high order moment computation of gray level images describes an efficient approach to calculate geometric moments of a 2 d gray level image it is shown both theoretically and experimentally that the new method compares favorably with previous techniques especially for high order moments

two dimensional local moment surface fitting and their fast computation abstract it is proved that local moments are related to their surface fitting coefficients by a linear transform and by this the surface fitting based feature extraction is converted into the computation of 2d local moments a npt recursive method is proposed to compute 2d local moments quickly and then to efficiently implement surface fitting the new method has a novel advantage that its computational complexity for both 2d local moment computation and surface fitting is independent of the mask size which much efficiently speed up the feature extraction

limit set trichotomy and dichotomy for skew product semiflows with applications abstract the limit set trichotomy and dichotomy for skew product semiflows are established under suitable monotone and concave assumptions as an application a delayed hopfield type neural network model is considered

multiresolution moment filters theory and applications we introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales when the window function satisfies a two scale relation we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet like algorithm we show that b splines are well suited window functions because in addition to being refinable they are positive symmetric separable and very nearly isotropic gaussian shape we present three applications of these multiscale local moments the first is a feature extraction method for detecting and characterizing elongated structures in images the second is a noise reduction method which can be viewed as a multiscale extension of savitzky golay filtering the third is a multiscale optical flow algorithm that uses a local affine model for the motion field extending the lucas kanade optical flow method the results obtained in all cases are promising

orthogonal moments based on exponent functions exponent fourier moments abstract in this paper we propose a new set of orthogonal moments based on exponent functions named exponent fourier moments efms which are suitable for image analysis and rotation invariant pattern recognition compared with zernike polynomials of the same degree the new radial functions have more zeros and these zeros are evenly distributed this property make efms have strong ability in describing image unlike zernike moments the kernel of computation of efms is extremely simple theoretical and experimental results show that exponent fourier moments perform very well in terms of image reconstruction capability and invariant recognition accuracy in noise free noisy and smooth distortion conditions the exponent fourier moments can be thought of as generalized orthogonal complex moments

a general approach to moment calculation for polygons and line segments abstract the calculation of moments from the boundary of a solid digital area is a quite useful technique used in pattern recognition a straightforward way of doing the calculation that does not have any numerical instabilities is presented all the symmetries available in moment calculations are preserved and it is shown how different moment formulas are related to each other this includes relations between different moments and the relationships among different methods for calculating moments finally a direct relation between the moment expansion of a polygonal area and a stick figure made from one dimensional lines embedded in a two dimensional plane is derived

invariance of stereo images via the theory of complex moments abstract for an approximate epipolar registration of stereo images the transformation between the images can be approximated from an affine one to a rotational and translational one this property can be used to find the point of correspondences of stereo images in conjunction with complex moments which are themselves rotational invariant corners are chosen as features and around them an intensity kernel is defined and complex moments are calculated then the correspondences are found from similar corners by finding the l2 norm of the invariances between the two images from the matched points the respective affine models of the images are also created

normalization and shape recognition of three dimensional objects by 3d moments abstract in this paper we are primarily concerned with the recognition of three dimensional 3d objects on the basis of the geometry of their physical surface the 3d moments are defined on the object s surface with two main aims 1 to derive a normalized version irrespective of position size and orientation for each imaged object and 2 to establish global descriptors extracted from normalized shapes in order to make up a representative feature vector to be used in recognition tasks the source data are multiple orthographic views of 3d objects with known viewpoint specifications a volume intersection procedure is used in the recovery of 3d object surface from two dimensional 2d data the problem of normalization to achieve recognition of 3d objects is dealt with and a heuristic solution is proposed to lessen the inherent ambiguity of the principal axes method for object orientation experimental results are described with ten object categories showing excellent percentage classification success rates by using only a small number of normalized moments as elements of the feature vector

