estimates on translations and taylor expansions in fractional sobolev spaces abstract in this paper we study how the normalised gagliardo semi norms u w s p r n control translations in particular we prove that u y u l p r n c u w s p r n y s for n 1 s 0 1 and p 1 where c depends only on n we then obtain a corresponding higher order version of this result we get fractional rates of the error term in the taylor expansion we also present relevant implications of our two results first we obtain a direct proof of several compact embedding of w s p r n where the frechet kolmogorov theorem is applied with known rates we also derive fractional rates of convergence of the convolution of a function with suitable mollifiers thirdly we obtain fractional rates of convergence of finite difference discretisations for w s p r n

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

an adaptable image retrieval system with relevance feedback using kernel machines and selective sampling this paper presents an adaptable content based image retrieval cbir system developed using regularization theory kernel based machines and fisher information measure the system consists of a retrieval subsystem that carries out similarity matching using image dependant information multiple mapping subsystems that adaptively modify the similarity measures and a relevance feedback mechanism that incorporates user information the adaptation process drives the retrieval error to zero in order to exactly meet either an existing multiclass classification model or the user high level concepts using reference model or relevance feedback learning respectively to facilitate the selection of the most informative query images during relevance feedback learning a new method based upon the fisher information is introduced model reference and relevance feedback learning mechanisms are thoroughly tested on a domain specific image database that encompasses a wide range of underwater objects captured using an electro optical sensor benchmarking results with two other relevance feedback learning methods are also provided

boundary behaviors for liouville s equation in planar singular domains abstract we study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary the conformal structure plays an essential role

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

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

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

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

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

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

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

a new characterization of sobolev spaces on mathbb r n in this paper we present a new characterization of sobolev spaces on mathbb r n our characterizing condition is obtained via a quadratic multiscale expression which exploits the particular symmetry properties of euclidean space an interesting feature of our condition is that depends only on the metric of mathbb r n and the lebesgue measure so that one can define sobolev spaces of any order of smoothness on any metric measure space

characterization of sobolev spaces on the sphere abstract we introduce a new characterization of the sobolev spaces h on the unit sphere s d 1 where the smoothness index is any positive real number and d 2 this characterization is in terms of a square function and it avoids the notion of differentiability the result is fully based on zonal fourier multipliers special functions and the subtle use of classical analysis tools

on the accuracy of zernike moments for image analysis we give a detailed analysis of the accuracy of zernike moments in terms of their discretization errors and the reconstruction power it is found that there is an inherent limitation in the precision of computing the zernike moments due to the geometric nature of a circular domain this is explained by relating the accuracy issue to a celebrated problem in analytic number theory of evaluating the lattice points within a circle

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

existence of compactly supported global minimisers for the interaction energy the existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses the main assumption on the potential is that it is catastrophic or not h stable which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics the proof is based on a uniform control on the local mass around each point of the support of a global minimiser together with an estimate on the size of the gaps it may have the class of potentials for which we prove the existence of global minimisers includes power law potentials and for some range of parameters morse potentials widely used in applications we also show that the support of local minimisers is compact under suitable assumptions

characterizations of sobolev spaces via averages on balls abstract in this paper the authors prove several equivalent characterizations of sobolev spaces of even integer orders on r n using the average b t f x 1 b x t b x t f y d y of a function f over the ball b x t y r n y x t with x r n and t 0 these characterizations rely only on the metric and the lebesgue measure on r n and are simpler than those obtained recently by alabern et al 2012 moreover these results may shed new light on the theory of high order sobolev spaces on spaces of homogeneous type

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

on image analysis by moments research has been performed investigating the use of moments for pattern recognition in recent years the basic problem of the influence of discretization and noise on moment accuracy as object descriptors has been barely investigated in this paper the detailed error analysis involved in the moment method is discussed several new techniques to increase the accuracy and efficiency of moment descriptor are proposed we utilize these results for the problem of image reconstruction from the orthogonal legendre moments computed from discrete and noisy data the automatic selection of an optimal number of moments is also discussed

littlewood paley functions and sobolev spaces abstract we establish littlewood paley characterizations of the sobolev spaces w p in euclidean spaces using several square functions defined via the spherical average the ball average the bochner riesz means and some other well known operators we provide a simple proof so that we are able to extend and improve many results published in recent papers

