maximal automorphism groups of symmetric riemann surfaces with small genus the genus of a finite group g is the smallest genus of its cayley graphs if g has genus g 1 then by theorems of tom tucker g 168 g 1 and this inequality is strict unless g can be generated by elements a b c satisfying a2 b2 c2 ab 2 bc 3 ac 7 1 with ab and bc generating a proper subgroup of g conversely any group g of the latter sort has genus g g 168 1 and moreover is faithfully representable as a group of homeomorphisms of a compact riemann surface s of the same genus g with half the elements of g reversing the surface s orientation this paper describes all such groups g which have order less than 2 million and gives for each corresponding value of g in the range 1 g 11 905 the number of distinct possibilities for the riemann surface s the classification follows that of all hurwitz groups of order less than 1 million obtained by the same author

pseudo real riemann surfaces and chiral regular maps a riemann surface is called pseudo real if it admits anticonformal automorphisms but no anticonformal involution in this paper we study general properties of the automorphism groups of such surfaces and their uniformizing nec groups in particular we prove that there exist pseudo real riemann surfaces of every possible genus g 2 we also study pseudo real surfaces of genus 2 and 3 further we establish a connection between pseudo real surfaces with maximal automorphism group and chiral 3 valent regular maps and use this to show there exist such surfaces for infinitely many genera by exhibiting infinite families of chiral regular maps of type 3 k for all k 7

affine invariant pattern recognition using multiscale autoconvolution this paper presents a new affine invariant image transform called multiscale autoconvolution msa the proposed transform is based on a probabilistic interpretation of the image function the method is directly applicable to isolated objects and does not require extraction of boundaries or interest points and the computational load is significantly reduced using the fast fourier transform the transform values can be used as descriptors for affine invariant pattern classification and in this article we illustrate their performance in various object classification tasks as shown by a comparison with other affine invariant techniques the new method appears to be suitable for problems where image distortions can be approximated with affine transformations

abelian covers of chiral polytopes abstract abstract polytopes are combinatorial structures with certain properties drawn from the study of geometric structures like the platonic solids and of maps on surfaces of particular interest are the polytopes with maximal possible symmetry subject to certain natural constraints symmetry can be measured by the effect of automorphisms on the flags of the polytope which are maximal chains of elements of increasing rank dimension an abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags such that two flags that differ in one element always lie in different orbits examples of chiral polytopes have been difficult to find and construct in this paper we introduce a new covering method that allows the construction of some infinite families of chiral polytopes with each member of a family having the same rank as the original but with the size of the members of the family growing linearly with one or more of the parameters making up its type schlafli symbol in particular we use this method to construct several new infinite families of chiral polytopes of ranks 3 4 5 and 6

foundations of semi differential invariants this paper elaborates the theoretical foundations of a semi differential framework for invariance semi differential invariants combine coordinates and their derivatives with respect to some contour parameter at several points of the image contour thus allowing for an optimal trade off between identification of points and the calculation of derivatives a systematic way of generating complete and independent sets of such invariants is presented it is also shown that invariance under reparametrisation can be cast in the same framework the theory is illustrated by a complete analysis of 2d affine transformations in a companion paper pauwels et al 1995 these affine semi differential invariants are implemented in the computer program form flat object recognition method for the recognition of planar contours under pseudo perspective projection

review of shape representation and description techniques more and more images have been generated in digital form around the world there is a growing interest in 1nding images in large collections or from remote databases in order to 1nd an image the image has to be described or represented by certain features shape is an important visual feature of an image searching for images using shape features has attracted much attention there are many shape representation anddescription techniques in the literature in this paper we classify and review these important techniques we examine implementation procedures for each technique and discuss its advantages and disadvantages some recent research results are also included and discussed in this paper finally we identify some promising techniques for image retrieval according to standard principles

estimation of general 2d affine motion using fourier descriptors abstract in the present paper a general affine motion estimation algorithm using the fourier descriptors is proposed it is able to estimate in addition to the translation scaling and rotation the stretching parameters or equivalently the scale the shift and the four coefficients of the general affine matrix a modified claire test sequence will serve as a useful test to evaluate the proposed motion algorithm this sequence was selected by international expert groups and was provided by the centre national d etudes des telecommunications cnet france

