curvature weighted gradient based shape orientation determining the orientation of a shape is a common task in many image processing applications it is usually part of the image preprocessing stages and further processing may rely on an adequate method to determine the orientation there are several methods for computing the orientation of a shape each of them with its own strengths and weaknesses a method which performs outstandingly for one application may have a poor performance for a different application in this paper we present a new method for computing shape orientation based on the projection of the tangent vectors of a shape onto a line and weighting them using a function of the curvature some of the results from zunic 2008 14 are particular cases of the results presented here

symmetry detection by generalized complex gc moments a close form solution this paper presents a unified method for detecting both reflection symmetry and rotation symmetry of 2d images based on an identical set of features i e the first three nonzero generalized complex gc moments this method is theoretically guaranteed to detect all the axes of symmetries of every 2d image if more nonzero gc moments are used in the feature set furthermore we establish the relationship between reflectional symmetry and rotational symmetry in an image which can be used to check the correctness of symmetry detection this method has been demonstrated experimentally using more than 200 images

boundary based shape orientation the computation of a shape s orientation is a common task in the area of computer vision and image processing being used for example to define a local frame of reference and is helpful for recognition and registration robot manipulation etc it is usually an initial step or a part of data preprocessing in many image processing and computer vision tasks thus it is important to have a good solution for shape orientation because an unsuitable solution could lead to a big cumulative error at the end of the computing process there are several approaches to the problem most of them could be understood as the area based ones or at least they do not take into account all the boundary points if a shape orientation measure is based on its encasing rectangle only the convex hull points count for example thus the demand for a pure boundary based method where the orientation of the shape is dependent on the boundary points seems to be very reasonable such a method is presented in this paper we are initially focused on the shapes having polygonal boundaries we define the orientation of a polygonal shape by the line that maximises the total sum of squared lengths of all the boundary edge projections onto this line the advantages and limitations of the new method are analysed next we suggested how the method can be adapted in order to be applicable to a wider class than the initial method is finally we introduced another modification of the method in such a way that the modified method can be applied to shapes with arbitrary boundaries several illustrative experiments are provided

universal principal axes an easy to construct tool useful in defining shape orientations for almost every kind of shape abstract a convenient tool for defining shape orientations called universal principal axes is introduced these axes are formed of half lines starting from the shape centroid with their directional angles expressed as functions of the polar angle of the first non zero complex number encountered in the sequence x iy 1 d a with 1 2 3 4 provided that the integration domain is taken to be the given two dimensional 2d shape and the origin is assumed to be the shape centroid universal principal axes are shown to be independent of the translation scaling and rotation of the coordinate system used and they are therefore qualified in defining shape orientations a major benefit of using the proposed universal principal axes is that they exist for almost every kind of shape another benefit is that there is no need to judge in advance whether a given shape is mirror symmetric rotationally symmetric irregular etc moreover in the case of rotationally symmetric shapes the universal principal axes make the preprocessing procedure for detecting the number of folds contained in the given shape unnecessary although such procedure is essential for many existing tools designed to define orientations of rotationally symmetric shapes defining shape orientations by universal principal axes is therefore quite convenient an algorithm describing the construction of universal principal axes and several examples showing the detected universal principal axes for some shapes of distinct types are given the relationship between universal and conventional principal axes is also discussed

generalized affine invariant image normalization we provide a generalized image normalization technique which basically solves all problems in image normalization the orientation of any image can be uniquely defined by at most three non zero generalized complex moments the correctness of our method is demonstrated theoretically as well as in practice by applying them to a number of degenerate images which have failed other previously reported techniques for image normalization

