invariant character recognition with zernike and orthogonal fourier mellin moments abstract in this paper we consider the use of orthogonal moments for invariant classification of alphanumeric characters of different size in addition to the zernike and pseudo zernike moments zms and pzms which have been previously proposed for invariant character recognition a new method of combining orthogonal fourier mellin moments ofmms with centroid bounding circle scaling is introduced which is shown to be useful in characterizing images with large variability through extensive experimentation using zms and ofmms as features different scaling methodologies and classifiers it is shown that ofmms give the best overall performance in terms of both image reconstruction and classification accuracy

on bargmann representations of the wigner function by using the localized character of canonical coherent states we give a straightforward derivation of the bargmann integral representation of the wigner function w a non integral representation is presented in terms of a quadratic form w v fv where f is a self adjoint matrix whose entries are tabulated functions and v is a vector depending in a simple recursive way on the derivatives of the bargmann function such a representation may be of use in numerical computations we discuss a relation involving the geometry of the wigner function and the spatial uncertainty of the coherent state basis we use to represent it

spaces of type w growth of hermite coefficients wigner distribution and bargmann transform abstract in their famous monograph on generalized functions gelfand and shilov introduce the function spaces w m w and w m these spaces consist of c functions and holomorphic functions respectively with growth behaviour specified by suitable convex functions m and in this paper we study the spaces wmmx where mx denotes young s dual function corresponding to m we characterize the hermite expansion coefficients of the functions in wmmx their fourier transforms wigner distributions and bargmann transforms in particular we prove that wmmx wm wmx

multichannel interpolation of nonuniform samples with application to image recovery abstract the multichannel trigonometric reconstruction from uniform samples was proposed recently it not only makes use of multichannel information about the signal but is also capable to generate various kinds of interpolation formulas according to the types and amounts of the collected samples the paper presents the theory of multichannel interpolation from nonuniform samples two distinct models of nonuniform sampling patterns are considered namely recurrent and generic nonuniform sampling each model involves two types of samples nonuniform samples of the observed signal and its derivatives numerical examples and quantitative error analysis are provided to demonstrate the effectiveness of the proposed algorithms additionally the proposed algorithms for recovering highly corrupted images are also investigated in comparison with the median filter and the shepard interpolation the proposed algorithms produce superior results with lower errors

boundary value problems with symmetry and their approximation by finite elements symmetry and related questions of group theory especially representation theory are well known popular topics among physicists this paper describes how these classical ideas can be put to use in the realm of finite element computations by substituting a family of independent problems on a reduced domain the symmetry cell for the original problem on the whole domain two specific difficulties appear one is how to set boundary conditions on symmetry planes representation theory gives the answer the other is how to proceed with the assembly of finite elements that constitute the symmetry cell to deal with this this paper introduces the concept of index of an element as part of the whole structure which is the number of its distinct symmetrical images the rule for assembly then becomes to perform as usual but to count each element as many times as its index

eigenvalues for equivariant matrices an equivariant matrix a commutes with a group of permutation matrices such matrices often arise in numerical applications where the computational domain exhibits geometrical symmetries for instance triangles cubes or icosahedra the theory for block diagonalizing equivariant matrices via the generalized fourier transform gft is reviewed and applied to eigenvalue computations for dense matrices which are equivariant under large symmetry groups we give theoretical estimates that show a substantial performance gain in case of cubic symmetry the gain is about 800 times which is verified by numerical results it is also shown how the multiplicity of the eigenvalues is determined by the symmetry which thereby restricts the number of distinct eigenvalues the inverse gft is used to compute the corresponding eigenvectors it is emphasized that the inverse transform in this case is very fast due to the sparseness of the eigenvectors in the transformed space

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

image analysis by circularly semi orthogonal moments various types of circularly orthogonal moments have been widely used for image reconstruction and rotation invariant classification however they suffer from two errors namely numerical integration error and geometric error which affect their reconstruction capability and pattern recognition accuracy in this paper a novel category of circular moments named circularly semi orthogonal moments is proposed in the proposed moment a set of orthogonal basis functions modulated by a negative power exponential envelope is utilized as the radial basis function for a given degree n the radial basis function possesses more compact bandwidth less cutoff frequency and more zeros compared with the frequently used circularly orthogonal moments including zernike and orthogonal fourier mellin moments and so the circularly semi orthogonal moment calculated with the zeroth order approximation is more robust to numerical error than the frequently used circularly orthogonal moments furthermore the capability of the semi orthogonal moment to describe high spatial frequency components of images is relatively higher than that of the frequently used circularly orthogonal moments experimental results demonstrate that the semi orthogonal moments calculated with the zeroth order approximation perform better than the frequently used circularly orthogonal moments in terms of image reconstruction capability and invariant recognition accuracy in noise free noisy and smooth distortion conditions it is also shown that the proposed high order moments are more numerically stable than the circularly orthogonal moments a kind of circular moments named circularly semi orthogonal moment is proposed it outperforms in image reconstruction over the circularly orthogonal moments it performs better in invariant image recognition under noisy and smooth distortions

classification of invariant image representations using a neural network a neural network nn based approach for classification of images represented by translation scale and rotation invariant features is presented the utilized network is a multilayer perceptron mlp classifier with one hidden layer back propagation learning is used for its training two types of features are used moment invariants derived from geometrical moments of the image and features based on zernlike moments which are the mapping of the image onto a set of complex orthogonal polynomials the performance of the mlp is compared to the bayes nearest neighbor and minimum mean distance statistical classifiers through extensive experimentation with noiseless as well as noisy binary images of all english characters 26 classes the following conclusions are reached 1 the mlp outperforms the other three classifiers especially when noise is present 2 the nearest neighbor classifier performs about the same as the nn for the noiseless case 3 the nn can do well even with a very small number of training samples 4 the nn has a good degree of fault tolerance and 5 the zernlike moment based features possess strong class separability power and are more powerful than moment invariants

