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

shape description and matching using integral invariants on eccentricity transformed images matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision to keep track of changes inside the breast for example it is important for a computer aided detection system to establish correspondences between regions of interest shape transformations computed both with integral invariants ii and with geodesic distance yield signatures that are invariant to isometric deformations such as bending and articulations integral invariants describe the boundaries of planar shapes however they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure conversely eccentricity transforms ecc can match shapes by signatures of geodesic distance histograms based on information from inside the shape but they ignore the boundary information we describe a method that combines the boundary signature of a shape obtained from ii and structural information from the ecc to yield results that improve on them separately

pattern matching with affine moment descriptors this paper proposes a method for matching images based on their higher order moments without knowing the point correspondences it is assumed that the disparity between the images can be explained by an affine transformation the second order statistics is used to transform the image points into canonical form which reduces the affine matching problem for determining an orthonormal transformation matrix between the two point sets next higher order complex moments are used to solve the remaining transformation these affine moment descriptors are expressed in terms of the central moments of the original data it is also shown that the resulting descriptors can be converted into affine moment invariants a general framework for deriving affine moment descriptors as well as moment invariants is described the experiments carried out with simulated data and real images indicate that the proposed method utilizing the second and third order statistics can provide good alignment results from noisy and spurious observations

3d object recognition based on curvature information of planar views in this paper a 3d object recognition algorithm is proposed objects are recognized by studying planar images corresponding to a sequence of views planar shape contours are represented by their adaptively calculated curvature functions which are decomposed in the fourier domain as a linear combination of a set of representative shapes finally sequences of views are identified by means of hidden markov models the proposed system has been tested for artificial and real objects distorted and noisy versions of the objects were correctly clustered together

probabilistic analysis of the application of the cross ratio to model based vision misclassification the cross ratio of four colinear points is of fundamental importance in model based vision because it is the simplest numerical property of an object that is invariant under projection to an image it provides a basis for algorithms to recognise objects from images without first estimating the position and orientation of the camera a quantitative analysis of the effectiveness of the cross ratio in model based vision is made a given imagei of four colinear points is classified by making comparisons between the measured cross ratio of the four image points and the cross ratios stored in the model database the imagei is accepted as a projection of an objecto with cross ratio if ntu wheren is the standard deviation of the image noise t is a threshold andu the performance of the cross ratio is described quantitatively by the probability of rejectionr the probability of false alarmf and the probability of misclassificationp defined for two model cross ratios the trade off between these different probabilities is determined byt it is assumed that in the absence of an object the image points have identical gaussian distributions and that in the presence of an object the image points have the appropriate conditional densities the measurements of the image points are subject to small random gaussian perturbations under these assumptions the trade offs betweenr f andp are given to a good approximation byr 2 1 t f r f t sqrt p sigma varsigma e sigma in t left sigma varsigma right 1 t 1 where e is the relative noise level is cumulative distribution function for the normal distribution r f is constant ande is a function of only the trade off betweenr andf is obtained in maybank 1994 in this paper the trade off betweenr andp is obtained it is conjectured that the general form of the above trade offs betweenr f andp is the same for a range of invariants useful in model based vision the conjecture prompts the following definition an invariant which has trade offs betweenr f p of the above form is said to benon degenerate for model based vision the consequences of the trade off betweenr andp are examined in particular it is shown that for a fixed overall probability of misclassification there is a maximum possible model cross ratio m and there is a maximum possible numbern of models approximate expressions for m andn are obtained they indicate that in practice a model database containing only cross ratio values can have a size of order at most ten for a physically plausible level of image noise and for a probability of misclassification of the order 0 1

perceptual grouping for generic recognition we address the problem of recognition of generic objects from a single intensity image this precludes the use of purely geometric methods which assume that models are geometrically and precisely designed instead we propose to use descriptions in terms of features and their qualitative geometric relationships to succeed it is clear that these features need to be high level rather than points or lines we propose to detect groups using perceptual organization criteria such as proximity symmetry parallelism and closure the detection of these features is performed in an efficient way using proximity indexing since many groups are created we also perform selection of relevant groups by organizing them into sets of similar perceptual content finally we present an implementation of a recognition system using these sets as primitives it is an efficient colored graph matching algorithm using the adjacency matrix representation of a graph using indexing we retrieve matching hypotheses which are verified against each other with respect to topological constraints groups of consistent hypotheses represent detected model instances in a scene the complete system is illustrated on real images we also discuss further extensions