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

recognition by symmetry derivatives and the generalized structure tensor we suggest a set of complex differential operators that can be used to produce and filter dense orientation tensor fields for feature extraction matching and pattern recognition we present results on the invariance properties of these operators that we call symmetry derivatives these show that in contrast to ordinary derivatives all orders of symmetry derivatives of gaussians yield a remarkable invariance they are obtained by replacing the original differential polynomial with the same polynomial but using ordinary coordinates x and y corresponding to partial derivatives moreover the symmetry derivatives of gaussians are closed under the convolution operator and they are invariant to the fourier transform the equivalent of the structure tensor representing and extracting orientations of curve patterns had previously been shown to hold in harmonic coordinates in a nearly identical manner as a result positions orientations and certainties of intricate patterns e g spirals crosses parabolic shapes can be modeled by use of symmetry derivatives of gaussians with greater analytical precision as well as computational efficiency since gaussians and their derivatives are utilized extensively in image processing the revealed properties have practical consequences for local orientation based feature extraction the usefulness of these results is demonstrated by two applications 1 tracking cross markers in long image sequences from vehicle crash tests and 2 alignment of noisy fingerprints

n dimensional moment invariants and conceptual mathematical theory of recognition n dimensional solids the proof of the generalized fundamental theorem of moment invariants gftmi is presented for n dimensional pattern recognition on the basis of gftmi the moment invariants of affine transformation and subgroups of affine transformation are constructed using these invariants the conceptual mathematical theory of recognition of geometric figures solids and their n dimensional generalizations is worked out by means of this theory it is possible for the first time to analyze scenes consisting not only of polygons and polyhedra but also scenes consisting of geometric figures and solids with curved contours and surfaces respectively in general it is the author s opinion that this theory is a useful step toward the essential development of robot vision and toward creating machine intelligence to make machines able to think by means of geometric concepts of different generalities and dimensions and by associations of these concepts

symmetric phase only matched filtering of fourier mellin transforms for image registration and recognition presents a new method to match a 2d image to a translated rotated and scaled reference image the approach consists of two steps the calculation of a fourier mellin invariant fmi descriptor for each image to be matched and the matching of the fmi descriptors the fmi descriptor is translation invariant and represents rotation and scaling as translations in parameter space the matching of the fmi descriptors is achieved using symmetric phase only matched filtering spomf the performance of the fmi spomf algorithm is the same or similar to that of phase only matched filtering when dealing with image translations the significant advantage of the new technique is its capability to match rotated and scaled images accurately and efficiently the innovation is the application of spomf to the fmi descriptors which guarantees high discriminating power and excellent robustness in the presence of noise this paper describes the principle of the new method and its discrete implementation for either image detection problems or image registration problems practical results are presented for various applications in medical imaging remote sensing fingerprint recognition and multiobject identification

discriminative wavelet shape descriptors for recognition of 2 d patterns in this paper we present a set of wavelet moment invariants together with a discriminative feature selection method for the classification of seemingly similar objects with subtle di erences these invariant features are selected automatically based on the discrimination measures defined for the invariant features using a minimum distance classifier our wavelet moment invariants achieved the highest classification rate for all four di erent sets tested compared with zernike s moment invariants and li s moment invariants for a test set consisting of 26 upper cased english letters wavelet moment invariants could obtain 100 classification rate when applied to 26 30 randomly generated noisy and scaled letters whereas zernike s moment invariants and li s moment invariants obtained only 98 7 and 75 3 respectively the theoretical and experimental analyses in this paper prove that the proposed method has the ability to classify many types of image objects and is particularly suitable for classifying seemingly similar objects with subtle di erences 1999 pattern recognition society published by elsevier science ltd all rights reserved

combined blur and affine moment invariants and their use in pattern recognition abstract the paper is devoted to the recognition of objects and patterns deformed by imaging geometry as well as by unknown blurring we introduce a new class of features invariant simultaneously to blurring with a centrosymmetric psf and to affine transformation as we prove in the paper they can be constructed by combining affine moment invariants and blur invariants derived earlier combined invariants allow to recognize objects in the degraded scene without any restoration