efficient computations for generalized zernike moments and image recovery abstract zernike moments are a set of orthogonal moments which have been successfully applied in the fields of image processing and pattern recognition an innovative calculation method for zernike moments named generalized zernike moments is presented in this study the generalized zernike moment is a variant of zernike moment in this paper we are proposing methods to calculate high order generalized zernike moments two kinds of recurrence for calculating generalized zernike moments were introduced with rigorous proofs through the usage of the symmetries operated by the dihedral group of order eight the proposed method is fast and stable the experimental results show that of the proposed method took 4 206s to compute the top 500 order generalized zernike moments of an image with 512 by 512 pixels furthermore by choosing the extra parameter in the recurrence the method enhanced the accuracy remarkably compared to the regular zernike moments its normalized mean square error is 0 00144067 when was set to 66 and the top 500 order moments were used to reconstruct the image this error is 40 47 smaller than the one obtained by using the regular zernike moments

generalized dual hahn moment invariants in this work we introduce a generalized expression of the weighted dual hahn moment invariants up to any order and for any value of their parameters in order for the proposed invariants to be formed the weighted dual hahn moments up to any order and for any value of their parameters are expressed as a linear combination of geometric ones for this reason a formula expressing the nth degree dual hahn polynomial for any value of its parameters as a linear combination of monomials c r x r is proved in addition a recurrent relation for the fast computation of the aforementioned monomials coefficients c r is also given moreover normalization aspects of the generalized weighted dual hahn moment invariants are discussed while a modification of them is proposed in order to avoid their numerical instabilities finally experimental results and classification scenarios including datasets of natural scenes evaluate the proposed methodology

quantum symmetries and the weyl wigner product of group representations in the usual formulation of quantum mechanics groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex hilbert space in accordance with wigner s theorem in the phase space formulation they have real true unitary representations in the space of square integrable functions on phase space each such phase space representation is a weyl wigner product of the corresponding hilbert space representation with its contragredient and these can be recovered by factorizing the weyl wigner product however not every real unitary representation on phase space corresponds to a group of automorphisms so not every such representation is in the form of a weyl wigner product and can be factorized the conditions under which this is possible are examined examples are presented

symmetrizing grids radial basis functions and chebyshev and zernike polynomials for the d 4 symmetry group interpolation within a squircle part i a domain is invariant under the eight element d 4 symmetry group if it is unchanged by reflection with respect to the x and y axes and also the diagonal line x y previous treatments of group theory for spectral methods have generally demanded a semester s worth of group theory we show this is unnecessary by providing explicit recipes for creating grids etc we show how to decompose an arbitrary function into six symmetry invariant components and thereby split the interpolation problem into six independent subproblems we also show how to make symmetry invariant basis functions from products of chebyshev polynomials from zernike polynomials and from radial basis functions rbfs of any species these recipes are completely general and apply to any domain that is invariant under the dihedral group d 4 these concepts are illustrated by rbf pseudospectral solutions of the poisson equation in a domain bounded by a squircle the square with rounded corners defined by x 2 y 2 1 0 where here 2 we also apply chebyshev polynomials to compute eigenmodes of the helmholtz equation on the square and show each mode belongs to one and only one of the six d 4 classes

self similarity inspired local descriptor for non rigid multi modal image registration the self similarity inspired local descriptor is proposed for non rigid multi modal registration the zernike moments is introduced into the proposed method to explore rotational self similarity the effective computation of self similarity based both intensities and edge texture features of the multi modal images is utilized in the proposed method the proposed method can achieve higher registration accuracy than the other state of the art image representation based registration methods non rigid multi modal image registration plays an important role in medical image analysis it remains a challenging problem due to the significant intensity distortion and the non rigid transformation between images existing registration methods based on the information theoretic measures and image representations cannot address this problem effectively in this paper we have proposed a novel self similarity inspired local descriptor to determine the similarity metrics a key component in image registration the self similarity is determined based on the zernike moments of image patches in a local neighborhood and it is utilized to generate the local descriptor the euclidean distance between the local descriptors computed in the reference and moving images is used as the similarity metrics distinctively the proposed local descriptor can provide an effective representation of complicated image features due to its robustness to noise and rotational invariance which provides the proposed method with good registration performance extensive experiments on both simulated and real multi modal image datasets demonstrate that the proposed method has higher registration accuracy appreciated by the target registration error tre than the state of the art registration methods based on the normalized mutual information nmi the sum of squared differences on entropy images essd the weber local descriptor wld and the modality independent neighborhood descriptor mind