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

exact enumeration approach to fractal properties of the percolation backbone and 1 σ expansion an exact numeration approach is developed for the backbone fractal of the incipient infinite cluster at the percolation threshold the authors use this approach to calculate exactly the first low density expansion of lbb p for arbitrary system dimensionality d where lbb p is the mean of backbone bonds and p is the bond occupation probability standard series extrapolation methods provide estimates of the fractal dimension of the backbone for all d these disagree with the sierpinski gasket model of the backbone they also calculate the first low density expansions of lmin p and lred p which are respectively the mean number of bonds in the minimum path between i and j and the mean number of singly connected red bonds

computing size functions from edge maps size functions are integer valued functions of two real variables which have been recently proposed for the representation and recognition of shape a main limitation of the theory of size functions appeared to be the fragility of the produced representation with respect to edge fragmentation in this paper it is shown that size functions can actually be defined without making assumptions on the topological structure of the viewed shape consequently size functions can be profitably used even in the presence of fragmented edge maps in order to demonstrate the potential of size functions for computer vision a system for shape recognition is described and tested on two different domains the very good performances of the system indicate that size functions are extremely effective for the analysis of shapes for which geometric models might be difficult to obtain

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

minimal representations of 3d models in terms of image parameters under calibrated and uncalibrated perspective indexing is a well known paradigm for object recognition in indexing each 3d model is represented as the set of values assumed by a given vector of image parameters in correspondence to all the possible images of the 3d model an open problem posed by jacobs 1992 concerned the minimum dimensionality of such sets under perspective this paper proves that under calibrated or uncalibrated perspective the minimum dimensionality of the set representing any 3d modeled point set is two two dimensional representations are found also for 3d curved objects

feature matching constrained by cross ratio invariance abstract the main aim of this work is to propose a new technique to solve the well known feature correspondence problem for motion estimation the problem is formulated as an optimization process whose energy function includes constraints based on projective invariance of cross ratio of five coplanar points starting from some approximated correspondences estimated by radiometric similarity for features with high directional variance optimal matches are obtained through an optimization technique the new contribution of this work consists of a matching process refining the raw measurements based on an energy function minimization technique converging to an optimal solution for most of the features by taking advantage of some good initial guess and in the use of cross ratio as geometrical invariant constraint to detect and correct the mismatches due to wrong radiometric similarity measures though the method is based on geometrical invariance of coplanar points it is not required that all features have to be coplanar or to preprocess the images to detect the planar regions experimental results are presented for real and synthetic images and the performance of the novel approach is evaluated on different image sequences and compared to well known techniques

a target recognition technique employing geometric invariants abstract in this paper we propose an efficient target recognition technique in the complex scenes for the viewpoint invariant matching the well known geometric invariants the invariants of five coplanar points and the invariants of five coplanar lines which are invariant under the projective transform are employed to alleviate the computational complexity the circular ordered matching technique and compatibility test are proposed utilizing the geometric properties moreover stability of the invariants for five coplanar points to the input noise is analyzed and a method to select the stable invariants is also suggested the performance of the proposed algorithm is evaluated on both various synthetic and real images the results of the experiments show that the proposed matching technique is both effective and stable to the input noise for the target recognition in complex images

ecological and evolutionary dynamics of two stage models of social insects with egg cannibalism abstract cannibalistic interactions between different developmental stages in a population are prevalent among many groups of animals and social insects cannibalism plays an important role in the population outcomes and represents an adaptive strategy in which parents consume some offspring to increase their current and or future reproductive success to understand how egg cannibalism affects ecological dynamics and evolutionary outcomes we propose a simple two stage ecological model and its evolutionary model by using the framework of evolutionary game theory our analytical study shows that 1 at the population level a large rate of egg cannibalism can lead to a forward transcritical bifurcation characterized by the emergence of a globally stable interior equilibrium while a small rate of egg cannibalism can lead to a backward subcritical bifurcation which generates strong allee effects 2 when the environment is harsh egg cannibalism can prevent extinction in both the ecological and the evolutionary setting in addition evolution preserves strong allee effects by choosing the trait with the smaller cannibalism rate 3 evolution may decrease or increase the fitness of the colony by decreasing or increasing the total population size 4 the trait function is very important since it can affect permanence of the system these results suggest that cannibalism behavior is indeed an adaptive strategy when the availability of food is scarce and the nutrient is limited additionally cannibalism may be averted by defensive behaviors or other adaptations that serve to their age distribution and it may be also a mechanism that can generate strong allee effects