complete invariants for robust face recognition in this paper we propose two complete sets of similarity invariant descriptors under the fourier mellin transform and the analytical fourier mellin transform afmt frameworks respectively the magnitude and phase spectra are jointly processed in our case and the presented invariants are complete and can be used to reconstruct the image their numerical properties are also revealed through image reconstruction in order to simplify the invariant feature data for recognition and discrimination a 2d pca approach is incorporated into the presented complete invariant descriptor the obtained compact representation through the 2d pca preserves the essential structure of the objects in an image we tested this compact form on the orl yale and bioid face databases for experimental verification and achieved a face verification under similarity transforms with a much inferior equal error rate eer compared to when the 2d pca based compact form is used without complete invariants

normalization of rotationally symmetric shapes for pattern recognition abstract image normalization is useful for image recognition by the matching method recently a normalization algorithm was developed to normalize images under the distortion of translation rotation scaling and skew however the method fails when the image is rotationally symmetric a method is introduced called the modified fourier descriptor to normalize rotationally symmetric shapes this method is simple and fast and can be combined with the preceding normalization algorithm to make it more general and realizable

robust detection of skewed symmetries by combining local and semi local affine invariants abstract affine invariant feature vector ip and shen image vision comput 16 2 1998 135 146 that captures both local and semi local geometric features around each point of the object boundary is applied here for the detection of skewed symmetries based on the affine invariant shape representation the problem of detecting symmetry axes has been formulated as a problem of detecting lines with known orientations in a local similarity matrix of an object since the feature vector extracts sufficient local and semi local shape information for every point along the object boundary the process of checking symmetric point pairs is thus robust against both noises and deformations moreover our technique is able to detect all the local reflectional symmetries contained in the object various experimental results have shown the robustness and effectiveness of our method in detecting skewed symmetries from both self symmetric objects and generalized objects

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

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

hammer hierarchical attribute matching mechanism for elastic registration a new approach is presented for elastic registration of medical images and is applied to magnetic resonance images of the brain experimental results demonstrate very high accuracy in superposition of images from different subjects there are two major novelties in the proposed algorithm first it uses an attribute vector i e a set of geometric moment invariants gmis that are defined on each voxel in an image and are calculated from the tissue maps to reflect the underlying anatomy at different scales the attribute vector if rich enough can distinguish between different parts of an image which helps establish anatomical correspondences in the deformation procedure it also helps reduce local minima by reducing ambiguity in potential matches this is a fundamental deviation of our method referred to as the hierarchical attribute matching mechanism for elastic registration hammer from other volumetric deformation methods which are typically based on maximizing image similarity second in order to avoid being trapped by local minima i e suboptimal poor matches hammer uses a successive approximation of the energy function being optimized by lower dimensional smooth energy functions which are constructed to have significantly fewer local minima this is achieved by hierarchically selecting the driving features that have distinct attribute vectors thus drastically reducing ambiguity in finding correspondence a number of experiments demonstrate that the proposed algorithm results in accurate superposition of image data from individuals with significant anatomical differences

automatic symmetry determination and normalization for rotationally symmetric 2d shapes and 3d solid objects abstract determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision application however the usual normalization methods are inapplicable when the shape is n fold rotationally symmetric n 3 in this paper an efficient and reliable new method using modified fourier transform is introduced to normalize such a 2d rotationally symmetric shape furthermore we extend this new method to the normalization of a n fold axial rotationally symmetric 3d object there are two obvious advantages when using this new method 1 it prevents the high computation load and the difficulty in normalizing the shape s orientation which occurred in the old methods using full matching or correlation functions and 2 the weakness introduced by the old methods using data reduction from 2d data to 1d data is remedied experiments will show the validity of our method

revisiting complex moments for 2 d shape representation and image normalization when comparing 2 d shapes a key issue is their normalization translation and scale are easily taken care of by removing the mean and normalizing the energy however defining and computing the orientation of a 2 d shape is not so simple in fact although for elongated shapes the principal axis can be used to define one of two possible orientations there is no such tool for general shapes as we show in the paper previous approaches fail to compute the orientation of even noiseless observations of simple shapes we address this problem and show how to uniquely define the orientation of an arbitrary 2 d shape in terms of what we call its principal moments we start by showing that a small subset of these moments suffices to describe the underlying 2 d shape i e that they form a compact representation which is particularly relevant when dealing with large databases then we propose a new method to efficiently compute the shape orientation principal moment analysis finally we discuss how this method can further be applied to normalize gray level images besides the theoretical proof of correctness we describe experiments demonstrating robustness to noise and illustrating with real images