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

invariant image recognition by zernike moments the problem of rotation scale and translation invariant recognition of images is discussed a set of rotation invariant features are introduced they are the magnitudes of a set of orthogonal complex moments of the image known as zernike moments scale and translation invariance are obtained by first normalizing the image with respect to these parameters using its regular geometrical moments a systematic reconstruction based method for deciding the highest order zernike moments required in a classification problem is developed the quality of the reconstructed image is examined through its comparison to the original one the orthogonality property of the zernike moments which simplifies the process of image reconstruction make the suggest feature selection approach practical features of each order can also be weighted according to their contribution to the reconstruction process the superiority of zernike moment features over regular moments and moment invariants was experimentally verified

orthogonal moment operators for subpixel edge detection abstract a new approach to detect step edges with subpixel accuracy is presented the proposed approach is based on a set of orthogonal complex moments of the image known as zernike moments an ideal two dimensional 2d step edge is modeled in terms of four parameters the background gray level the step size the distance of the edge from the center of the mask and the orientation of the edge discrete zernike moments are used to obtain a total of three complex masks to compute all the edge parameters for subpixel detection for pixel level edge detection only two masks one real and one complex are required the theoretical analysis of the influence of noise on the location and the orientation of an edge is presented this analysis reveals that the accuracy of the proposed approach is virtually unaffected by the additive noise the technique is effective in detecting both the pixel level and subpixel level edges experimental results are presented to demonstrate the efficacy of the proposed technique

notes on shape based tools for treating the objects ellipticity issues we establish the average shape ellipticity measure as an average value of the estimated similarities between the shape considered and all the ellipses whose axes length ratio belongs to a certain predefined interval we derive the maximal shape ellipticity measure as the maximal similarity estimate between a given shape and all the ellipses in this paper we put under the same umbrella several well known results and establish several new results related to the shape based tools which treat the objects ellipticity issues we start with a derivation of an explicit and closed formula for the computation of the ellipticity measures from an infinite family introduced recently the new formula enables a fast computation of such measures since it does not require any optimizing procedure for the computation as it was the case before in addition the established formula enables an easy theoretical manipulation as a result we have discovered new shape features as they are i the average shape ellipticity measure which might be interpreted as an average value of the estimated similarities between the shape considered and all the ellipses whose axes length ratio belongs to a certain predefined interval ii the maximal shape ellipticity measure which might be understood as the maximal possible similarity estimate between a given shape and an ellipses some of the other results obtained relate strongly to the well known measures and methods broadly used in shape based object analysis tasks

numerical stability of fast computation algorithms of zernike moments abstract a detailed comparative study of the numerical stability of the recursive algorithms widely used to calculate the zernike moments of an image is presented in this paper while many papers introducing fast algorithms for the computation of zernike moments have been presented in the literature there is not any work studying the numerical behaviour of these methods these algorithms have been in the past compared to each other only according to their computational complexity without been given the appropriate attention as far as their numerical stability is concerned being the most significant part of the algorithms reliability the present contribution attempts to fill this gap in the literature since it mainly demonstrates that the usefulness of a recursive algorithm is defined not only by its low computational complexity but most of all by its numerical robustness this paper exhaustively compares some well known recursive algorithms for the computation of zernike moments and sets the appropriate conditions in which each algorithm may fall in an unstable state the experiments show that any of these algorithms can be unstable under some conditions and thus the need to develop more stable algorithms is of major importance

improved quality of reconstructed images using floating point arithmetic for moment calculation zernike moments which are superior to geometric moments because of their special properties of image reconstruction and immunity to noise suffer from several discretization errors these errors lead to poor quality of reconstructed image and wide variations in the numerical values of the moments the predominant factor as observed in this paper is due to the discrete integer implementation of the steps involved in moment calculation it is shown in this paper that by modifying the algorithms to include discrete float implementation the quality of the reconstructed image improves significantly and the first order moment becomes zero low order zernike moments have been found to be stable under linear transformations while the high order moments have large variations the large variations in high order moments however do not greatly affect the quality of the reconstructed image implying that they should be ignored when numerical values of moments are used as features the 11 functions based on geometric moments have also been found to be stable under linear transformations and thus these can be used as features pixel level analysis of the images has been carried out to strengthen the results

zernike functions rigged hilbert spaces and potential applications we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing we revise the symmetries of the zernike polynomials that determine the lie algebra su 1 1 su 1 1 we show how they induce discrete as well as continuous bases that coexist in the framework of rigged hilbert spaces we also discuss some other interesting properties of zernike polynomials and zernike functions one of the areas of interest of zernike functions has been their applications in optics here we suggest that operators on the spaces of zernike functions may play a role in optical image processing

orthogonal variant moments features in image analysis moments are statistical measures used to obtain relevant information about a certain object under study e g signals images or waveforms e g to describe the shape of an object to be recognized by a pattern recognition system invariant moments e g the hu invariant set are a special kind of these statistical measures designed to remain constant after some transformations such as object rotation scaling translation or image illumination changes in order to e g improve the reliability of a pattern recognition system the classical moment invariants methodology is based on the determination of a set of transformations or perturbations for which the system must remain unaltered although very well established the classical moment invariants theory has been mainly used for processing single static images i e snapshots and the use of image moments to analyze images sequences or video from a dynamic point of view has not been sufficiently explored and is a subject of much interest nowadays in this paper we propose the use of variant moments as an alternative to the classical approach this approach presents clear differences compared to the classical moment invariants approach that in specific domains have important advantages the difference between the classical invariant and the proposed variant approach is mainly but not solely conceptual invariants are sensitive to any image change or perturbation for which they are not invariant so any unexpected perturbation will affect the measurements i e is subject to uncertainty on the contrary a variant moment is designed to be sensitive to a specific perturbation i e to measure a transformation not to be invariant to it and thus if the specific perturbation occurs it will be measured hence any unexpected disturbance will not affect the objective of the measurement confronting thus uncertainty furthermore given the fact that the proposed variant moments are orthogonal i e uncorrelated it is possible to considerably reduce the total inherent uncertainty the presented approach has been applied to interesting open problems in computer vision such as shape analysis image segmentation tracking object deformations and object motion tracking obtaining encouraging results and proving the effectiveness of the proposed approach