3 d object recognition using a new invariant relationship by single view abstract we propose a new method for recognizing three dimensional objects using a three dimensional invariant relationship and geometric hashing by single view we develop a special structure consisting of four co planar points and any two non planar points with respect to the plane we derive an invariant relationship for the structure which is represented by a plane equation for the recognition of three dimensional objects using the geometric hashing a set of points on the plane thereby satisfying the invariant relationship are mapped into a set of points intersecting the plane and the unit sphere since the structure is much more general than the previous structures proposed by rothwell et al oxford university tr ouel 1927 92 1992 and zhu et al proceedings of the 12th international conference on robotics and automation nagoya japan 1995 pp 1726 1731 it gives enough many voting to generate hypotheses we also show that from the proposed invariant relationship an invariant for the structure by two view and an invariant for a structure proposed by zhu et al proceedings of the 12th international conference on robotics and automation nagoya japan 1995 pp 1726 1731 can also be derived experiments using three dimensional polyhedral objects are carried out to demonstrate the feasibility of our method for three dimensional objects

invasion percolation new algorithms and universality classes employing highly efficient algorithms for simulating invasion percolation ip whose execution time scales as o mlog m or better for a cluster of m sites and for determining the backbone of the cluster we obtain precise estimates for the fractal dimensions of the sample spanning cluster the backbone and the minimal path in order to identify the universality classes of four different ip processes site and bond ip with and without trapping in two dimensions ip is characterized by two universality classes one each for ip without trapping and site and bond ip with trapping in a three dimensional site ip with and without trapping is in the universality class of random percolation while bond ip with trapping is in a distinct universality class which may be the same as that of optimal paths in strongly disordered media

shape quantization and recognition with randomized trees we explore a new approach to shape recognition based on a virtually infinite family of binary features queries of the image data designed to accommodate prior information about shape invariance and regularity each query corresponds to a spatial arrangement of several local topographic codes or tags which are in themselves too primitive and common to be informative about shape all the discriminating power derives from relative angles and distances among the tags the important attributes of the queries are a natural partial ordering corresponding to increasing structure and complexity semi invariance meaning that most shapes of a given class will answer the same way to two queries that are successive in the ordering and stability since the queries are not based on distinguished points and substructures no classifier based on the full feature set can be evaluated and it is impossible to determine a priori which arrangements are informative our approach is to select informative features and build tree classifiers at the same time by inductive learning in effect each tree provides an approximation to the full posterior where the features chosen depend on the branch that is traversed due to the number and nature of the queries standard decision tree construction based on a fixed length feature vector is not feasible instead we entertain only a small random sample of queries at each node constrain their complexity to increase with tree depth and grow multiple trees the terminal nodes are labeled by estimates of the corresponding posterior distribution over shape classes an image is classified by sending it down every tree and aggregating the resulting distributions the method is applied to classifying handwritten digits and synthetic linear and nonlinear deformations of three hundred l at e x symbols stateof the art error rates are achieved on the national institute of standards and technology database of digits the principal goal of the experiments on l at e x symbols is to analyze invariance generalization error and related issues and a comparison with artificial neural networks methods is presented in this context

vanishing gaps in 1d bandstructures the criterion for vanishing or zero energy gaps in the spectrum of 1d electronic bandstructures is obtained from perturbation theory it is in agreement with examples of periodic potentials which never exhibit zero gaps as well as with those which may including certain smooth periodic potentials the results have relevance to recent work in the field of semiconductor layered heterostructures or superlattices the analysis should also have implications for other problems that involve hill s determinant

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

the space requirements of indexing under perspective projections object recognition systems can be made more efficient through the use of table lookup to match features the cost of this indexing process depends on the space required to represent groups of model features in such a lookup table we determine the space required to perform indexing of arbitrary sets of 3d model points for lookup from a single 2d image formed under perspective projection we show that in this case one must use a 3d surface to represent model groups and we provide an analytic description of such a surface this is in contrast to the cases of scaled orthographic or affine projection in which only a 2d surface is required to represent a group of model features this demonstrates a fundamental way in which the recognition of objects under perspective projection is more complex than is recognition under other projection models

when is it possible to identify 3d objects from single images using class constraints one approach to recognizing objects seen from arbitrary viewpoint is by extracting invariant properties of the objects from single images such properties are found in images of 3d objects only when the objects are constrained to belong to certain classes e g bilaterally symmetric objects existing studies that follow this approach propose how to compute invariant representations for a handful of classes of objects a fundamental question regarding the invariance approach is whether it can be applied to a wide range of classes to answer this question it is essential to study the set of classes for which invariance exists this paper introduces a new method for determining the existence of invariant functions for classes of objects together with the set of images from which these invariants can be computed we develop algebraic tests that determine whether the objects in a given class can be identified from single images these tests apply to classes of objects undergoing affine projection in addition these tests allow us to determine the set of views of the objects which are degenerate we apply these tests to several classes of objects and determine which of them is identifiable and which of their views are degenerate

3d object classification using multi object kohonen networks the problem of three dimensional planar faced opaque object recognition from one single perspective view is addressed in this paper a view parameterization based on hough transform is described classification is accomplished by multiple multi object kohonen networks a new object grouping criterion has been investigated to assign disjoint subsets of objects to each kohonen network of the system on the basis of input space topology recognition tests are presented on both synthetic and real world 3d objects even in the presence of partial occlusion

