efficient parallel algorithms for the computation of two dimensional image moments abstract moments constitute an important set of parameters for image analysis the low order moments contain significant information about a simple object they have been used in finding the location and orientation of an object moment invariants have been used as features for pattern recognition to compute moments of a two dimensional image a large number of multiplications and additions are required in a direct approach multiplications which are the most time consuming operations in simple processors can be completely avoided in the proposed algorithms for low order moments in this paper parallel algorithms are proposed for efficient implementation in processor arrays the basic idea is to decompose a 2 d moment into many vertical moments and a horizontal moment and to use the data parallelism for the vertical moments and the task parallelism for the horizontal moment

2d affine invariant contour matching using b spline model this paper presents a new affine invariant matching algorithm based on b spline modeling which solves the problem of the nonuniqueness of b spline in curve matching this method first smoothes the b spline curve by increasing the degree of the curve it is followed by a reduction of the curve degree using the least square error lse approach to construct the curvature scale space css image css matching is then carried out our method combines the advantages of b spline that are continuous curve representation and the robustness of css matching with respect to noise and affine transformation it avoids the need for other matching algorithms that have to use the resampled points on the curve thus the curve matching error is reduced the proposed algorithm has been tested by matching similar shapes from a prototype database the experimental results showed the robustness and accuracy of the proposed method in b spline curve matching

synthesized affine invariant function for 2d shape recognition by defining the weighted wavelet synthesis the synthesized feature signals of an interesting shape are extracted to derive the innovative synthesized affine invariant function saif the synthesized feature signals hold the shape information with minimum loss by excluding simply the translation dependent and noise contaminated bands the saif is shown excellent in the invariance property and representative in describing the original shape for automated recognition experimental results demonstrate that automated shape recognition based on the saif achieves high correctness and significantly outperforms those using conventional wavelet affine invariant functions

the revised fundamental theorem of moment invariants a revised fundamental theorem of moment invariants for pattern recognition which corrects the fundamental theorem proposed by m k hu 1962 is presented the correction affects neither similitude scale nor rotation invariants derived using the original theorem but it does affect features invariant to general linear transformations four of the latter invariants were presented originally by hu these are revised to take the correction to the fundamental theorem into account furthermore these four invariants are combined to yield three new invariants which are additionally invariant to changes in the illumination of an image

an efficient evolutionary algorithm for accurate polygonal approximation an optimization problem for polygonal approximation of 2 d shapes is investigated in this paper the optimization problem for a digital contour of n points with the approximating polygon of k vertices has a search space of c n k instances i e the number of ways of choosing k vertices out of n points a genetic algorithm based method has been proposed for determining the optimal polygons of digital curves and its performance is better than that of several existing methods for the polygonal approximation problems this paper proposes an efficient evolutionary algorithm eea with a novel orthogonal array crossover for obtaining the optimal solution to the polygonal approximation problem it is shown empirically that the proposed eea outperforms the existing genetic algorithm based method under the same cost conditions in terms of the quality of the best solution average solution variance of solutions and the convergence speed especially in solving large polygonal approximation problems

cross weighted moments and affine invariants for image registration and matching a framework for deriving a class of new global affine invariants for both object matching and positioning based on a novel concept of cross weighted moments with fractional weights is presented the fractional weight factor allows for a more flexible range to balance between the capability to discriminate between objects that differ only in small shape details and the sensitivity of small shape details to the presence of the noise moreover it makes it possible to arrive at low order zero order affine invariants that are more robust than those derived from higher order regular moments the affine transformation parameters are recovered from the zero and the first order cross weighted moments without requiring any feature point correspondence information the equations used to find the affine transformation parameters are linear algebraic the sensitivity of the cross weighted moment invariants to noise missing data and perspective effects is shown on real images

pattern recognition by affine moment invariants abstract the paper deals with moment invariants which are invariant under general affine transformation and may be used for recognition of affine deformed objects our approach is based on the theory of algebraic invariants the invariants from second and third order moments are derived and shown to be complete the paper is a significant extension and generalization of recent works several numerical experiments dealing with pattern recognition by means of the affine moment invariants as the features are described