phase space representation of quantum state vectors phase space representation of quantum state vectors is obtained within the framework of the relative state formulation for this purpose the hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system relative position state and relative momentum state are defined in the extended hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators acting on these states are obtained phase space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative position states and the relative momentum states the absolute square of the phase space function represents the probability distribution of the phase space variables time evolution of a quantum system is investigated in terms of the phase space functions the relations to the phase space representations formulated by the other methods are obtained

accurate calculation of zernike moments zernike moments zms are very effective global image descriptors which are used in many digital image processing applications the digitization process compromises the accuracy of the moments and therefore several of its properties are affected there are two major discretization errors namely the geometric error and numerical integration error in this paper we propose two new algorithms which eliminate these errors the first algorithm performs the exact computation of geometric moments gms over a unit disk and then uses gms to zms relationship to compute the latter this algorithm is computationally more expensive and it becomes numerically instable for higher order moments therefore we develop a second algorithm based on gaussian quadrature numerical integration the second algorithm reduces both the errors simultaneously and its accuracy increases as the degree of gaussian quadrature numerical integration increases the proposed algorithms are observed to provide very accurate zms which result in improved image reconstruction reduction in reconstruction error and improvement in rotation and scale invariance exhaustive experiments are provided to support improved accuracy of zms and time complexity analysis is performed for the existing and the proposed methods

stereo effect of image converted from planar research on planar to stereo image conversion has significant theoretical and practical implications for image processing in this paper we use a method which segments an image into sub blocks to perform this kind of conversion parameters such as random variables are used for conversion control we use two quantitative criteria cross entropy and root mean square error to evaluate the stereo effect furthermore the stereo effect that the random variables create is discussed the results of the experiment show that i when all random variables have the same distribution different values of these random variables only slightly affect the stereo effect and ii when different distributions are applied to the random variables the cross entropies or root mean square errors are slightly different which indicates different distributions have a small influence on the stereo effect generally we recommend normal distribution for better stereo effect in most cases

model space results for the gabor and wavelet transforms we prove that the unique gabor window leading to an analytic model space is the gaussian one and we give an analogous result for the wavelet transform for the general case we give a new approach to study irregular gabor and wavelet frames we improve some known results for gabor windows in the feichtinger algebra and for a special class of wavelets

fast and accurate method for high order zernike moments computation abstract zernike moments zms which belong to a class of circularly orthogonal rotation invariant moments are very useful image descriptors for various image processing applications they however suffer from various errors such as discretization error geometric error and numerical integration error in this paper we present an approach for their computation which reduces geometric error and numerical integration error geometric error is reduced by performing exact mapping of a square grid domain into a circular grid by using both the square and arc grids numerical integration error is reduced by resorting to gaussian numerical integration over the square and arc grids it is observed that the contribution of numerical integration error to the total error is much more than the geometric error the effect of these two errors is more significant in small images as compared to large images the use of numerical integration however enhances the time complexity this problem is resolved by using recursive algorithms for the computation of radial polynomials 8 way symmetry of radial polynomials and 8 way symmetry anti symmetry of angular part of the kernel function exhaustive experiments on synthetic and real images are performed to demonstrate the effectiveness of the proposed approach

moment based local image watermarking via genetic optimization a totally optimized image watermarking methodology that manages to enhance its local behaviour by applying the krawtchouk moments is presented through this paper the introduced technique is making use of a simple genetic algorithm in order to optimize the set of parameters that significantly influences the locality properties alongside with the overall performance of the watermarking procedure herein the watermarking process is tackled as an optimization procedure in this context the appropriate set of configuration parameters is being searched for ensuring high quality watermarked images and low bit error rates at the extraction stage for this purpose several traditional and newly introduced image quality measures are used in order to quantify the influence of the examined set of parameters as far as the quality of the watermarked image and the accuracy of the extracted watermark are concerned the proposed method produces watermarked images of high quality and ensures high detection rates under several non geometric attacks by using less prior knowledge at the detector s side extensive experiments have shown that by handling the watermarking process as an optimization problem more robust and accurate watermarking schemes can be derived

generalized q functions the modulus squared of a class of wavefunctions defined on phase space is used to define a generalized family of q or husimi functions a parameter lambda specifies orderings in a mapping from the operator psi sigma to the corresponding phase space wavefunction where sigma is a given fiducial vector the choice lambda 0 specifies the weyl mapping and the q function so obtained is the usual one when sigma is the vacuum state more generally any choice of of lambda in the range 1 1 corresponds to orderings varying between standard and anti standard for all such orderings the generalized q functions are non negative by construction they are shown to be proportional to expectation of the system state rho with respect to a generalized displaced squeezed state which depends on lambda and position p q in phase space thus when a system has been prepared in the state rho a generalized q function is proportional to the probability of finding it in the generalized squeezed state any such q function can also be written as the smoothing of the wigner function for the system state rho by convolution with the wigner function for the generalized squeezed state

bounds on integrals of the wigner function the integral of the wigner function over a subregion of the phase space of a quantum system may be less than zero or greater than one it is shown that for systems with 1 degree of freedom the problem of determining the best possible upper and lower bounds on such an integral over an possible states reduces to the problem of finding the greatest and least eigenvalues of a hermitian operator corresponding to the subregion the problem is solved exactly in the case of an arbitrary elliptical region these bounds provide checks on experimentally measured quasiprobability distributions