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

simple algorithms for partial point set pattern matching under rigid motion this paper presents simple and deterministic algorithms for partial point set pattern matching in 2d given a set p of n points called sample set and a query set q of k points n k the problem is to find a matching of q with a subset of p under rigid motion the match may be of two types exact and approximate if an exact matching exists then each point in q coincides with the corresponding point in p under some translation and or rotation for an approximate match some translation and or rotation may be allowed such that each point in q lies in a predefined e neighborhood region around some point in p the proposed algorithm for the exact matching needs o n 2 space and o n 2logn preprocessing time the existence of a match for a given query set q can be checked in o email protected time in the worst case where a is the maximum number of equidistant pairs of point in p for a set of n points a may be o n 4 3 in the worst case some applications of the partial point set pattern matching are then illustrated experimental results on random point sets and some fingerprint databases show that in practice the computation time is much smaller than the worst case requirement the algorithm is then extended for checking the exact match of a set of k line segments in the query set with a k subset of n line segments in the sample set under rigid motion in o knlogn time next a simple version of the approximate matching problem is studied where one point of q exactly matches with a point of p and each of the other points of q lie in the e neighborhood of some point of p the worst case time and space complexities of the proposed algorithm are o n 2k 2logn and o n respectively the proposed algorithms will find many applications to fingerprint matching image registration and object recognition

invariants of six points and projective reconstruction from three uncalibrated images there are three projective invariants of a set of six points in general position in space it is well known that these invariants cannot be recovered from one image however an invariant relationship does exist between space invariants and image invariants this invariant relationship is first derived for a single image then this invariant relationship is used to derive the space invariants when multiple images are available this paper establishes that the minimum number of images for computing these invariants is three and the computation of invariants of six points from three images can have as many as three solutions algorithms are presented for computing these invariants in closed form the accuracy and stability with respect to image noise selection of the triplets of images and distance between viewing positions are studied both through real and simulated images applications of these invariants are also presented both the results of faugeras 1992 and hartley et al 1992 for projective reconstruction and sturm s method 1869 for epipolar geometry determination from two uncalibrated images with at least seven points are extended to the case of three uncalibrated images with only six points

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

backbone and elastic backbone of percolation clusters obtained by the new method of burning we introduce a method of obtaining information about percolation clusters in which the clusters are burned in different ways this method yields the backbone and the elastic backbone their number of loops shortest path and other quantities it works in any dimension and can be applied also to other clusters we obtain accurate estimates for the backbone exponents in two and three dimensions

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

geometric invariants and object recognition we discuss the role of the general invariance concept in object recognition and review the classical and recent literature on projective invariance invariants help solve major problems of object recognition for instance different images of the same object often differ from each other because of the different viewpoint from which they were obtained to match the two images common methods thus need to find the correct viewpoint a difficult problem that can involve search in a high dimensional space of all possible points of view and or finding point correspondences geometric invariants are shape descriptors computed from the geometry of the shape that remain unchanged under geometric transformations such as changing the viewpoint thus they can be matched without search deformations of objects are another important class of geometric changes for which invariance is useful

evolutionary dynamics of single species model with allee effect abstract by considering a continuous strategy evolutionary game theory egt a single species population subject to additive allee effect is investigated we assume that the carrying capacity and predator evasion ability following a gaussian distribution this assumption shows that evolution prevents extinction even if population has strong allee effect the ratio of competition and anti predation phenotypes has an important effect on determining global dynamics of the egt model the numerical simulations present the model can have nine equilibria and the model has multiple evolutionarily stable strategy equilibria ess

dynamics of a single species evolutionary model with allee effects abstract we investigate the evolutionary outcomes of a single species population subject to allee effects within the framework of a continuous strategy evolutionary game theory egt model our model assumes a single trait creates a phenotypic trade off between carrying capacity i e competition and predator evasion ability following a gaussian distribution this assumption contributes to one of our interesting findings that evolution prevents extinction even when population exhibits strong allee effects however the extinction equilibrium can be an ess under some special distributions of anti predation phenotypes the ratio of variation in competition and anti predation phenotypes plays an important role in determining global dynamics of our egt model a evolution may suppress strong allee effects for large values of this ratio b evolution may preserve strong allee effects for small values of this ratio by generating a low density evolutionary stable strategy ess equilibrium which can serve as a potential allee threshold and c intermediate values of this ratio can result in multiple ess equilibria

planar object recognition using projective shape representation we describe a model based recognition system called lewis for the identification of planar objects based on a projectively invariant representation of shape the advantages of this shape description include simple model acquisition direct from images no need for camera calibration or object pose computation and the use of index functions we describe the feature construction and recognition algorithms in detail and provide an analysis of the combinatorial advantages of using index functions index functions are used to select models from a model base and are constructed from projective invariants based on algebraic curves and a canonical projective coordinate frame examples are given of object recognition from images of real scenes with extensive object libraries successful recognition is demonstrated despite partial occlusion by unmodelled objects and realistic lighting conditions