the fuzzy intersection rule in variational analysis with applications the fuzzy intersection rule for frechet normal cones in asplund spaces was established by mordukhovich and the author using the extremal principle which appears more convenient to apply in some applications in this paper we present a complete discussion of this rule in various aspects we show that the fuzzy intersection rule is another characterization of the asplund property of the space various applications are considered as well in particular a complete set of fuzzy calculus rules for general lower semicontinuous functions are established

automorphism groups of cyclic p gonal pseudo real riemann surfaces abstract in this article we prove that the full automorphism group of a cyclic p gonal pseudo real riemann surface of genus g is either a semidirect product c n c p or a cyclic group where p is a prime 2 and g p 1 2 we obtain necessary and sufficient conditions for the existence of a cyclic p gonal pseudo real riemann surface with full automorphism group isomorphic to a given finite group finally we describe some families of cyclic p gonal pseudo real riemann surfaces where the order of the full automorphism group is maximal and show that such families determine real 2 manifolds embedded in the branch locus of moduli space

on the fuzzy intersection rule abstract the paper provides a complete exposition of the fuzzy intersection rule in variational analysis and its applications to generalized differentiation theory new forms of the rule and further applications are presented

learning affine transformations under the assumption of weak perspective two views of the same planar object are related through an affine transformation in this paper we consider the problem of training a simple neural network to learn to predict the parameters of the affine transformation although the proposed scheme has similarities with other neural network schemes its practical advantages are more profound first of all the views used to train the neural network are not obtained by taking pictures of the object from different viewpoints instead the training views are obtained by sampling the space of affine transformed views of the object this space is constructed using a single view of the object fundamental to this procedure is a methodology based on singular value decomposition svd and interval arithmetic ia for estimating the ranges of values that the parameters of affine transformation can assume second the accuracy of the proposed scheme is very close to that of a traditional least squares approach with slightly better space and time requirements a front end stage to the neural network based on principal components analysis pca shows to increase its noise tolerance dramatically and also to guides us in deciding how many training views are necessary in order for the network to learn a good noise tolerant mapping the proposed approach has been tested using both artificial and real data

metric regularity tangential distances and generalized differentiation in banach spaces abstract in this paper we establish new generalized differentiation rules in general banach spaces regarding normal cones to set images under functions coderivatives of compositions of set valued mappings as well as calculus results for normal compactness of sets and their images in addition to the metric regularity of mappings our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations basic properties as well as relations to similar notions some related results are also established

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

groups psl 3 p and nonorientable regular maps a 2 cell embedding of a graph on a nonorientable closed surface is called regular if its automorphism group acts regularly on its flags this paper gives a classification of nonorientable regular maps with the automorphism groups psl 3 p for a prime p equivalently we determine the representatives of orbits of aut psl 3 p acting on the set of involutory generating triples t r l of psl 3 p such that t l l t

epi lipschitz like sets in banach space theorems and examples on considere un espace de banach x et on etudie quelles proprietes un ensemble c dans x doit avoir pour que le cone tangent de clarke t c x et le cone normal n c x mesurent adequatement le comportement a la frontiere de x dans c

perceptually relevant and piecewise linear matching of silhouettes in this paper a novel alignment method for silhouettes is proposed this method is based on the establishment of correspondences between landmarks on their boundaries and in turn on the establishment of correspondences of the boundary pieces in between these landmarks the method yields more correct correspondences than conventional methods that scale the arc length descriptions of silhouettes to align them the selection of landmarks is investigated as to the robustness of their localization and their perceptual relevance matching of silhouettes is then achieved by quantifying the dissimilarity of a pair of silhouette boundaries based on a novel dissimilarity metric the matching procedure is evaluated based on retrieval experiments and it is concluded that the precision of the results is higher than that obtained by conventional pointwise comparison methods

the pseudo real genus of a group abstract a compact riemann surface is called pseudo real if it admits anti conformal orientation reversing automorphisms but no anti conformal automorphism of order 2 or equivalently if the surface is reflexible but not definable over the reals it is known that there exist pseudo real surfaces of genus g for every integer g 2 and the number of automorphisms of any such surface is bounded above by 12 g 1 in this paper we extend the concepts of symmetric genus strong symmetric genus and symmetric cross cap genus of a group by defining and investigating two new parameters as follows 1 the pseudo real genus g of a finite group g is the smallest genus of those pseudo real surfaces on which g acts faithfully as a group of automorphisms some of which might reverse orientation and 2 the strong pseudo real genus g of g is the smallest genus of those pseudo real surfaces on which g acts faithfully as a group of automorphisms some of which do reverse orientation when there exists such a surface for g our main theorem is that for every integer g 2 there exists a finite group g for which g g g and hence that the range of each of the functions and is the set of all integers g 2 we also give an example of a group g for which g is defined but g g

skew symmetry detection via invariant signatures we propose a new approach to skew symmetry detection based on the theory of invariant signatures for planar objects invariant signatures associated to object boundaries are generalizations of the curvature versus arclength description of curves invariant under geometric transformations more complex than the euclidean ones we show that symmetries of objects and hence of closed boundaries translate into simple structures in the invariant signature functions and are therefore in principle readily detectable