quantum mechanics as an approximation to classical mechanics in hilbert space classical mechanics is formulated in complex hilbert space with the introduction of a commutative product of operators an antisymmetric bracket and a quasidensity operator that is not positive definite these are analogues of the star product the moyal bracket and the wigner function in the phase space formulation of quantum mechanics quantum mechanics is then viewed as a limiting form of classical mechanics as planck s constant approaches zero rather than the other way around the forms of semiquantum approximations to classical mechanics analogous to semiclassical approximations to quantum mechanics are indicated

geometric invariance in image watermarking surviving geometric attacks in image watermarking is considered to be of great importance in this paper the watermark is used in an authentication context two solutions are being proposed for such a problem both geometric and invariant moments are used in the proposed techniques an invariant watermark is designed and tested against attacks performed by stirmark using the invariant moments on the other hand an image normalization technique is also proposed which creates a normalized environment for watermark embedding and detection the proposed algorithms have the advantage of being robust computationally efficient and no overhead needs to be transmitted to the decoder side the proposed techniques have proven to be highly robust to all geometric manipulations filtering compression and slight cropping which are performed as part of stirmark attacks as well as noise addition both gaussian and salt pepper

a comparative analysis of algorithms for fast computation of zernike moments abstract this paper details a comparative analysis on time taken by the present and proposed methods to compute the zernike moments z pq the present method comprises of direct belkasim s prata s kintner s and coefficient methods we propose a new technique denoted as q recursive method specifically for fast computation of zernike moments it uses radial polynomials of fixed order p with a varying index q to compute zernike moments fast computation is achieved because it uses polynomials of higher index q to derive the polynomials of lower index q and it does not use any factorial terms individual order of moments can be calculated independently without employing lower or higher order moments this is especially useful in cases where only selected orders of zernike moments are needed as pattern features the performance of the present and proposed methods are experimentally analyzed by calculating zernike moments of orders 0 to p and specific order p using binary and grayscale images in both the cases the q recursive method takes the shortest time to compute zernike moments

an efficient illumination invariant face recognition framework via illumination enhancement and dd dtcwt filtering in this paper it is shown that multiscale analysis of facial structure and features of face images leads to superior recognition rates for images under varying illumination the proposed method which is computationally cost effective significantly suppresses illumination effects the problem is defined as how better to extract the reflectance portion from a given image this can be used directly as an input to a dimensionality reduction unit followed by a classifier for recognition purposes we first assume that an image i x y is a black box consisting of a combination of illumination and reflectance a new approximation is proposed to enhance the illumination removal phase as illumination resides in the low frequency part of image it is reasonable to consider the use of a high performance multiresolution transformation to first accurately separate the frequency components of an image the double density dual tree complex wavelet transform dd dtcwt possesses three core advantages i e the transformation is i shift invariant ii directionally selective with no checkerboard effect iii enriched by extra wavelets interpreted as double density the output of the first phase is sent to a dd dtcwt unit to be decomposed into frequency subbands high frequency subbands are thresholded and an inverse dd dtcwt is then applied to subbands to construct a low frequency raw image which is followed by a fine tuning process finally after extracting a mask feature vector is formed and the principal component analysis pca is used for dimensionality reduction which is then proceeded by the extreme learning machine elm as a classifier to evaluate the performance of the proposed algorithm for face recognition under varying illumination unlike similar works the proposed method is free of any prior information about the face shape it is systematic and easy to implement and it can be applied separately on each image furthermore the proposed method which is significantly faster than similar techniques presents a robust behavior against the reduction in the number of images required for the training cycle several experiments are performed employing the available well known databases such as the yale b extended yale b cmu pie feret at t and the labeled faces in the wild lfw illustrative examples are given and the results compare favorably to the current results in the literature

existence of solutions for the abstract nonlinear equation btu f u on considere l equation non lineaire abstraite btu f u ou b t sont des operateurs lineaires clos dans un espace de banach e on suppose toujours que b admet un inverse b 1 7b l e et 0 est un pole isole pour la resolvante t i 1 d ordre m 1 on suppose que f est une fonction continue de e dans e on etudie l existence d une solution classique u e a l aide d une decomposition de e en une somme directe induite par t

a unified methodology for computing accurate quaternion color moments and moment invariants in this paper a general framework for computing accurate quaternion color moments and their corresponding invariants is proposed the proposed unified scheme arose by studying the characteristics of different orthogonal polynomials these polynomials are used as kernels in order to form moments the invariants of which can easily be derived the resulted scheme permits the usage of any polynomial like kernel in a unified and consistent way the resulted moments and moment invariants demonstrate robustness to noisy conditions and high discriminative power additionally in the case of continuous moments accurate computations take place to avoid approximation errors based on this general methodology the quaternion tchebichef krawtchouk dual hahn legendre orthogonal fourier mellin pseudo zernike and zernike color moments and their corresponding invariants are introduced a selected paradigm presents the reconstruction capability of each moment family whereas proper classification scenarios evaluate the performance of color moment invariants

novel moment invariants for improved classification performance in computer vision applications a novel set of moment invariants based on the krawtchouk moments are introduced in this paper these moment invariants are computed over a finite number of image intensity slices extracted by applying an innovative image representation scheme the image slice representation isr method based on this technique an image is decomposed to a several non overlapped intensity slices which can be considered as binary slices of certain intensity this image representation gives the advantage to accelerate the computation of image s moments since the image can be described in a number of homogenous rectangular blocks which permits the simplification of the computation formulas the moments computed over the extracted slices seem to be more efficient than the corresponding moments of the same order that describe the whole image in recognizing the pattern under processing the proposed moment invariants are exhaustively tested in several well known computer vision datasets regarding their rotation scaling and translation rst invariant recognition performance by resulting to remarkable outcomes