two dimensional polar harmonic transforms for invariant image representation this paper introduces a set of 2d transforms based on a set of orthogonal projection bases to generate a set of features which are invariant to rotation we call these transforms polar harmonic transforms phts unlike the well known zernike and pseudo zernike moments the kernel computation of phts is extremely simple and has no numerical stability issue whatsoever this implies that phts encompass the orthogonality and invariance advantages of zernike and pseudo zernike moments but are free from their inherent limitations this also means that phts are well suited for application where maximal discriminant information is needed furthermore phts make available a large set of features for further feature selection in the process of seeking for the best discriminative or representative features for a particular application

recognitive aspects of moment invariants moment invariants are evaluated as a feature space for pattern recognition in terms of discrimination power and noise tolerance the notion of complex moments is introduced as a simple and straightforward way to derive moment invariants through this relation properties of complex moments are used to characterize moment invariants aspects of information loss suppression and redundancy encountered in moment invariants are investigated and significant results are derived the behavior of moment invariants in the presence of additive noise is also described

fast computation of geometric moments using a symmetric kernel this paper presents a novel set of geometric moments with symmetric kernel sgm obtained using an appropriate transformation of image coordinates by using this image transformation the computational complexity of geometric moments gm is reduced significantly through the embedded symmetry and separability properties in addition it minimizes the numerical instability problem that occurs in high order gm computation the novelty of the method proposed in this paper lies in the transformation of gm kernel from interval 0 to interval 1 1 the transformed gm monomials are symmetry at the origin of principal cartesian coordinate axes and hence possess symmetrical property the computational complexity of sgm is reduced significantly from order o n 4 using the original form of computation to order o n 3 for the proposed symmetry separable approach experimental results show that the percentage of reduction in computation time of the proposed sgm over the original gm is very significant at about 75 0 and 50 0 for square and non square images respectively furthermore the invariant properties of translation scaling and rotation in hu s moment invariants are maintained the advantages of applying sgm over gm in zernike moments computation in terms of efficient representation and computation have been shown through experimental results

3d rotation invariants by complex moments a generalization of the complex moments from 2d to 3d is described group representation theory is used to construct 3d rotation invariants from them the algorithm for automatic generating of the invariants of higher orders is proposed an algorithm for automatic generation of higher order invariants is proposed the linearly dependent invariants are eliminated the invariants are experimentally tested on 3d graphical models and also on real volumetric data

combined invariants to similarity transformation and to blur using orthogonal zernike moments the derivation of moment invariants has been extensively investigated in the past decades in this paper we construct a set of invariants derived from zernike moments which is simultaneously invariant to similarity transformation and to convolution with circularly symmetric point spread function psf two main contributions are provided the theoretical framework for deriving the zernike moments of a blurred image and the way to construct the combined geometric blur invariants the performance of the proposed descriptors is evaluated with various psfs and similarity transformations the comparison of the proposed method with the existing ones is also provided in terms of pattern recognition accuracy template matching and robustness to noise experimental results show that the proposed descriptors perform on the overall better

moment of cepstrum and its applications it is shown that the moments of a signal sequence and of its corresponding cepstral sequence are related in a way which obviates the need for the direct calculation of the cepstral coefficients hence an ideal calculation for the moments of the cepstrum is possible even if the duration of the cepstrum is infinite it is also possible to describe all the properties of a signal sequence from its moments the effectiveness of this method is demonstrated in the problems of homomorphic deconvolution and echo removal it is also shown that in linear filtering problems the moments of the minimum phase sequence are calculated from the corresponding linear phase sequence without actually calculating the minimum phase sequence for a discrete time real signal sequence with finite duration complete equivalence is demonstrated between the signal sequence and its moments this permits a new description and a new set of design specifications based on moments for such systems

degraded image analysis an invariant approach analysis and interpretation of an image which was acquired by a nonideal imaging system is the key problem in many application areas the observed image is usually corrupted by blurring spatial degradations and random noise classical methods like blind deconvolution try to estimate the blur parameters and to restore the image we propose an alternative approach we derive the features for image representation which are invariant with respect to blur regardless of the degradation psf provided that it is centrally symmetric as we prove in the paper there exist two classes of such features the first one in the spatial domain and the second one in the frequency domain we also derive so called combined invariants which are invariant to composite geometric and blur degradations knowing these features we can recognize objects in the degraded scene without any restoration