cartan subalgebras in c algebras existence and uniqueness we initiate the study of cartan subalgebras in c algebras with a particular focus on existence and uniqueness questions for homogeneous c algebras these questions can be analyzed systematically using the theory of fiber bundles for group c algebras while we are able to find cartan subalgebras in c algebras of many connected lie groups there are classes of discrete groups for instance non abelian free groups whose reduced group c algebras do not have any cartan subalgebras moreover we show that uniqueness of cartan subalgebras usually fails for classifiable c algebras however distinguished cartan subalgebras exist in some cases for instance in nuclear uniform roe algebras

fast computation of moment invariants abstract the subject of moment invariants has found wide application in pattern recognition since it was proposed in the present paper we propose a fast algorithm for the calculation of moment invariants firstly we propose using green s theorem for double integral to transform moment calculation to a simple integration along the boundary which reduces the complexity from o n 2 to o n secondly the relation between the monomials for successive pixels along a curve is analysed and the pascal triangle matrix transform is proposed to calculate the monomials for a pixel from those for the preceding one an iterative algorithm is thus proposed for moment calculation which needs no multiplications and the number of additions needed is reduced to o n moreover a very simple systolic structure is proposed to implement the pascal triangle transformation and the proposed algorithm can thus be realized by simple hardware to further accelerate the calculation comparison of the computational complexity with some known methods is also given which shows that our algorithm significantly reduces the complexity

improved moment invariants for shape discrimination abstract moment invariants have been frequently used as features for shape recognition they are computed based on the information provided by both the shape boundary and its interior region although several fast algorithms for computing traditional moment invariants have been proposed none has ever shown the theoretical results of moment invariants computed based on the shape boundary only this paper proposes improved moment invariants computed using the shape boundary only which tremendously reduces computations the new moment invariants called improved moment invariants are mathematically proved to be invariant to scaling translation and rotation graphical plots of the first two improved moment invariants for six country maps and four industrial tools using improved moment invariants are also given the results suggest that improved moment invariants can be used as effective features for shape discrimination or recognition

image normalization by complex moments the role of moments in image normalization and invariant pattern recognition is addressed the classical idea of the principal axes is analyzed and extended to a more general definition the relationship between moment based normalization moment invariants and circular harmonics is established invariance properties of moments as opposed to their recognition properties are identified using a new class of normalization procedures the application of moment based normalization in pattern recognition is demonstrated by experiment

friedland hersonsky problem for matrix algebra abstract in this paper we show that the answer to problem 3 9 in duke math j 69 1993 593 is positive when n 2 and negative when n 3

simple and fast computation of moments abstract in this paper we address the problem of efficient computation of moments from the boundary of a digital area boundary based computation is superior to usual region based approaches as the data dimension of boundary representations is substantially smaller than that of region representations we investigate the inter order relationship of moments one of our results is that moments of higher order can be computed from moments of lower order based on this relationship a simple iterative algorithm is proposed for the computation of moments from a polygonal approximation of the boundary in comparison with a direct computation method our algorithm is simpler to program the memory requirement is minimum simulation results show that a speed up of factor 8 can be achieved using our algorithm a special version of the algorithm can be utilized to compute moments from the run length chain code of the boundary our algorithm can be applied to compute the most popular geometric moments as well as other types of moments like legendre zernike rotational and complex moments

robust automatic selection of optimal views in multi view free form object recognition this paper addresses the issue of automatic selection of the best and the optimum number of representative views for each object in a database that can enable the accurate recognition of that object from any single arbitrary view of the object the object boundary in each view is represented by its curvature scale space css image the css representation has been selected for mpeg 7 standardisation as a contour shape descriptor the paper also presents a novel method for fusion of results from combined shape descriptors the utilisation of this method for multi view three dimensional 3 d object recognition has been explored the object boundary of each view is represented effectively using the css technique moment invariants and fourier descriptors it has been shown that the results obtained from the fusion method are superior to the results obtained from any single technique the method has been tested on a collection of free form 3 d objects each object has been modelled using an optimal number of silhouette contours obtained from different viewpoints this number varies depending on the complexity of the object and the measure of expected accuracy a comprehensive analysis of the performance of the system has been given