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

statistical bias of conic fitting and renormalization introducing a statistical model of noise in terms of the covariance matrix of the n vector we point out that the least squares conic fitting is statistically biased we present a new fitting scheme called renormalization for computing an unbiased estimate by automatically adjusting to noise relationships to existing methods are discussed and our method is tested using real and synthetic data

recognition of object classes from range data abstract we develop techniques for recognizing instances of 3d object classes which may consist of multiple and or repeated sub parts with internal degrees of freedom linked by parameterized transformations from sets of 3d feature observations recognition of a class instance is structured as a search of an interpretation tree in which geometric constraints on pairs of sensed features not only prune the tree but are used to determine upper and lower bounds on the model parameter values of the instance a real valued constraint propagation network unifies the representations of the model parameters model constraints and feature constraints and provides a simple and effective mechanism for accessing and updating parameter values recognition of objects with multiple internal degrees of freedom including non uniform scaling and stretching articulations and sub part repetitions is demonstrated and analysed for two different types of real range data 3d edge fragments from a stereo vision system and position surface normal data derived from planar patches extracted from a range image

matching 3 d models to 2 d images we consider the problem of analytically characterizing the set of all 2 d images that a group of 3 d features may produce and demonstrate that this is a useful thing to do our results apply for simple point features and point features with associated orientation vectors when we model projection as a 3 d to 2 d affine transformation we show how to represent the set of images that a group of 3 d points produces with two lines 1 d subspaces one in each of two orthogonal high dimensional spaces where a single image group corresponds to one point in each space the images of groups of oriented point features can be represented by a 2 d hyperbolic surface in a single high dimensional space the problem of matching an image to models is essentially reduced to the problem of matching a point to simple geometric structures moreover we show that these are the simplest and lowest dimensional representations possible for these cases we demonstrate the value of this way of approaching matching by applying our results to a variety of vision problems in particular we use this result to build a space efficient indexing system that performs 3 d to 2 d matching by table lookup this system is analytically built and accessed accounts for the effects of sensing error and is tested on real images we also derive new results concerning the existence of invariants and non accidental properties in this domain finally we show that oriented points present unexpected difficulties indexing requires fundamentally more space with oriented than with simple points we must use more images in a motion sequence to determine the affine structure of oriented points and the linear combinations result does not hold for oriented points

a simple method for fitting of bounding rectangle to closed regions in this paper we introduce a new approach for fitting of a bounding rectangle to closed regions in this approach the coordinates of the vertices are computed directly using a closed form solution this approach is based on simple coordinate geometry and uses the boundary points of regions using a least square approach we determine the directions of major and minor axes of the object which gives the orientation of the object the four vertexes of the bounding rectangle are computed by pair wise solving the four straight lines examples from synthetic data and some real life data show that the approach is both accurate and efficient

a precise characterization of three dimensional percolating backbones the backbones of three dimensional critical percolation clusters are extracted and the fractal dimension is accurately calculated using various mass scaling and box counting techniques to be dfb 1 855 or 0 015 the eigenvalue spectrum of the brownian transition probability matrix is studied numerically which results in the estimates dwb 3 13 or 0 03 for the walk dimension and dsb 1 18 or 0 01 for the spectral dimension of the backbone independent calculation of all three quantities allows the conductivity exponent mu to be determined and also allows the validity of the scaling relation ds 2df dw proposed by alexander and orbach 1982 to be tested on the backbone

vertex based features for recognition of projectively deformed polygons the paper deals with features of a general polygon which are invariant with respect to projective transform first some properties of the area of a triangle under projective transform are discussed new projective triangular invariants of polygons are derived as the quotient of two different products of the areas of triangles formed by the vertices of the polygon the features are proved to be invariant to numbering of the vertices of the polygon the number of projective triangular invariants for polygons with a given number of vertices is derived a method for reduction of the number of invariants in the case of polygons with a high number of vertices is presented numerical experiments dealing with three octagons and one polygon with 18 vertices deformed by projective transforms are described