invariant pattern recognition a review in this document we review and compare some of the classical and modern techniques for solving the problem of invariant pattern recognition such techniques include integral transforms construction of algebraic moments and the use of structured neural networks in all cases we assume that the nature of the invariance group is known a priori many of the methods described apply to specific geometrical transformation groups however some of the techniques are highly general and applicable to large classes of invariance groups we also review some results regarding the existence and structure of invariants under certain kinds of groups

a simple way to compute the existence region of 1d chaotic attractors in 2d maps abstract we study a new approach to computing the existence region of self synchronized states in the error feedback synchronization scheme for discrete time chaotic systems the overall system is modelled by a triangular 2d map in which the nonlinearity is a symmetric tent map we compute the stability region d c of one dimensional chaotic attractors using the boundaries in the bifurcation diagram of the 1d symmetric tent map also called critical lines we show that with a modest number of these critical lines about 16 the approximation of d c is quite accurate

digital watermarking robust to geometric distortions in this paper we present two watermarking approaches that are robust to geometric distortions the first approach is based on image normalization in which both watermark embedding and extraction are carried out with respect to an image normalized to meet a set of predefined moment criteria we propose a new normalization procedure which is invariant to affine transform attacks the resulting watermarking scheme is suitable for public watermarking applications where the original image is not available for watermark extraction the second approach is based on a watermark resynchronization scheme aimed to alleviate the effects of random bending attacks in this scheme a deformable mesh is used to correct the distortion caused by the attack the watermark is then extracted from the corrected image in contrast to the first scheme the latter is suitable for private watermarking applications where the original image is necessary for watermark detection in both schemes we employ a direct sequence code division multiple access approach to embed a multibit watermark in the discrete cosine transform domain of the image numerical experiments demonstrate that the proposed watermarking schemes are robust to a wide range of geometric attacks

a unifying approach to moment based shape orientation and symmetry classification in this paper the problem of moment based shape orientation and symmetry classification is jointly considered a generalization and modification of current state of the art geometric moment based functions is introduced the properties of these functions are investigated thoroughly using fourier series analysis and several observations and closed form solutions are derived we demonstrate the connection between the results presented in this work and symmetry detection principles suggested from previous complex moment based formulations the proposed analysis offers a unifying framework for shape orientation symmetry detection in the context of symmetry classification and matching the second part of this work presents a frequency domain method aiming at computing a robust moment based feature set based on a true polar fourier representation of image complex gradients and a novel periodicity detection scheme using subspace analysis the proposed approach removes the requirement for accurate shape centroid estimation which is the main limitation of moment based methods operating in the image spatial domain the proposed framework demonstrated improved performance compared to state of the art methods

detection of generalized principal axes is rotationally symmetric shapes abstract automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper the inapplicability of a well known equation which is used to compute the direction of the principal axis of a given shape using moment functions to a class of so called degenerate shapes is first pointed out rotationally symmetric shapes often encountered in real applications are shown to belong to this class this problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions analytic equations for computing the direction of high order principal axes are derived they include the well known equation for computing the direction of the second order principal axis as a special case some experimental results are included finally to show the effectiveness of the derived analytic equations

image registration by super curves we solve the 2 d affine image registration problem by curve matching and alignment our approach starts with a super curve which is formed by superimposing two affine related curves in one coordinate system we use b spline fusion technique to find a single b spline approximation of the super curve and a registration between the two curves simultaneously this approach achieves superior accuracy and efficiency in curve matching and alignment we then address the occlusion problem by finding partial match between the curves segmented using inflections and cusps which are affine invariant the combination of edge detection and curve alignment lead to accurate image registration