a comment on fast and numerically stable methods for the computation of zernike moments by singh et al pattern recognition 43 2010 pages 2497 2506 the aim of the present comment is to point out that the q recursive relation introduced by singh et al is wrong these errors have been corrected in the comment the numerical experiments show that the correct recurrence relation is consistent with the original q recursive method

symplectically covariant schrodinger equation in phase space a classical theorem of stone and von neumann states that the schrodinger representation is up to unitary equivalences the only irreducible representation of the heisenberg group on the hilbert space of square integrable functions on configuration space using the wigner moyal transform we construct an irreducible representation of the heisenberg group on a certain hilbert space of square integrable functions defined on phase space this allows us to extend the usual weyl calculus into a phase space calculus and leads us to a quantum mechanics in phase space equivalent to standard quantum mechanics we also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space schrodinger equation

the quantum state vector in phase space and gabor s windowed fourier transform representations of quantum state vectors by complex phase space amplitudes complementing the description of the density operator by the wigner function have been defined by applying the weyl wigner transform to dyadic operators linear in the state vector and anti linear in a fixed window state vector here aspects of this construction are explored and a connection is established with gabor s windowed fourier transform the amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations generalized bargmann representations of the state vector appear as special cases associated with gaussian windows for every choice of window amplitudes lie in a corresponding linear subspace of square integrable functions on phase space a generalized born interpretation of amplitudes is described with both the wigner function and a generalized husimi function appearing as quantities linear in an amplitude and anti linear in its complex conjugate schrodinger s time dependent and time independent equations are represented on phase space amplitudes and their solutions described in simple cases

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

fast computation of legendre and zernike moments this paper presents recursive algorithms for fast computation of legendre and zernike moments of a grey level image intensity distribution for a binary image a contour integration method is developed for the evaluation of legendre moments using only the boundary information a method for recursive calculation of zernike polynomial coefficients is also given a square to circular image transformation scheme is introduced to minimize the computation involved in zernike moment functions the recursive formulae can also be used in inverse moment transforms to reconstruct the original image from moments the mathematical framework of the algorithms is given in detail and illustrated with binary and grey level images

new computational methods for full and subset zernike moments the computation of zernike radial polynomials contributes most of the computation time in computing the zernike moments due to the involvement of factorial terms the common approaches used in fast computation of zernike moments are kintner s prata s coefficient and q recursive methods in this paper we propose faster methods to derive the full set of zernike moments as well as a subset of zernike moments a hybrid algorithm that uses prata s simplified kintner s and coefficient methods is used to derive the full set of zernike moments in the computation of a subset of zernike moments we propose using the combination of prata s simplified kintner s coefficient and q recursive methods fast computation is achieved by using the recurrence relations between the zernike radial polynomials of successive order without any involvement of factorial terms in the first and second experiments we show both the hybrid algorithms take lesser computation time than the existing methods in computing the full set of zernike moments and a selected subset of zernike moments which are not in successive sequence both hybrid algorithms have been applied in real world application in the classification of rice grains using full set and subset of zernike moments the classification performance using optimal subset of zernike moments is better than using full set of zernike moments

fast computation of exact zernike moments using cascaded digital filters zernike moments have been extensively used and have received much research attention in a number of fields object recognition image reconstruction image segmentation edge detection and biomedical imaging however computation of these moments is time consuming thus we present a fast computation technique to calculate exact zernike moments by using cascaded digital filters the novelty of the method proposed in this paper lies in the computation of exact geometric moments directly from digital filter outputs without the need to first compute geometric moments the mathematical relationship between digital filter outputs and exact geometric moments is derived and then they are used in the formulation of exact zernike moments a comparison of the speed of performance of the proposed algorithm with other state of the art alternatives shows that the proposed algorithm betters current computation time and uses less memory

a novel algorithm for fast computation of zernike moments abstract zernike moments zms have been successfully used in pattern recognition and image analysis due to their good properties of orthogonality and rotation invariance however their computation by a direct method is too expensive which limits the application of zms in this paper we present a novel algorithm for fast computation of zernike moments by using the recursive property of zernike polynomials the inter relationship of the zernike moments can be established as a result the zernike moment of order n with repetition m znm can be expressed as a combination of zn 2 m and zn 4 m based on this relationship the zernike moment znm for n m can be deduced from zmm to reduce the computational complexity we adopt an algorithm known as systolic array for computing these latter moments using such a strategy the multiplication number required in the moment calculation of zmm can be decreased significantly comparison with known methods shows that our algorithm is as accurate as the existing methods but is more efficient

a coordinate independent formulation of the weyl wigner transform theory abstract we present the weyl wigner ww transform theory in a much more compact way than usual by introducing the basis in an intrinsic form this permits the derivation of new identities and also leads to generalizations like the inclusion of finite dimensional systems in the ww theory which is also discussed we show in this case some striking differences in the structure of finite phase space depending on the underlying dimension of quantum space being an even or odd integer

fast and numerically stable methods for the computation of zernike moments zernike moments zms are used in many image processing applications due to their superior performance over other moments however they suffer from high computation cost and numerical instability at high order of moments in the past many recursive methods have been developed to improve their speed performance and considerable success has been achieved the analysis of numerical stability has also gained momentum as it affects the accuracy of moments and their invariance property there are three recursive methods which are normally used in zms calculation prata s kintner s and q recursive methods the earlier studies have found the q recursive method outperforming the two other methods in this paper we modify prata s method and present a recursive relation which is proved to be faster than the q recursive method numerical instability is observed at high orders of moments with the q recursive method suffering from the underflow problem while the modified prata s method suffering from finite precision error the modified kintner s method is the least susceptible to these errors keeping in view the better numerical stability we further make the modified kintner s method marginally faster than the q recursive method we recommend the modified prata s method for low orders 90 and kintner s fast method for high orders 90 of zms