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

numerical studies of critical percolation in three dimensions presents results of high statistics simulations of the spreading of 3d percolation close to the critical point the main results are i a more precise estimate of the spreading resp chemical distance dimension ii a novel heuristic scaling theory for percolation near surfaces and iii more precise values for critical surface and edge exponents in addition the author verifies the very precise results of ziff and stell 1988 for the percolation thresholds and studies in detail corrections to scaling the latter is made possible by simulating bond and site percolation in a number of different geometries in addition in most cases one could measure more than one observable during each simulation which provided quite severe internal consistency constraints in some cases the author found quite important deviations from scaling which are not easily described by a single power or logarithmic correction term

recognition of planar shapes by a semi differential invariant for moving triangles consider the recognition of planar contours of several moving triangles using pictures taken by camera our new idea is that we take pictures of not only rotated triangles but also image trajectories of circles placed properly we apply this method to the recognition of three dimensional tetrahedrons by showing that each angle between two face planes is measured only from the circles on face triangles in addition to recognizing face triangles the method uses the semi differential invariance for pseudo perspective projection which can recognize local figures

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

classical diffusion on eden trees we study an aggregation process which gives rise to compact clusters with no loops a generalization of the node counting theorem applicable to such branched graphs trees is proved this is used to determine the spectral dimension of these clusters on square and cubic embedding lattices by monte carlo simulations the results are explained in terms of the geometric structure of these trees

corrections to scaling for diffusion exponents on three dimensional percolation systems at criticality recent results of monte carlo simulations of the ant in the labyrinth method in three dimensional percolation lattices are reanalyzed in the light of more accurate corrections to scaling ansatz motivated by inconsistent results that have appeared in the literature the results are observed to be sensitive to the form of the scaling correction terms using a single correction term we estimate the valuek 0 197 0 004 for the anomalous diffusion exponent at criticality when two correction terms are included k 0 200 0 002 is obtained these new estimates are consistent with known theoretical bounds with recent series expansion results and with numerical calculations of the conductance of random resistor networks above criticality

approximate geometric pattern matching under rigid motions we present techniques for matching point sets in two and three dimensions under rigid body transformations we prove bounds on the worst case performance of these algorithms to be within a small constant factor of optimal and conduct experiments to show that the average performance of these matching algorithms is often better than that predicted by the worst case bounds

geometric hashing with line features abstract geometric hashing is a model based object recognition technique for detecting objects which can be partially overlapping or partly occluded in its simplest form the geometric hashing method assumes relatively noise free data and is applied to objects with points as local features however extracting of the locations of point features is inherently error prone line features compared with point features can generally be extracted with greater accuracy the application of line features for geometric hashing to the recognition of two dimensional 2d or flat 3d objects undergoing various geometric transformations is investigated

a precise determination of the backbone fractal dimension on two dimensional percolation clusters the backbone fractal dimension dbf is calculated on two dimensional percolation clusters at the percolation threshold studies are carried out on backbones defined in three different ways bus bar point to point and with fully periodic boundaries the estimates of dbf are obtained by measuring the variation of mass with radius for all clusters and by means of finite size scaling on the bus bar backbones all cases imply a value of dbf 1 64 or 0 01 for the backbone fractal dimension because of the high degree of self consistency of all the results the authors believe that this estimate represents a considerably improved accuracy

point based projective invariants abstract the paper deals with features of a 2 d point set which are invariant with respect to a projective transform first projective invariants for five point sets which are simultaneously invariant to the projective transform and to permutation of the points are derived they are expressed as functions of five point cross ratios then the invariants for more than five points are derived the algorithm for searching the correspondence between the points of two 2 d point sets is presented the algorithm is based on the comparison of two projective and permutation invariants of five tuples of the points the best matched five tuples are then used for the computation of the projective transformation and that with the maximum of corresponding points is used stability and discriminability of the features and behavior of the searching algorithm are demonstrated by numerical experiments

robust algorithms for object localization model based localization the task of estimating an object s pose from sensed and corresponding model features is a fundamental task in machine vision exact constant time localization algorithms have been developed for the case where the sensed features and the model features are the same type still it is not uncommon for the sensed features and the model features to be of different types i e sensed data points may correspond to model faces or edges previous localization approaches have handled different model and sensed features of different types via sampling and synthesizing virtual features to reduce the problem of matching features of dissimilar types to the problem of matching features of similar types unfortunately these approaches may be suboptimal because they introduce artificial errors other localization approaches have reformulated object localization as a nonlinear least squares problem where the error is between the sensed data and model features in image coordinates the euclidean image error metric unfortunately all of the previous approaches which minimized the euclidean image error metric relied on gradient descent methods to find the global minima and gradient descent methods may suffer from problems of local minima in this paper we describe an exact efficient solution to the nonlinear least squares minimization problem based upon resultants linear algebra and numerical techniques on a sparc 20 our localization algorithm runs in a few microseconds for rectilinear polygonal models a few milliseconds for generic polygonal models and one second for generalized polygonal models models composed of linear edges and circular arcs

a new one parametric fitting method for planar objects this paper is an extension of the already published paper by voss and suesse 1997 we developed a new region based fitting method using the method of normalization there we demonstrated the zero parametric fitting of lines triangles parallelograms circles and ellipses we discuss this normalization idea for fitting of closed regions using circular segments elliptical segments and super ellipses as features we use the area based low order moments we show that we have to solve only one dimensional optimization problems in these cases

