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

multiple window parallel adaptive boundary finding in computer vision the problem considered in this paper is the estimation of highly variable object boundaries in noisy images boundaries may be those of a tank in an ir image a spinal canal in a cat scan a cloud in a visible light image etc or they may be internal to an object such as the boundary between a spherical surface and a cylindrical surface in a manufactured object the focus of the paper is on parallel multiple window boundary estimation algorithms here the image field is parti tioned into an array of rectangular windows and boundary finders are run simultaneously within the windows the boundary segments found within the windows are then seamed together to obtain meaningful global boundaries the entire procedure is treated within a maximum likelihood estimation framework that we have developed for boundary finding although our multiple window estimation approach can be used with a number of local boundary finding algorithms we concen trate on one which is based on dynamic programming and will produce the true maximum likelihood boundary some theoretical considera tions for boundary model design and boundary finding runtime are covered included is the use of a low computational cost f test for test ing whether a window contains a boundary and an analytical treatment which shows that use of coarse pixels with a chi square test or an f test improves the probability of correctly recognizing whether a boundary is present in a window

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

topologically faithful fitting of simple closed curves implicit representations of curves have certain advantages over explicit representation one of them being the ability to determine with ease whether a point is inside or outside the curve inside outside functions however save for some special cases it is not known how to construct implicit representations which are guaranteed to preserve the curve s topology as a result points may be erroneously classified with respect to the curve the paper offers to overcome this problem by using a representation which is guaranteed to yield the correct topology of a simple closed curve by using homeomorphic mappings of the plane to itself if such a map carries the curve onto the unit circle then a point is inside the curve if and only if its image is inside the unit circle

parts of visual form computational aspects underlying recognition is an organization of objects and their parts into classes and hierarchies a representation of parts for recognition requires that they be invariant to rigid transformations robust in the presence of occlusions stable with changes in viewing geometry and be arranged in a hierarchy these constraints are captured in a general framework using notions of a part line and a partitioning scheme a proposed general principle of form from function motivates a particular partitioning scheme involving two types of parts neck based and limb based neck based parts arise from narrowings in shape or the local minima in distance between two points on the boundary while limb based parts arise from a pair of negative curvature minima which have co circular tangents in this paper we present computational support for the limb based and neck based parts by showing that they are invariant robust stable and yield a hierarchy of parts examples illustrate that the resulting decompositions are robust in the presence of occlusion and clutter for a range of man made and natural objects and lead to natural and intuitive parts which can be used for recognition

optimal edge based shape detection we propose an approach to accurately detecting two dimensional 2 d shapes the cross section of the shape boundary is modeled as a step function we first derive a one dimensional 1 d optimal step edge operator which minimizes both the noise power and the mean squared error between the input and the filter output this operator is found to be the derivative of the double exponential dode function originally derived by ben arie and rao 1994 we define an operator for shape detection by extending the dode filter along the shape s boundary contour the responses are accumulated at the centroid of the operator to estimate the likelihood of the presence of the given shape this method of detecting a shape is in fact a natural extension of the task of edge detection at the pixel level to the problem of global contour detection this simple filtering scheme also provides a tool for a systematic analysis of edge based shape detection we investigate how the error is propagated by the shape geometry we have found that under general assumptions the operator is locally linear at the peak of the response we compute the expected shape of the response and derive some of its statistical properties this enables us to predict both its localization and detection performance and adjust its parameters according to imaging conditions and given performance specifications applications to the problem of vehicle detection in aerial images human facial feature detection and contour tracking in video are presented

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

bifurcation of limit cycles and integrability of planar dynamical systems in complex form we describe a new algorithm for the calculation of liapunov quantities for two dimensional systems in terms of the complex form of the system it is more transparent conceptually and more efficient computationally we demonstrate the advantages of this approach by deriving the integrability conditions and information about the bifurcation of limit cycles for some cubic systems

recognition using region correspondences recognition systems attempt to recover information about the identity of observed objects and their location in the environment a fundamental problem in recognition is pose estimation this is the problem of using a correspondence between some portions of an object model and some portions of an image to determine whether the image contains an instance of the object and in case it does to determine the transformation that relates the model to the image the current approaches to this problem are divided into methods that use global properties of the object e g centroid and moments of inertia and methods that use local properties of the object e g corners and line segments global properties are sensitive to occlusion and specifically to self occlusion local properties are difficult to locate reliably and their matching involves intensive computation we present a novel method for recognition that uses region information in our approach the model and the image are divided into regions given a match between subsets of regions without any explicit correspondence between different pieces of the regions the alignment transformation is computed the method applies to planar objects under similarity affine and projective transformations and to projections of 3 d objects undergoing affine and projective transformations the new approach combines many of the advantages of the previous two approaches while avoiding some of their pitfalls like the global methods our approach makes use of region information that reflects the true shape of the object but like local methods our approach can handle occlusion