the use of force histograms for affine invariant relative position description affine invariant descriptors have been widely used for recognition of objects regardless of their position size and orientation in space examples of color texture and shape descriptors abound in the literature however many tasks in computer vision require looking not only at single objects or regions in images but also at their spatial relationships in an earlier work we showed that the relative position of two objects can be quantitatively described by a histogram of forces here we study how affine transformations affect this descriptor the position of an object with respect to another changes when the objects are affine transformed we analyze the link between 1 the applied affinity 2 the relative position before transformation described through a force histogram and 3 the relative position after transformation we show that any two of these elements allow the third one to be recovered moreover it is possible to determine whether or how well two relative positions are actually related through an affine transformation if they are not the affinity that best approximates the unknown transformation can be retrieved and the quality of the approximation assessed

matching shapes with self intersections application to leaf classification we address the problem of two dimensional 2 d shape representation and matching in presence of self intersection for large image databases this may occur when part of an object is hidden behind another part and results in a darker section in the gray level image of the object the boundary contour of the object must include the boundary of this part which is entirely inside the outline of the object the curvature scale space css image of a shape is a multiscale organization of its inflection points as it is smoothed the css based shape representation method has been selected for mpeg 7 standardization we study the effects of contour self intersection on the curvature scale space image when there is no self intersection the css image contains several arch shape contours each related to a concavity or a convexity of the shape self intersections create contours with minima as well as maxima in the css image an efficient shape representation method has been introduced in this paper which describes a shape using the maxima as well as the minima of its css contours this is a natural generalization of the conventional method which only includes the maxima of the css image contours the conventional matching algorithm has also been modified to accommodate the new information about the minima the method has been successfully used in a real world application to find for an unknown leaf similar classes from a database of classified leaf images representing different varieties of chrysanthemum for many classes of leaves self intersection is inevitable during the scanning of the image therefore the original contributions of this paper is the generalization of the curvature scale space representation to the class of 2 d contours with self intersection and its application to the classification of chrysanthemum leaves

fast algorithm for the computation of moment invariants abstract moment invariants have been used as feature descriptors in a variety of object recognition applications when assuming a continuous image function moments calculated using a double integral formulation are invariant to variations in translation rotation and size of the object however due to the recursive nature of the calculations and the limited speed of microprocessors the moments were not computable in real time in this paper we present fast invariant moment computations using the delta method as a means of scene representation this method computes moments of contiguous images by using the contribution of each line rather than individual pixel

affine curve moment invariants for shape recognition this paper presents a method of affine curve moment invariants for shape recognition the proposed method extends affine moment invariants from an area domain to a curve domain first a new type of curve moments is defined on a parameterized boundary description of an object a set of affine moment invariants are then derived based on the theory of algebraic invariants the proposed moment invariants only deal with pixels on a shape boundary and are mathematically proven to be invariant under arbitrary affine transformations furthermore after a parameterized object boundary description is acquired through curve fitting the newly defined curve moments can be represented by an integration of the fitting function so that the computation of moment invariants can be obtained from either one by one computation of boundary pixels or a direct integration of the fitting function which makes the computation efficient while at the same time reducing noise effect affine curve moment invariants work well in recognition of affine deformed objects which is demonstrated by processing various 2 d shapes under affine transformations

3 d moment forms their construction and application to object identification and positioning the 3 d moment method is applied to object identification and positioning a general theory of deriving 3 d moments invariants is proposed the notion of complex moments is introduced complex moments are defined as linear combinations of moments with complex coefficients and are collected into multiplets such that each multiplet transforms irreducibly under 3 d rotations the application of the 3 d moment method to motion estimation is also discussed using group theoretic techniques various invariant scalars are extracted from compounds of complex moments via clebsch gordon expansion twelve moment invariants consisting of the second order and third order moments are explicitly derived based on a perturbation formula it is shown that the second order moment invariants can be used to predict whether the estimation using noisy data is reliable or not the new derivation of vector forms also facilities the calculation of motion estimation in a tensor approach vectors consisting of the third order moments can be derived in a similar manner