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

variational analysis in nonsmooth optimization and discrete optimal control this paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite dimensional spaces the main focus is on the study of problems with nonsmooth structures which require the usage of advanced tools of generalized differentiation in this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete time inclusions in infinite dimensions the principal difference between finite dimensional and infinite dimensional frameworks of optimization and control consists of the lack of compactness in infinite dimensions which leads to imposing certain normal compactness properties and developing their comprehensive calculus together with appropriate calculus rules of generalized differentiation on the other hand one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems in particular we fully avoid the requirements of this type imposed on target endpoint sets in infinite dimensional optimal control for discrete time inclusions

schwarps locally projective image warps based on 2d schwarzian derivatives image warps or just warps capture the geometric deformation existing between two images of a deforming surface the current approach to enforce a warp s smoothness is to penalize its second order partial derivatives bookstein in ieee trans pattern anal mach intell 11 567 585 1989 rueckert et al in ieee trans med imaging 18 712 721 1999 because this favors locally affine warps this fails to capture the local projective component of the image deformation this may have a negative impact on applications such as image registration and deformable 3d reconstruction we propose a novel penalty designed to smooth the warp while capturing the deformation s local projective structure our penalty is based on equivalents to the schwarzian derivatives which are projective differential invariants exactly preserved by homographies we propose a methodology to derive a set of partial differential equations with only homographies as solutions we call this system the schwarzian equations and we explicitly derive them for 2d functions using differential properties of homographies we name as schwarp a warp which is estimated by penalizing the residual of schwarzian equations experimental evaluation shows that schwarps outperform existing warps in modeling and extrapolation power and lead to better results in three deformable reconstruction methods namely shape reconstruction in shape from template camera calibration in shape from template and non rigid structure from motion

the local projective shape of smooth surfaces and their outlines this article examines projectively invariant local geometric properties of smooth curves and surfaces oriented projective differential geometry is proposed as a general framework for establishing such invariants and characterizing the local projective shape of surfaces and their outlines it is applied to two problems 1 the projective generalization of koenderink s famous characterization of convexities concavities and inflections of the apparent contours of solids bounded by smooth surfaces and 2 the image based construction of rim meshes which provide a combinatorial description of the arrangement induced on the surface of an object by the contour generators associated with multiple cameras observing it

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

the genus of compact riemann surfaces with maximal automorphism group any compact riemann surface with genus g 1 has at most 84 g 1 conformal automorphisms in this paper it is shown that there are just 32 integers g in the range 1 g 11905 for which there exist compact riemann surfaces of genus g with this maximum possible number of automorphisms the automorphism group of each such surface is isomorphic to a quotient n of the 2 3 7 triangle group x y x2 y3 xy 7 1 where n is one of 92 proper normal subgroups of with index less than 106 these normal subgroups are found using elementary group theoretic techniques

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

recognition of planar shapes under affine distortion methods for the recognition ofplanar shapes from arbitrary viewpoints are described the adopted model of projection is orthographic the invariant descriptions derived for this group are one dimensional shape signatures comparable to the well known curvature as a function of arc length description of euclidean geometry since the use of such differential invariants in the affine case would lead to unacceptably high orders of derivatives affine invariant descriptions based onsemi differential invariants are proposed as an alternative a systematic discussion of different types of these invariants is given the usefulness and viability of this methodology is demonstrated on a database containing more than 40 objects

on symmetries of cayley graphs and the graphs underlying regular maps abstract by definition cayley graphs are vertex transitive and graphs underlying regular or orientably regular maps on surfaces are arc transitive this paper addresses questions about how large the automorphism groups of such graphs can be in particular it is shown how to construct 3 valent cayley graphs that are 5 arc transitive in answer to a question by cai heng li and cayley graphs of valency 3 t 1 that are 7 arc transitive for all t 0 the same approach can be taken in considering the graphs underlying regular or orientably regular maps leading to classifications of all such maps having a 1 4 or 5 arc regular 3 valent underlying graph in answer to questions by cheryl praeger and sanming zhou

