adaptive noise smoothing filter for images with signal dependent noise in this paper we consider the restoration of images with signal dependent noise the filter is noise smoothing and adapts to local changes in image statistics based on a nonstationary mean nonstationary variance nmnv image model for images degraded by a class of uncorrelated signal dependent noise without blur the adaptive noise smoothing filter becomes a point processor and is similar to lee s local statistics algorithm 16 the filter is able to adapt itself to the nonstationary local image statistics in the presence of different types of signal dependent noise for multiplicative noise the adaptive noise smoothing filter is a systematic derivation of lee s algorithm with some extensions that allow different estimators for the local image variance the advantage of the derivation is its easy extension to deal with various types of signal dependent noise film grain and poisson signal dependent restoration problems are also considered as examples all the nonstationary image statistical parameters needed for the filter can be estimated from the noisy image and no a priori information about the original image is required

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

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

analysis for the reconstruction of a noisy signal based on orthogonal moments suppose that we have a set of discrete data of a noisy signal and we reconstruct the original signal from its orthogonal moments based on legendre polynomials in this paper we derive a new error estimation of the reconstruction which reveals the relationships between the number of moments the signal smoothness and sampling rate numerical simulations provided are consistent with our analysis

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

aggregation error in bayesian analysis of reliability systems perfect aggregation in bayesian system reliability analysis has been shown to be extremely unlikely in other words aggregation error is almost inevitable consequently analysts have to deal with the following dilemma on one hand an aggregate analysis i e an analysis at the system level while relatively inexpensive may be misleading on the other hand a disaggregate analysis i e at the component level provides more accurate results but may be costly and impractical therefore simple techniques to estimate the size of aggregation error are necessary to help analysts choose the most appropriate level of detail for an analysis in this paper reasonable bounds on the aggregation error are derived for a variety of reliability models in particular these bounds will never be more than twice the actual error tools to compute these bounds and in some cases the actual error are also provided

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

ambiguity and fuzziness measures defined on the set of closed intervals in 0 1 starting from the idea of activity orders we define a family of orders on the set of closed intervals of a distributive lattice l that will allow us to set up some preorders associated to the ambiguity and fuzziness in order to measure the ambiguity and fuzziness degree of any closed interval in 0 1

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

a moment based variational approach to tomographic reconstruction we describe a variational framework for the tomographic reconstruction of an image from the maximum likelihood ml estimates of its orthogonal moments we show how these estimated moments and their correlated error statistics can be computed directly and in a linear fashion from given noisy and possibly sparse projection data moreover thanks to the consistency properties of the radon transform this two step approach moment estimation followed by image reconstruction can be viewed as a statistically optimal procedure furthermore by focusing on the important role played by the moments of projection data we immediately see the close connection between tomographic reconstruction of nonnegative valued images and the problem of nonparametric estimation of probability densities given estimates of their moments taking advantage of this connection our proposed variational algorithm is based on the minimization of a cost functional composed of a term measuring the divergence between a given prior estimate of the image and the current estimate of the image and a second quadratic term based on the error incurred in the estimation of the moments of the underlying image from the noisy projection data we show that an iterative refinement of this algorithm leads to a practical algorithm for the solution of the highly complex equality constrained divergence minimization problem we show that this iterative refinement results in superior reconstructions of images from very noisy data as compared with the classical filtered back projection fbp algorithm

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

on the shape from moments problem and recovering edges from noisy radon data we consider the problem of reconstructing a planar convex set from noisy observations of its moments an estimation method based on pointwise recovering of the support function of the set is developed we study intrinsic accuracy limitations in the shape from moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error it is shown that the proposed estimator is near optimal in the sense of the order an application to tomographic reconstruction is discussed and it is indicated how the proposed estimation method can be used for recovering edges from noisy radon data

image analysis by modified legendre moments in the paper a new set of orthogonal moments based on the modified legendre polynomials is introduced three properties of the modified legendre polynomials which are orthogonality orthogonal invariance and the characteristic that an interval on the center of 1 1 covers more zeros than do that on the edge of 1 1 are discussed detailedly the orthogonality of the proposed moments ensures minimal information redundancy in a moment set the orthogonal invariance of the proposed polynomials makes the proposed moments have the property of translation invariance and the third property ensures that the modified legendre moments have superior feature representation capabilities over legendre moments in analyzing small images for small images the description by the modified legendre moments is better than that by the legendre moments and the chebyshev moments in terms of image reconstruction errors theoretical and experimental measures of performance are carried out to investigate the image representation capabilities of the proposed moments for images and noisy images the computational aspects of the moments using recurrence integral symmetry and translation invariance are also discussed experimental results are shown

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

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

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

on the 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

template matching in rotated images a rotationally invariant template matching using normalized invariant moments is described it is shown that if normalized invariant moments in circular windows are used then template matching in rotated images becomes similar to template matching in translated images a speedup technique based on the idea of two stage template matching is also described in this technique the zeroth order moment is used in the first stage to determine the likely match positions and the second and third order moments are used in the second stage to determine the best match position among the likely ones

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

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

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

recursive computation of tchebichef moment and its inverse transform tchebichef moment is a novel set of orthogonal moment applied in the fields of image analysis and pattern recognition less work has been made for the computation of tchebichef moment and its inverse moment transform in this paper both a direct recursive algorithm and a compact algorithm are developed for the computation of tchebichef moment the effective recursive algorithm for inverse tchebichef moment transform is also presented clenshaw s recurrence formula was used in this paper to transform kernels of the forward and inverse tchebichef moment transform there is no need for the proposed algorithms to compute the tchebichef polynomial values the approaches presented are more efficient compared with the straightforward methods and particularly suitable for parallel vlsi implementation due to their regular and simple filter structures

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

optimal estimation of contour properties by cross validated regularization the problem of estimating the properties of smooth continuous contours from discrete noisy samples is used as vehicle to demonstrate the robustness of cross validated regularization applied to a vision problem a method for estimation of contour properties based on smoothing spline approximations is presented generalized cross validation is to devise an automatic algorithm for finding the optimal value of the smoothing regularization parameter from the data the cross validated smoothing splines are then used to obtain optimal estimates of the derivatives of quantized contours experimental results are presented which demonstrate the robustness of the method applied to the estimation of curvature of quantized contours under variable scale rotation and partial occlusion these results suggest the application of generalized cross validation to other computer vision algorithms involving regularization