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

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

the number of n point digital discs a digital disc is the set of all integer points inside some given disc let dn be the number of different digital discs consisting of n points different up to translation the upper bound d n o n2 was shown recently no corresponding lower bound is known in this paper we refine the upper bound to dn o n which seems to be the true order of magnitude and we show that the average dn d1 d2 dn n has upper and lower bounds which are of polynomial growth in n

littlewood paley characterizations of triebel lizorkin morrey spaces via ball averages abstract let 1 p u and q 1 in this article the authors establish equivalent characterizations of the triebel lizorkin morrey space e u q p r n with smoothness order 0 2 in terms of the lusin area function and the g function generated by the difference between f x and its ball average b 2 j f x 1 b x 2 j b x 2 j f y d y j 1 2 x r n where for any j 1 2 and x r n b x 2 j y r n y x 2 j as applications the authors obtain several characterizations of e u p r n via pointwise inequalities involving ball averages

characterizations of besov and triebel lizorkin spaces via averages on balls abstract let l n and p 1 in this article the authors prove that the sequence f b l 2 k f k z consisting of the differences between f and the ball average b l 2 k f characterizes the besov space b p q r n with q 0 and the triebel lizorkin space f p q r n with q 1 when the smoothness order 0 2 l more precisely it is proved that f b l 2 k f plays the same role as the approximation to the identity 2 k f appearing in the definitions of b p q r n and f p q r n the corresponding results for inhomogeneous besov and triebel lizorkin spaces are also obtained these results for the first time give a way to introduce besov and triebel lizorkin spaces with any smoothness order in 0 2 l on spaces of homogeneous type where l n

some equivalent definitions of high order sobolev spaces on stratified groups and generalizations to metric spaces recently in the article lw the authors use the notion of polynomials in metric spaces mathcal s rho mu of homogeneous type in the sense of coifman weiss to prove a relationship between high order poincare inequalities and representation formulas involving fractional integrals of high order assuming only that mu is a doubling measure and that geodesics exist motivated by this and by recent work in h fhk ks and flw about first order sobolev spaces in metric spaces we define sobolev spaces of high order in such metric spaces mathcal s rho mu we prove that several definitions are equivalent if functions of polynomial type exist in the case of stratified groups where polynomials do exist we show that our spaces are equivalent to the sobolev spaces defined by folland and stein in fs our results also give some alternate definitions of sobolev spaces in the classical euclidean case

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

orthogonal moment features for use with parametric and non parametric classifiers this research examines a variety of approaches for using two dimensional orthogonal polynomials for the recognition of handwritten arabic numerals it also makes use of parametric and non parametric statistical and neural network classifiers polynomials including legendre zernike and pseudo zernike are used to generate moment based features which are invariant to location size and optionally rotation an efficient method for computing the moments via geometric moments is presented a side effect of this method also yields scale invariance a new approach to location invariance using a minimum bounding circle is presented and a detailed analysis of the rotational properties of the moments is given data partitioning tests are performed to evaluate the various feature types and classifiers for rotational invariant character recognition the highest percentage of correctly classified characters was 91 7 and for non rotational invariant recognition it was 97 6 this compares with a previous effort using the same data and test conditions of 94 8 the techniques developed here should also be applicable to other areas of shape recognition

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

orthogonal rotation invariant moments for digital image processing orthogonal rotation invariant moments orims such as zernike moments are introduced and defined on a continuous unit disk and have been proven powerful tools in optics applications these moments have also been digitized for applications in digital image processing unfortunately digitization compromises the orthogonality of the moments and therefore digital orims are incapable of representing subtle details in images and cannot accurately reconstruct images typical approaches to alleviate the digitization artifact can be divided into two categories 1 careful selection of a set of pixels as close approximation to the unit disk and using numerical integration to determine the orim values and 2 representing pixels using circular shapes such that they resemble that of the unit disk and then calculating orims in polar space these improvements still fall short of preserving the orthogonality of the orims in this paper in contrast to the previous methods we propose a different approach of using numerical optimization techniques to improve the orthogonality we prove that with the improved orthogonality image reconstruction becomes more accurate our simulation results also show that the optimized digital orims can accurately reconstruct images and can represent subtle image details