conic reconstruction and correspondence from two views conics are widely accepted as one of the most fundamental image features together with points and line segments the problem of space reconstruction and correspondence of two conics from two views is addressed in this paper it is shown that there are two independent polynomial conditions on the corresponding pair of conics across two views given the relative orientation of the two views these two correspondence conditions are derived algebraically and one of them is shown to be fundamental in establishing the correspondences of conics a unified closed form solution is also developed for both projective reconstruction of conics in space from two uncalibrated camera views and metric reconstruction from two calibrated camera views experiments are conducted to demonstrate the discriminality of the correspondence conditions and the accuracy and stability of the reconstruction both for simulated and real images

indexing hierarchical structures using graph spectra hierarchical image structures are abundant in computer vision and have been used to encode part structure scale spaces and a variety of multiresolution features in this paper we describe a framework for indexing such representations that embeds the topological structure of a directed acyclic graph dag into a low dimensional vector space based on a novel spectral characterization of a dag this topological signature allows us to efficiently retrieve a promising set of candidates from a database of models using a simple nearest neighbor search we establish the insensitivity of the signature to minor perturbation of graph structure due to noise occlusion or node split merge to accommodate large scale occlusion the dag rooted at each nonleaf node of the query votes for model objects that share that part effectively accumulating local evidence in a model dag s topological subspaces we demonstrate the approach with a series of indexing experiments in the domain of view based 3d object recognition using shock graphs

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

diffusion and spectral dimension on eden tree the authors calculate the eigenspectrum of random walks on the eden tree in two and three dimensions from this they calculate the spectral dimension ds and the walk dimension dw and test the scaling relation ds 2df dw 2d dw for an eden tree finite size induced crossovers are observed whereby the system crosses over from a short time regime where this relation is violated particularly in two dimensions to a long time regime where the behaviour appears to be complicated and dependent on dimension even qualitatively

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

scaling of cluster mass between two lines in 3d percolation abstract we consider the cluster mass distribution between two lines of arbitrary orientations and lengths in porous media in three dimensions and model the porous media by bond percolation at the percolation threshold p c we observe that for many geometrical configurations the mass probability distribution presents power law behavior we determine how the characteristic mass of the distribution scales with such geometrical parameters as the line length w the minimal distance between lines r and the angle between the lines the fractal dimensions of the cluster mass are independent of w r and the slope of the power law regime of the cluster mass is unaffected by changes in these three variables however the characteristic mass of the cluster depends upon we propose new scaling functions that reproduce the dependence of the characteristic mass found in the simulations

the curvature primal sketch in this paper we introduce a novel representation of the significant changes in curvature along the bounding contour of planar shape we call the representation the curvature primal sketch because of the close analogy to the primal sketch representation advocated by marr for describing significant intensity changes we define a set of primitive parameterized curvature discontinuities and derive expressions for their convolutions with the first and second derivatives of a gaussian we describe an implemented algorithm that computes the curvature primal sketch by matching the multiscale convolutions of a shape and illustrate its performance on a set of tool shapes several applications of the representation are sketched

a spherical representation for recognition of free form surfaces introduces a new surface representation for recognizing curved objects the authors approach begins by representing an object by a discrete mesh of points built from range data or from a geometric model of the object the mesh is computed from the data by deforming a standard shaped mesh for example an ellipsoid until it fits the surface of the object the authors define local regularity constraints that the mesh must satisfy the authors then define a canonical mapping between the mesh describing the object and a standard spherical mesh a surface curvature index that is pose invariant is stored at every node of the mesh the authors use this object representation for recognition by comparing the spherical model of a reference object with the model extracted from a new observed scene the authors show how the similarity between reference model and observed data can be evaluated and they show how the pose of the reference object in the observed scene can be easily computed using this representation the authors present results on real range images which show that this approach to modelling and recognizing 3d objects has three main advantages 1 it is applicable to complex curved surfaces that cannot be handled by conventional techniques 2 it reduces the recognition problem to the computation of similarity between spherical distributions in particular the recognition algorithm does not require any combinatorial search and 3 even though it is based on a spherical mapping the approach can handle occlusions and partial views

analysis of a stochastic hybrid population model with allee effect abstract this research formulates and studies a stochastic population model with allee effect and markovian switching sufficient criteria for extinction stochastic permanence and the existence of a unique ergodic invariant measure of the model are established the result indicates that these long term behaviors of the model are intimately associated with the random perturbations