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

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

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

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

an estimation based approach for range image segmentation on the reliability of primitive extraction this paper presents a new algorithm for estimation based range image segmentation aiming at surface primitive extraction from range data we focus on the reliability of the primitive representation in the process of region estimation we introduce an optimal description of surface primitives by which the uncertainty of a region estimate is explicitly represented with a covariance matrix then the reliability of an estimate is interpreted in terms of measure of uncertainty the segmentation approach follows the region growing scheme in which the regions are estimated in an iterative way with the probabilistic model proposed in this paper surface homogeneity is defined and tested by an optimal criterion a notable feature of the algorithm is that the order of merging is organized to lead the growth towards the most reliable representation of the merged region concerned with man made objects in the scene we restrict the class of surface primitives to be quadric or planar the proposed algorithm has been applied to real data and synthetic data and demonstrated with experimental results

stable fitting of 2d curves and 3d surfaces by implicit polynomials this work deals with fitting 2d and 3d implicit polynomials ips to 2d curves and 3d surfaces respectively the zero set of the polynomial is determined by the ip coefficients and describes the data the polynomial fitting algorithms proposed in this paper aim at reducing the sensitivity of the polynomial to coefficient errors errors in coefficient values may be the result of numerical calculations when solving the fitting problem or due to coefficient quantization it is demonstrated that the effect of reducing this sensitivity also improves the fitting tightness and stability of the proposed two algorithms in fitting noisy data as compared to existing algorithms like the well known 3l and gradient one algorithms the development of the proposed algorithms is based on an analysis of the sensitivity of the zero set to small coefficient changes and on minimizing a bound on the maximal error for one algorithm and minimizing the error variance for the second simulation results show that the proposed algorithms provide a significant reduction in fitting errors particularly when fitting noisy data of complex shapes with high order polynomials as compared to the performance obtained by the above mentioned existing algorithms

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

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

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

computing integrability conditions for a cubic differential system abstract the integrability conditions for certain systems of differential equations p y y q y where p and q are cubic polynomials are obtained using two distinct computational approaches such conditions are required in the investigation of hilbert s 16 th problem and in the development of algorithms for the automatic solution of differential equations

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

bifurcation of limit cycles and integrability conditions for 6 parameter families of polynomial vector fields of arbitrary degree abstract we characterize the center variety and the maximum order of any focus of the 6 parameter family of real planar polynomial vectors given in complex notation by z i z a z n 1 z j 1 b z n 2 z j 2 c z n 3 z j 3 where a b c c 0 n 1 j 1 n 2 j 2 n 3 j 3 n k j k 1 for k 1 2 3 n 1 j 1 n 2 j 2 n 3 j 3 1 n 3 j 3 1 n 2 j 2 1 n 1 j 1 and 1 n 1 j 1 1 n 2 j 2 0

on deformable models for visual pattern recognition this paper reviews model based methods for non rigid shape recognition these methods model match and classify non rigid shapes which are generally problematic for conventational algorithms using rigid models issues including model representation optimization criteria formulation model matching and classification are examined in detail with the objective to provide interested researchers a roadmap for exploring the field this paper emphasizes on 2d deformable models their potential applications and future research directions particularly on deformable pattern classification are discussed

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

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

continuous maximin knapsack problems with glb constraints this paper introduces the continuous minimax knapsack problem with generalized lower bound constraints and describes an algorithm that solves this problem in o n logn time we also discuss the related problem with generalized upper bound constraints

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

recognition of 3d objects based on implicit polynomials closed form expressions for a new set of 3d rotation invariants that are linear quadratic and angular combinations of implicit polynomial ip coefficients are developed based on these invariants we propose a 3d object recognition method that outperforms recognition based on ip fitting after pose estimation and the mpeg 7 ssd technique

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

the 3l algorithm for fitting implicit polynomial curves and surfaces to data we introduce a completely new approach to fitting implicit polynomial geometric shape models to data and to studying these polynomials the power of these models is in their ability to represent nonstar complex shapes in two 2d and three dimensional 3d data to permit fast repeatable fitting to unorganized data which may not be uniformly sampled and which may contain gaps to permit position invariant shape recognition based on new complete sets of euclidean and affine invariants and to permit fast stable single computation pose estimation the algorithm represents a significant advancement of implicit polynomial technology for four important reasons first it is orders of magnitude taster than existing fitting methods for implicit polynomial 2d curves and 3d surfaces and the algorithms for 2d and 3d are essentially the same second it has significantly better repeatability numerical stability and robustness than current methods in dealing with noisy deformed or missing data third it can easily fit polynomials of high such as 14th or 16th degree fourth additional linear constraints can be easily incorporated into the fitting process and general linear vector space concepts apply

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

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