on some abstract degenerate problems of parabolic type 2 the nonlinear case on etudie certaines caracteristiques du probleme aux valeurs initiales degenere d m t u t dt l t u t f t t 0 m t u t t 0 w 0 ou m t l t sont des operateurs lineaires clos de y dans x deux espaces de banach et f est continue de 0 0 dans x w 0 x

using zernike moments for the illumination and geometry invariant classification of multispectral texture we develop a method for recognizing color texture independent of rotation scale and illumination color texture is modeled using spatial correlation functions defined within and between sensor bands using a linear model for surface spectral reflectance with the same number of parameters as the number of sensor classes we show that illumination and geometry changes in the scene correspond to a linear transformation of the correlation functions and a linear transformation of their coordinates a several step algorithm that includes scale estimation and correlation moment computation is used to achieve the invariance the key to the method is the new result that illumination rotation and scale changes in the scene correspond to a specific transformation of correlation function zernike moment matrices these matrices can be estimated from a color image this relationship is used to derive an efficient algorithm for recognition the algorithm is substantiated using classification results on over 200 images of color textures obtained under various illumination conditions and geometric configurations

a unified methodology for the efficient computation of discrete orthogonal image moments a novel methodology is proposed in this paper to accelerate the computation of discrete orthogonal image moments the computation scheme is mainly based on a new image representation method the image slice representation isr method according to which an image can be expressed as the outcome of an appropriate combination of several non overlapped intensity slices this image representation decomposes an image into a number of binary slices of the same size whose pixels come in two intensities black or any other gray level value therefore the image block representation can be effectively applied to describe the image in a more compact way once the image is partitioned into intensity blocks the computation of the image moments can be accelerated as the moments can be computed by using decoupled computation forms the proposed algorithm constitutes a unified methodology that can be applied to any discrete moment family in the same way and produces similar promising results as has been concluded through a detailed experimental investigation

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

on group fourier analysis and symmetry preserving discretizations of pdes in this paper we review some group theoretic techniques applied to discretizations of pdes inspired by the recent years active research in lie group and exponential time integrators for differential equations we will in the first part of the paper present algorithms for computing matrix exponentials based on fourier transforms on finite groups as an example we consider spherically symmetric pdes where the discretization preserves the 120 symmetries of the icosahedral group this motivates the study of spectral element discretizations based on triangular subdivisions in the second part of the paper we introduce novel applications of multivariate non separable chebyshev polynomials in the construction of spectral element bases on triangular and simplicial sub domains these generalized chebyshev polynomials are intimately connected to the theory of root systems and weyl groups used in the classification of semi simple lie algebras and these polynomials share most of the remarkable properties of the classical chebyshev polynomials such as near optimal lebesgue constants for the interpolation error the existence of fft based algorithms for computing interpolants and pseudo spectral differentiation and existence of gaussian integration rules the two parts of the paper can be read independently

degenerate and singular evolution equations in banach space on etudie l equation operateur ba 1 u a 0 u h dans des espaces de banach sous certaines hypotheses permettant d appliquer les resultats obtenus pour le probleme aux valeurs initiales d a 1 t u t dt a 0 t u t h t 0 t t lim t 0 a 1 t u t 0 ou a i t i 0 1 est un bon operateur lineaire clos entre espaces de banach

a systematic method for efficient computation of full and subsets zernike moments a new method is proposed for fast and accurate computation of zernike moments this method presents a novel formula for computing exact zernike moments by using exact complex moments where the exact values of complex moments are computed by mathematical integration of the monomials over digital image pixels the proposed method is applicable to compute the full set of zernike moments as well as the subsets of individual order repetition and an individual moment a comparison with other conventional methods is performed the results show the superiority of the proposed method

a new class of zernike moments for computer vision applications a modified direct method for the computation of the zernike moments is presented in this paper the presence of many factorial terms in the direct method for computing the zernike moments makes their computation process a very time consuming task although the computational power of the modern computers is impressively increasing the calculation of the factorial of a big number is still an inaccurate numerical procedure the main concept of the present paper is that by using stirling s approximation formula for the factorial and by applying some suitable mathematical properties a novel factorial free direct method can be developed the resulted moments are not equal to those computed by the original direct method but they are a sufficiently accurate approximation of them besides their variability does not affect their ability to describe uniquely and distinguish the objects they represent this is verified by pattern recognition simulation examples

group theory and quasiprobability integrals of wigner functions the integral of the wigner function of a quantum mechanical system over a region or its boundary in the classical phase plane is called a quasiprobability integral unlike a true probability integral its value may lie outside the interval 0 1 it is characterized by a corresponding selfadjoint operator to be called a region or contour operator as appropriate which is determined by the characteristic function of that region or contour the spectral problem is studied for commuting families of region and contour operators associated with concentric discs and circles of given radius a their respective eigenvalues are determined as functions of a in terms of the gauss laguerre polynomials these polynomials provide a basis of vectors in a hilbert space carrying the positive discrete series representation of the algebra su 1 1 so 2 1 the explicit relation between the spectra of operators associated with discs and circles with proportional radii is given in terms of the discrete variable meixner polynomials

