tangency and one higher order tangency a sub 1 a sub 3 the geometry of the 3d medial axis thus consists of sheets a sub 1 sup 2 bounded by one type of curve a sub 3 on their free end which corresponds to ridges on the surface and attached to two other sheets at another type of curve a sub 1 sup 3 which support a generalized cylinder description the a sub 3 curves can only end in a sub 1 a sub 3 points where they must meet an a sub 1 sup 3 curve the a sub 1 sup 3 curves meet together in fours at an a sub 1 sup 4 point this formal result leads to a compact representation for 3d shape referred to as the medial axis hypergraph representation consisting of nodes a sub 1 sup 4 and a sub 1 a sub 3 points links between pairs of nodes a sub 1 sup 3 and a sub 3 curves and hyperlinks between groups of links a sub 1 sup 2 sheets the description of the local geometry at nodes by itself is sufficient to capture qualitative aspects of shapes in analogy to 2d we derive a pointwise reconstruction formula to reconstruct a surface from this medial axis hypergraph together with the radius function thus this information completely characterizes 3d shape and lays the theoretical foundation for its use in recognition morphing design and manipulation of shapes

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

a robust competitive clustering algorithm with applications in computer vision this paper addresses three major issues associated with conventional partitional clustering namely sensitivity to initialization difficulty in determining the number of clusters and sensitivity to noise and outliers the proposed robust competitive agglomeration rca algorithm starts with a large number of clusters to reduce the sensitivity to initialization and determines the actual number of clusters by a process of competitive agglomeration noise immunity is achieved by incorporating concepts from robust statistics into the algorithm rca assigns two different sets of weights for each data point the first set of constrained weights represents degrees of sharing and is used to create a competitive environment and to generate a fuzzy partition of the data set the second set corresponds to robust weights and is used to obtain robust estimates of the cluster prototypes by choosing an appropriate distance measure in the objective function rca can be used to find an unknown number of clusters of various shapes in noisy data sets as well as to fit an unknown number of parametric models simultaneously several examples such as clustering mixture decomposition line plane fitting segmentation of range images and estimation of motion parameters of multiple objects are shown

median spheres theory algorithms applications a generalized hypersphere is either a hyperplane or a hypersphere which consists of all points equidistant from a center geometrically a weighted median hypersphere minimizes a weighted average of the distances from it to finitely many data points as proved here for each finite data set there exists at least one weighted median generalized hypersphere moreover denote the sums of the weights of the data points inside by w outside by w and on the hypersphere by w 0 the present results show that each weighted median hypersphere is a weighted pseudo halving hypersphere in the sense that w w w 0 and passes through at least two distinct data points combinatorically a hypersphere is blocked if and only if it passes through data points in general position in the sense that no other hypersphere passes through the same data points a hypersphere is a halving hypersphere if and only if it is blocked contains exactly k data points inside confines exactly l data points outside and k l 1 in the plane the present results also show that if a median circle is not a halving circle then moving its center along a median between two data points on it until it passes through the next data point yields a halving circle relative to the center if the direction cosines of the external and internal data points have the same mean and variance then the median circle must be blocked and stays so under sufficiently small perturbations of the data moreover for every set of four points at least one unweighted median circle is blocked these results lend credence to a variant of a method used by archaeologists and explain some findings from operations research

2 d object recognition using invariant contour descriptor and projective refinement abstract this paper presents an efficient model based recognition method to recognize 2 d objects and to obtain correspondences between models and scene boundaries with a subpixel positioning error as a shape signature for a contour we propose a descriptor consisting of five point invariants which are used to index a hash table also we propose a projective refinement as a verification method to compute exact correspondences between models and scene contour points this method repeatedly computes projective transformation using a weighted pseudo inverse we present an error model for five point invariants which are used to define a similarity between two descriptors to determine a searching bound in indexing and to obtain the weights in the projective refinement in experiments using seriously distorted images of forty models this method led to the recognition of planar curved objects a transformation using the correspondence between the model and scene points on contours was also obtained

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

numerical solution of the schrodinger equation in a wavelet basis for hydrogen like atoms an iterative method is proposed to solve the schrodinger eigenvalue problem in a wavelet framework orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix this representation called the nonstandard ns form is obtained by means of the beylkin coifman rokhlin bcr algorithm and simplifies the numerical calculations problems due to the one dimensional mathematical model and to the discretization process receive special attention