brief review of invariant texture analysis methods this paper considers invariant texture analysis texture analysis approaches whose performances are not affected by translation rotation affine and perspective transform are addressed existing invariant texture analysis algorithms are carefully studied and classified into three categories statistical methods model based methods and structural methods the importance of invariant texture analysis is presented first each approach is reviewed according to its classification and its merits and drawbacks are outlined the focus of possible future work is also suggested

affine permutation invariance of 2 d shapes shapes provide a rich set of clues on the identity and topological properties of an object in many imaging environments however the same object appears to have different shapes due to distortions such as translation rotation reflection scaling or skewing further the order by which the object s feature points are scanned changes i e the order of the pixels may be permuted relating two dimensional shapes of the same object distorted by different affine and permutation transformations is a challenge we introduce a shape invariant that we refer to as the intrinsic shape of an object and describe an algorithm blaiser to recover it the intrinsic shape is invariant to affine permutation distortions it is a uniquely defined representative of the equivalence class of all affine permutation distortions of the same object blaiser computes the intrinsic shape from any arbitrarily affine permutation distorted image of the object without prior knowledge regarding the distortions or the undistorted shape of the object the critical step of blaiser is the determination of the shape orientation and we provide a detailed discussion on this topic the operations of blaiser are based on low order moments of the input shape and thus robust to error and noise examples illustrate the performance of the algorithm

on the detection of the axes of symmetry of symmetric and almost symmetric planar images a simple algorithm is presented for finding all the axes of symmetry of symmetric and almost symmetric planar images having nonuniform gray level intensity images the technique which can also be used in conjunction with planar curves is based on the identification of the centroids of the given image and other related sets of points followed by a maximization of a specially defined coefficient of symmetry owing to the nature of the required procedures which are strictly based on the computation of mean values the method is not particularly affected by the presence of noise and is especially suited for application in conjunction with digitized figures

notes on shape orientation where the standard method does not work in this paper we consider some questions related to the orientation of shapes with particular attention to the situation where the standard method does not work there are irregular and non symmetric shapes whose orientation cannot be computed in a standard way but in the literature the most studied situations are those where the shape under consideration has more than two axes of symmetry or where it is an n fold rotationally symmetric shape with n 2 the basic reference for our work is w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 we give a very simple proof of the main result from w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 and suggest a modification of the proposal on how the principal axes of rotationally symmetric shapes should be computed we show some desirable property in defining the orientation of such shapes if the modified approach is applied also we give some comments on the problems that arise when computing shape elongation

detection of rotationally symmetric shape orientations by fold invariant shape specific points abstract two new types of shape specific points called fold invariant centroid fic and fold invariant radius weighted mean firwm are introduced for detecting the orientations of rotationally symmetric shapes a given rotationally symmetric shape is first transformed into a new fold invariant shape the centroid and the radius weighted mean of the new shape are then transformed back to obtain the proposed fic and firwm respectively fic and firwm points are proved to be invariant to translation rotation and scaling and so can be used for matching rotationally symmetric shapes using the proposed fold invariant shape specific points to detect the orientations of rotationally symmetric shapes is easier to implement than that of using the fold principal axes some illustrative examples and a method to avoid the case where the proposed fold invariant shape specific points are too close to the centroid are also given

automatic rotational symmetry determination for shape analysis abstract the determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications in this paper a simple but effective algorithm to analyse the rotational symmetry of a given closed curve shape s is proposed a circle c with the centroid of s as the circle center and the average radius of s as the circle radius is superimposed on s resulting in the intersection of c and s at a set of points by theoretical analysis the relationship between the order n s of the rotational symmetry of s and the number of intersection points between c and s is established all the possible values for n s are determined and finally n s is determined by evaluating the similarity between s and its rotated versions in the proposed algorithm only simple pixel operations and second order moment function computation are involved several problems caused by the use of discrete image coordinates are analysed and solved for correct decision making in the algorithm good experimental results are also included