circularly orthogonal moments for geometrically robust image watermarking circularly orthogonal moments such as zernike moments zms and pseudo zernike moments pzms have attracted attention due to their invariance properties however we find that for digital images the invariance properties of some zms pzms are not perfectly valid this is significant for applications of zms pzms by distinguishing between the good and bad zms pzms in terms of their invariance properties we design image watermarks with good zms pzms to achieve watermark s robustness to geometric distortions which has been considered a crucial and difficult issue in the research of digital watermarking simulation results show that the embedded information can be decoded at low error rates robust against image rotation scaling flipping as well as a variety of other common manipulations such as lossy compression additive noise and lowpass filtering

fast generic polar harmonic transforms generic polar harmonic transforms have recently been proposed to extract rotation invariant features from images and their usefulness has been demonstrated in a number of pattern recognition problems however direct computation of these transforms from their definition is inefficient and is usually slower than some efficient computation strategies that have been proposed for other methods this paper presents a number of novel computation strategies to compute these transforms rapidly the proposed methods are based on the inherent recurrence relations among complex exponential and trigonometric functions used in the definition of the radial and angular kernels of these transforms the employment of these relations leads to recursive and addition chain based strategies for fast computation of harmonic function based kernels experimental results show that the proposed method is about 10 times faster than direct computation and 5 times faster than fast computation of zernike moments using the q recursive strategy thus among all existing rotation invariant feature extraction methods polar harmonic transforms are the fastest

a novel approach to the fast computation of zernike moments this paper presents a novel approach to the fast computation of zernike moments from a digital image most existing fast methods for computing zernike moments have focused on the reduction of the computational complexity of the zernike 1 d radial polynomials by introducing their recurrence relations instead in our proposed method we focus on the reduction of the complexity of the computation of the 2 d zernike basis functions as zernike basis functions have specific symmetry or anti symmetry about the x axis the y axis the origin and the straight line y x we can generate the zernike basis functions by only computing one of their octants as a result the proposed method makes the computation time eight times faster than existing methods the proposed method is applicable to the computation of an individual zernike moment as well as a set of zernike moments in addition when computing a series of zernike moments the proposed method can be used with one of the existing fast methods for computing zernike radial polynomials this paper also presents an accurate form of zernike moments for a discrete image function in the experiments results show the accuracy of the form for computing discrete zernike moments and confirm that the proposed method for the fast computation of zernike moments is much more efficient than existing fast methods in most cases

a parallel recurrence method for the fast computation of zernike moments abstract this paper presents a parallel recursive method for the computation of zernike moments from a digital image the proposed method can reduce the computational complexity of the zernike radial polynomials by introducing a novel recurrence relation and be applicable to either the computation of a single zernike moment or entire set of zernike moments the fast computation is achieved because it involves less addition and multiplication operations and is executed in parallel moreover the single zernike moment can be obtained with employing partial zernike moments of lower orders the experiments are carried out to evaluate the performance of the proposed method using binary and grayscale images the experimental results show that the proposed method takes the shortest time in computing the zernike moments of a specific order 28 as well as the entire zernike moments of orders 70

stable fast computation of high order zernike moments using a recursive method zernike moments and zernike polynomials have been widely applied in the fields of image processing and pattern recognition when high order zernike moments are computed both computing speed and numerical accuracy become inferior the main purpose of this study is to propose a stable fast method for computing high order zernike moments based on the recursive formulas for computing zernike radial polynomials this study develops stable fast algorithms to compute zernike moments symmetry under group action and farey sequence are both applied to shorten the computing time the experimental results show that the proposed method took 5 292 seconds to compute the top 500 order zernike moments of an image with 512 512 pixels the normalized mean square error is 0 00124846 if 450 order moments are used to reconstruct the image when computing the high order zernike moments the proposed method outperformed other compared methods in terms of speed and accuracy this study has proposed a recursive method for fast computation of zernike moments the idea of pascal s triangle is introduced to pre calculate the binomial coefficients symmetry property and farey sequence are applied to speed up the computation the proposed method can yield accurate values of the high order zernike moments the proposed took 5 398 seconds to compute the top 500 order zernike moments

fast and efficient method for computing art angular radial transform art which is the region based shape descriptor of mpeg 7 has desirable properties for representing shape information in a small number of features with no redundancy however in order for art to be useful especially in limited computing environments the computational cost of art must be greatly reduced in this paper we derive symmetric antisymmetric properties from the basis functions of art and present a fast and efficient method to compute the art coefficients using these properties the proposed method significantly reduces the number of sinusoidal operations and multiplications in computing the coefficients of art moreover the memory requirements needed to store the art basis functions in lookup tables are only 25 of the conventional method the experimental results are presented to show the effectiveness of the proposed method

efficient computation of radial moment functions using symmetrical property the applications of radial moment functions such as orthogonal zernike and pseudo zernike moments in real world have been limited by the computational complexity of their radial polynomials the common approaches used in reducing the computational complexity include the application of recurrence relations between successive radial polynomials and coefficients in this paper a novel approach is proposed to further reduce the computation complexity of zernike and pseudo zernike polynomials based on the symmetrical property of radial polynomials by using this symmetrical property the real valued radial polynomials computation is reduced to about one eighth of the full set polynomials while the computation of the exponential angle values is reduced by half this technique can be integrated with existing fast computation methods to further improve the computation speed besides significant reduction in computation complexity it also provides vast reduction in memory storage

multi dimensional option pricing using radial basis functions and the generalized fourier transform we show that the generalized fourier transform can be used for reducing the computational cost and memory requirements of radial basis function methods for multi dimensional option pricing we derive a general algorithm including a transformation of the black scholes equation into the heat equation that can be used in any number of dimensions numerical experiments in two and three dimensions show that the gain is substantial even for small problem sizes furthermore the gain increases with the number of dimensions