classification of partial 2 d shapes using fourier descriptors we present a method for the classification of 2 d partial shapes using fourier descriptors we formulate the problem as one of estimating the fourier descriptors of the unknown complete shape from the observations derived from an arbitrarily rotated and scaled shape with missing segments the method used for obtaining the estimates of the fourier descriptors minimizes a sum of two terms the first term of which is a least square fit to the given data subject to the condition that the number of missing boundary points is not known and the second term is the perimeter2 area of the unknown shape experiments with synthetic and real boundaries show that estimates closer to the true values of fourier descriptors of complete boundaries are obtained also classification experiments performed using real boundaries indicate that reasonable classification accuracies are obtained even when 20 30 percent of the data is missing

necessary suboptimality and optimality conditions via variational principles the paper aims to develop some basic principles and tools of nonconvex variational analysis with applications to necessary suboptimality and optimality conditions for constrained optimization problems in infinite dimensions we establish a certain subdifferential variational principle as a new characterization of asplund spaces this result is different from conventional support forms of variational principles and appears to be convenient for applications to nonsmooth optimization based on the subdifferential variational principle we obtain new necessary conditions for suboptimal solutions in general nonsmooth optimization problems with equality inequality and set constraints in asplund spaces in this way we establish the so called sequential normal compactness properties of constraint sets that play an essential role in infinite dimensional variational analysis and its applications as a by product of our approach we derive various forms of necessary optimality conditions for nonsmooth constrained problems in infinite dimensions which extend known results in that direction

robust symbolic representation for shape recognition and retrieval a new method for shape recognition and retrieval is proposed here the suggested algorithm is based on several steps the algorithm analyzes the contour of pairs of shapes their contours are recovered and represented by a pair of n points obtained by linear interpolation given two points p i and q j from the two shapes the cost of their matching is evaluated by using the shape context and by using dynamic programming the best matching between the point sets is obtained dynamic programming not only recovers the best matching but also identifies occlusions i e points in the two shapes which cannot be properly matched given the correspondence between the two point sets the two contours are aligned using procrustes analysis after alignment each contour is transformed into a string of symbols and a modified version of edit distance is used to compute the similarity between strings of symbols finally recognition and retrieval are obtained by a simple nearest neighbor procedure the algorithm has been tested on a large set of shape databases kimia mpeg 7 natural silhouette database gesture database marine database swedish leaf database diatom database eth 80 3d object database providing performances for both in recognition and in retrieval superior to most of previously proposed approaches

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

sequential normal compactness versus topological normal compactness in variational analysis abstract we study relationships between two normal compactness properties of sets in banach spaces that play an essential role in many aspects of variational analysis and its applications particularly in calculus rules for generalized differentiation necessary optimality and suboptimality conditions for optimization problems etc both properties automatically hold in finite dimensional spaces and reveal principal features of the infinite dimensional variational theory similar formulations of these properties involve the weak convergence of sequences and nets respectively containing generalized normal cones in duals to banach spaces we prove that these properties agree for a large class of banach spaces that include weakly compactly generated spaces we also show that they are always different in banach spaces whose unit dual ball is not weak sequentially compact moreover the sequential and topological normal compactness properties may not coincide even in non separable asplund spaces that admit an equivalent c smooth norm

variational analysis of extended generalized equations via coderivative calculus in asplund spaces abstract this paper is devoted to the development of variational analysis and generalized differentiation in the framework of asplund spaces we mainly concern the study of a special class of set valued mapping given in the form s x y y 0 f x y q x y x x where both f and q are set valued mappings between asplund spaces models of this type are associated with solutions maps to the so called extended generalized equations and play a significant role in many aspects of variational analysis and its applications to optimization stability control theory etc in this paper we conduct a local variational analysis of such extended solution maps s and their remarkable specifications based on dual space generalized differential constructions of the coderivative type the major part of our analysis revolves around coderivative calculus largely developed and implemented in this paper and then applied to establishing verifiable conditions for robust lipschitzian stability of extended generalized equations and related objects