a study of affine matching with bounded sensor error affine transformations of the plane have been used in a number of model based recognition systems because the underlying mathematics are based on exact data in practice various heuristics are used to adapt the methods to real data where there is positional uncertainty this paper provides a precise analysis of affine point matching under uncertainty we obtain an expression for the range of affine invariant values that are consistent with a given set of four points where each image point lies in an disc of uncertainty this range is shown to depend on the actualx y positions of the data points in other words given uncertainty in the data there are no representations that are invariant with respect to the cartesian coordinate system of the data this is problematic for methods such as geometric hashing that are based on affine invariant representations we also analyze the effect that uncertainty has on the probability that recognition methods using affine transformations will find false positive matches we find that there is a significant probability of false positives with even moderate levels of sensor error suggesting the importance of good verification techniques and good grouping techniques

determination of fractal dimensions for geometrical multifractals abstract two independent approaches the box counting and the sand box methods are used for the determination of the generalized dimensions d q associated with the geometrical structure of growing deterministic fractals we find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated d q s to their true values our study demonstrates that the above mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres in this case the latter approach provides better estimates of the generalized dimensions

investigating the interactions between allee effect and harvesting behaviour of a single species model an evolutionary dynamics approach abstract in this study we propose an ecological model which is based on the smith growth model and subject to multiplicative allee effect we then investigate the equilibrium bifurcation induced by the harvesting behaviour and the allee effect and pitch fork bifurcation and saddle node bifurcation may occur assuming that the carrying capacity and harvesting rate obey a gaussian distribution we then consider the evolutionary dynamics by using continuous strategy evolutionary game theory our study shows that strong allee effect and low harvesting rate can induce bistability it also suggests that evolution prevents extinction even if population has strong allee effect numerical analysis shows that the evolutionary model can admit up to nine equilibrium points including up to four evolutionary stable strategy equilibria

projective alignment with regions we have previously proposed basri and jacobs 1999 and jacobs and basri 1999 an approach to recognition that uses regions to determine the pose of objects while allowing for partial occlusion of the regions regions introduce an attractive alternative to existing global and local approaches since unlike global features they can handle occlusion and segmentation errors and unlike local features they are not as sensitive to sensor errors and they are easier to match the region based approach also uses image information directly without the construction of intermediate representations such as algebraic descriptions which may be difficult to reliably compute we further analyze properties of the method for planar objects undergoing projective transformations in particular we prove that three visible regions are sufficient to determine the transformation uniquely and that for a large class of objects two regions are insufficient for this purpose however we show that when several regions are available the pose of the object can generally be recovered even when some or all regions are significantly occluded our analysis is based on investigating the flow patterns of points under projective transformations in the presence of fixed points

probabilistic analysis of the application of the cross ratio to model based vision the probability density function for the cross ratio is obtained under the hypothesis that the four image points have independent identical gaussian distributions the density function has six symmetries which are closely linked to the six different values of the cross ratio obtained by permuting the quadruple of points from which the cross ratio is calculated the density function has logarithmic singularities corresponding to values of the cross ratio for which two of the four points are coincident the cross ratio forms the basis of a simple system for recognising or classifying quadruples of collinear image points the performance of the system depends on the choice of rule for deciding whether four image points have a given cross ratio a rule is stated which is computationally straightforward and which takes into account the effects on the cross ratio of small errors in locating the image points two key properties of the rule are the probabilityr of rejection and the probabilityf of a false alarm the probabilitiesr andf depend on a thresholdt in the decision rule there is a trade off betweenr andf obtained by varyingt it is shown that the trade off is insensitive to the given cross ratio letf w max o f thenr f w are related approximately by sqrt ln r 1 sqrt 2 varepsilon r f 1 f w provided e 1f w 4 in the equation e is the accuracy with which image points can be located relative to the width of the image andr f is a constant known as the normalised false alarm rate in the range e 1f w 4 the probabilitiesr andf w are related approximately by r 1 sqrt 2 pi 1 varepsilon 1 r f 1 f w the value ofr f is 14 37 the consequences of these relations between r and fw are discussed it is conjectured that the above general form of the trade off betweenr andf w holds for a wide class of scalar invariants that could be used for model based object recognition invariants with the same type of trade off between the probability of rejection and the probability of false alarm are said to be nondegenerate for model based vision

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

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

pose determination of a three dimensional object using triangle pairs a procedure is presented to estimate the unconstrained three dimensional location and orientation of an object with a known shape when it is visible in a single image using a generalized hough transform all six parameters of the object position are estimated from the distribution of values determined by matching triples of points on the object to possibly corresponding triples in the image most likely candidates for location are found and then the remaining rotational parameters are evaluated two solutions are generally admitted to the distribution by every match of triangles the number of possible matches is reduced by checking a simple geometric relation among triples even with partial occlusion experiments indicate that the procedure is very reliably accurate although optimization can further improve estimates of the parameters