shepard bernoulli operators we introduce the shepard bernoulli operator as a combination of the shepard operator with a new univariate interpolation operator the generalized taylor polynomial some properties and the rate of convergence of the new combined operator are studied and compared with those given for classical combined shepard operators an application to the interpolation of discrete solutions of initial value problems is given

reforming the theory of invariant moments for pattern recognition abstract the relationship between the algebraic invariants and the fourier mellin spectrum of an image irradiance distribution is clarified and a new approach for generating the two dimensional 2d invariant moments is proposed fifty two invariant functions of the second to the ninth order image moments are tabulated which are invariant with respect to image translation shift orientation and size changes and can be used as library functions in computer programs for automatic image identification the invariant functions derived in this study are shown useful in a numerical example concerned with a character recognition model

radial and angular moment invariants for image identification radial and angular moments of images are presented and methods are shown for deriving moment functions that are invariant with respect to rotation translation reflection and size changes without the aid of the theory of algebraic invariants hu s invariants are expressed in terms of these radial and angular moments and it is claimed that this facilitates visual inspection of invariance properties

on the reconstruction aspects of moment descriptors the problem of reconstruction of an image from discrete and noisy data by the method of moments is examined the set of orthogonal moments based on legendre polynomials is employed a general class of signal dependent noise models is taken into account an asymptotic expansion for the global reconstruction error is established this reveals mutual relationships between a number of moments the image smoothness sampling rate and noise model characteristics the problem of an automatic data driven section of an optimal number of moments is studied this is accomplished with the help of cross validation techniques

pattern recognition using invariants defined from higher order spectra one dimensional inputs a new approach to pattern recognition using in variant parameters based on higher order spectra is presented in particular invariant parameters derived from the bispec trum are used to classify one dimensional shapes the bispec trum which is translation invariant is integrated along straight lines passing through the origin in bifrequency space the phase of the integrated bispectrum is shown to be scale and amplifi cation invariant as well a minimal set of these invariants is selected as the feature vector for pattern classification and a minimum distance classifier using a statistical distance measure is used to classify test patterns the classification technique is shown to distinguish two similar but different bolts given their one dimensional profiles pattern recognition using higher or der spectral invariants is fast suited for parallel implementa tion and has high immunity to additive gaussian noise sim ulation results show very high classification accuracy even for low signal to noise ratios

analyzing looming motion components from their spatiotemporal spectral signature this paper addresses the use of spatio temporal transform methods applied to the analysis of dynamic image sequences and the characterization of image motion the image motion including a divergent component resulting from a looming camera component is analyzed in the spatio temporal mellin transform mt domain resulting in the separation of the spectrum into two parts a structural term corresponding to the spatial mt of the static image and a kinematic term depending on time to collision a motion support we examine potential applications of this property for the recovery of image motion from integral image brightness measurements and the computation of time to collision using spatio temporal mt analysis

implicit moment invariants the use of traditional moment invariants in object recognition is limited to simple geometric transforms such as rotation scaling and affine transformation of the image this paper introduces so called implicit moment invariants implicit invariants measure the similarity between two images factorized by admissible image deformations for many types of image deformations traditional invariants do not exist but implicit invariants can be used as features for object recognition in the paper we present implicit moment invariants with respect to polynomial transform of spatial coordinates describe their stable and efficient implementation by means of orthogonal moments and demonstrate their performance in artificial as well as real experiments

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

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

on the independence of rotation moment invariants abstract the problem of the independence and completeness of rotation moment invariants is addressed in this paper first a general method for constructing invariants of arbitrary orders by means of complex moments is described as a major contribution of the paper it is shown that for any set of invariants there exists a relatively small basis by means of which all other invariants can be generated the method how to construct such a basis and how to prove its independence and completeness is presented some practical impacts of the new results are mentioned at the end of the paper