geometric moment invariants this paper derives moment invariants in an intuitive way by multiple integrals of invariant geometric primitives like distance area and volume many existing 2 d and 3 d moment invariants are re expressed to prove the correctness of this method furthermore this construction method can be easily extended to higher dimensional space and get higher order moment invariants explicit expressions of 3 d moment invariants are generated automatically with the help of symbolic computation software and can be regarded as a kind of shape descriptor for the representation of solid objects

shape similarity retrieval under affine transforms the maxima of curvature scale space css image have already been used to represent 2 d shapes in different applications the representation has shown robustness under the similarity transformations scaling orientation changes translation and even noise can be easily handled by the representation and its associated matching algorithm in this paper we examine the robustness of the representation under general affine transforms we have a database of 1100 images of marine creatures the contours in this database demonstrate a great range of shape variation a database of 5000 contours has been constructed using 500 real object boundaries and 4500 contours which are the affine transformed versions of real objects the css representation is then used to find similar shapes from this prototype database the results provide substantial evidence of stability of the css image and its contour maxima under affine transformation the method is also evaluated objectively through a large classified database and its performance is compared with the performance of two well known methods namely fourier descriptors and moment invariants the css shape descriptor has been selected for mpeg 7 standardization

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

applications of tensor theory to object recognition and orientation determination a method is developed by which images resulting from orthogonal projection of rigid planar patch objects arbitrarily oriented in three dimensional 3 d space may be used to form systems of linear equations which are solved for the affine transform relating the images the technique is applicable to complete images and to unlabeled feature sets derived from images and with small modification may be used to transform images of unknown objects such that they represent images of those objects from a known orientation for use in object identification no knowledge of point correspondence between images is required theoretical development of the method and experimental results are presented the method is shown to be computationally efficient requiring o n multiplications and additions where depending on the computation algorithm n may equal the number of object or edge picture elements

bilipschitz equivalence is not equivalent to quasi isometric equivalence for finitely generated groups we show that certain lamplighter groups that are quasi isometric to each other are not bilipschitz equivalent this gives a positive answer to a question of de la harpe

affine invariants of convex polygons in this correspondence we prove that the affine invariants for image registration and object recognition proposed recently by yang and cohen see ibid vol 8 no 7 p 934 46 july 1999 are algebraically dependent we show how to select an independent and complete set of the invariants the use of this new set leads to a significant reduction of the computing complexity without decreasing the discrimination power

image registration and object recognition using affine invariants and convex hulls this paper is concerned with the problem of feature point registration and scene recognition from images under weak perspective transformations which are well approximated by affine transformations and under possible occlusion and or appearance of new objects it presents a set of local absolute affine invariants derived from the convex hull of scattered feature points e g fiducial or marking points corner points inflection points etc extracted from the image the affine invariants are constructed from the areas of the triangles formed by connecting three vertices among a set of four consecutive vertices quadruplets of the convex hull and hence do make direct use of the area invariance property associated with the affine transformation because they are locally constructed they are very well suited to handle the occlusion and or appearance of new objects these invariants are used to establish the correspondences between the convex hull vertices of a test image with a reference image in order to undo the affine transformation between them a point matching approach for recognition follows this the time complexity for registering l feature points on the test image with n feature points of the reference image is of order o n spl times l the method has been tested on real indoor and outdoor images and performs well

affine similar shape retrieval application to multiview 3 d object recognition the feasibility of representing a three dimensional 3 d object with a small number of standard views is studied the object boundary of each view is considered as a two dimensional 2 d shape and is represented by the locations of the maxima of its curvature scale space css image contours the idea is to identify an unknown object from an image taken from a random view by using the stored descriptions of the standard views the css image has been selected for mpeg 7 standardization the maxima of css image have already been used to represent 2 d shapes in different applications under similarity transforms since the new application involves affine transforms we first examine the effects of general affine transforms on the representation and show that the locations of the maxima of the css image do not move dramatically even under large affine transformations our system for shape based retrieval from large image databases is then applied to multiview 3 d object representation and recognition our collection of 3 d objects consists of 18 aircraft of different shapes three silhouette contours corresponding to random views are separately used as input for each object results indicate that robust and efficient 3 d free form object recognition through multiview representation can be achieved using the css representation

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