affine invariant geometry for non rigid shapes shape recognition deals with the study geometric structures modern surface processing methods can cope with non rigidity by measuring the lack of isometry deal with similarity or scaling by multiplying the euclidean arc length by the gaussian curvature and manage equi affine transformations by resorting to the special affine arc length definition in classical equi affine differential geometry here we propose a computational framework that is invariant to the full affine group of transformations similarity and equi affine thus by construction it can handle non rigid shapes technically we add the similarity invariant property to an equi affine invariant one and establish an affine invariant pseudo metric as an example we show how diffusion geometry can encapsulate the proposed measure to provide robust signatures and other analysis tools for affine invariant surface matching and comparison

surface symmetries and psl_2 p we classify up to conjugacy all orientation preserving actions of psl 2 p on closed connected orientable surfaces with spherical quotients this classification is valid in the topological pl smooth conformal geometric and algebraic categories and is related to the inverse galois problem

upward slopes and inf convolutions nonsmooth analysis and exact penalty methods give much prominence to functions that exit from certain lower level sets with positive upward slope a novel weak version of such slopes is formalized here and studied in terms of differential properties applied to inf convolutions those properties help to recover and extend recent results on nonconvex separation cooperative production or market games and welfare economics

on extendability of group actions on compact riemann surfaces the question of whether a given group g which acts faithfully on a compact riemann surface x of genus g 2 is the full group of automorphisms of x or some other such surface of the same genus is considered conditions are derived for the extendability of the action of the group g in terms of a concrete partial presentation for g associated with the relevant branching data using singerman s list of signatures of fuchsian groups that are not finitely maximal by way of illustration the results are applied to the special case where g is a non cyclic abelian group

an affine symmetric image model and its applications natural images contain considerable redundancy some of which is successfully captured using recently developed directional wavelets in this paper an affine symmetric image model is considered it provides a flexible scheme to exploit geometric redundancy a patch of texture from an image is rotated scaled and sheared to approximate other similar parts in the image revealing the self similarity relation the general scheme is derived as follows a texture model is required that identifies structural patterns then the affine symmetry is exploited between structural textures at a local level the objective being to find the minimum residual error by estimating the affine transform relating two patches of texture having developed a local model the methodology is extended to the whole image to estimate the global affine relation this global model is further developed in a multiresolution framework for multiscale analysis by which the self similarity of the image is exploited across space and scale the multiresolution model can be applied to a series of practical problems experimental evaluation demonstrates the effectiveness of the approach in affine invariant texture segmentation and image approximation

pose estimation of 3d free form contours in this article we discuss the 2d 3d pose estimation problem of 3d free form contours in our scenario we observe objects of any 3d shape in an image of a calibrated camera pose estimation means to estimate the relative position and orientation containing a rotation and translation of the 3d object to the reference camera system the fusion of modeling free form contours within the pose estimation problem is achieved by using the conformal geometric algebra the conformal geometric algebra is a geometric algebra which models entities as stereographically projected entities in a homogeneous model this leads to a linear description of kinematics on the one hand and projective geometry on the other hand to model free form contours in the conformal framework we use twists to model cycloidal curves as twist depending functions and interpret n times nested twist generated curves as functions generated by 3d fourier descriptors this means we use the twist concept to apply a spectral domain representation of 3d contours within the pose estimation problem we will show that twist representations of objects can be numerically efficient and easily be applied to the pose estimation problem the pose problem itself is formalized as implicit problem and we gain constraint equations which have to be fulfilled with respect to the unknown rigid body motion several experiments visualize the robustness and real time performance of our algorithms

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

quasi invariant parameterisations and matching of curves in images in this paper we investigate quasi invariance on a smooth manifold and show that there exist quasi invariant parameterisations which are not exactly invariant but approximately invariant under group transformations and do not require high order derivatives the affine quasi invariant parameterisation is investigated in more detail and exploited for defining general affine semi local invariants from second order derivatives only the new invariants are implemented and used for matching curve segments under general affine motions and extracting symmetry axes of objects with 3d bilateral symmetry

application of affine invariant fourier descriptors to recognition of 3 d objects the method of fourier descriptors is extended to produce a set of normalized coefficients which are invariant under any affine transformation translation rotation scaling and shearing the method is based on a parameterized boundary description which is transformed to the fourier domain and normalized there to eliminate dependencies on the affine transformation and on the starting point invariance to affine transforms allows considerable robustness when applied to images of objects which rotate in all three dimensions as is demonstrated by processing silhouettes of aircraft maneuvering in three space