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

on image analysis by the methods of moments various types of moments have been used to recognize image patterns in a number of applications a number of moments are evaluated and some fundamental questions are addressed such as image representation ability noise sensitivity and information redundancy moments considered include regular moments legendre moments zernike moments pseudo zernike moments rotational moments and complex moments properties of these moments are examined in detail and the interrelationships among them are discussed both theoretical and experimental results are presented

geometrically invariant image watermarking using polar harmonic transforms this paper presents an invariant image watermarking scheme by introducing the polar harmonic transform pht which is a recently developed orthogonal moment method similar to zernike moment zm and pseudo zernike moment pzm approaches pht is defined on a circular domain the magnitudes of phts are invariant to image rotation and scaling furthermore the phts are free of numerical instability so they are more suitable for watermarking in this paper the invariant properties of phts are investigated during embedding a subset of the accurate phts are modified according to the binary watermark sequence then a compensation image is formatted by reconstructing the modified pht vector the final watermarked image is obtained by adding the compensation image to the original image in the decoder the watermark can be retrieved from the magnitudes of the phts directly experimental results illustrate that the proposed scheme outperforms zm pzm based schemes in terms of embedding capacity and watermark robustness and is also robust to both geometric and signal processing based attacks

n folded symmetries by complex moments in gabor space and their application to unsupervised texture segmentation complex moments of the gabor power spectrum yield estimates of the n folded symmetry of the local image content at different frequency scales that is they allow to detect linear rectangular hexagonal triangular and so on structures with very fine to very coarse resolutions results from experiments on the unsupervised segmentation of real textures indicate their importance for image processing applications real geometric moments computed in gabor space also provide for very powerful texture features but lack the clear geometrical interpretation of complex moments

errata and comments on generic orthogonal moments jacobi fourier moments for invariant image description abstract ping et al z ping h ren j zou y sheng and w bo generic orthogonal moments jacobi fourier moments for invariant image description pattern recognition 40 4 2007 1245 1254 made a landmark contribution to the theory of two dimensional orthogonal moments confined to the unit disk by unifying the radial kernels of existing polynomial based circular orthogonal moments under the roof of shifted jacobi polynomials however the work contains some errata that result mainly from the confusion between the two slightly different definitions of shifted jacobi polynomials in the literature taking into account the great importance and the high impact of the work in the pattern recognition community this paper points out the confusing points corrects the errors and gives some other relevant comments the corrections developed in this paper are illustrated by some experimental evidence

on the recovery of a function on a circular domain we consider the problem of estimating a function f x y on the unit disk f x y x sup 2 y sup 2 spl les 1 given a discrete and noisy data recorded on a regular square grid an estimate of f x y based on a class of orthogonal and complete functions over the unit disk is proposed this class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate for radial functions the orthogonal set has a particularly simple form being related to the classical legendre polynomials we give the statistical accuracy analysis of the proposed estimate of f x y in the sense of the l sub 2 metric it is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain this is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle in fact the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain

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

moment forms invariant to rotation and blur in arbitrary number of dimensions we present the construction of combined blur and rotation moment invariants in arbitrary number of dimensions moment invariants to convolution with an arbitrary centrosymmetric filter are derived first and then their rotationally invariant forms are found by means of group representation theory to achieve the desired combined invariance several examples of the invariants are calculated explicitly to illustrate the proposed procedure their invariance robustness and capability of using in template matching and in image registration are demonstrated on 3d mri data and 2d indoor images

quaternion fourier mellin moments for color images the quaternion fourier mellin moments for describing color images are introduced which can be seen as the generalization of traditional fourier mellin moments for gray level images then the quaternion fourier mellin moment invariants are derived which could be a useful tool in color object recognition tasks that require the similarity invariance in addition the problem of color image registration using quaternion fourier mellin moments is discussed the registration method can match color images differing in rotation and scaling the advantage of our method is that it can process color image directly without losing color information experimental results validate the effectiveness of the method we proposed