Miroslaw Pawlak

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.

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.

Nonparametric identification of Hammerstein systems

A discrete-time nonlinear Hammerstein system is identified, and the correlation and frequency-domain methods for identification of its linear subsystem are presented. The main results concern the estimation of the nonlinear memoryless subsystem. No conditions concerning the functional form of the transform characteristic of the subsystem are made, and an algorithm for estimation of the characteristic is given. The algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data. It is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable. Pointwise as well as global consistencies are established. For Lipschitz characteristics the rate of the convergence in probability is O(n/sup -1/3/). >

Accurate Computation of Zernike Moments in Polar Coordinates

An algorithm for high-precision numerical computation of Zernike moments is presented. The algorithm, based on the introduced polar pixel tiling scheme, does not exhibit the geometric error and numerical integration error which are inherent in conventional methods based on Cartesian coordinates. This yields a dramatic improvement of the Zernike moments accuracy in terms of their reconstruction and invariance properties. The introduced image tiling requires an interpolation algorithm which turns out to be of the second order importance compared to the discretization error. Various comparisons are made between the accuracy of the proposed method and that of commonly used techniques. The results reveal the great advantage of our approach

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. >

On Nonparametric Identification of Wiener Systems

In this paper, a new method for the identification of the Wiener nonlinear system is proposed. The system, being a cascade connection of a linear dynamic subsystem and a nonlinear memoryless element, is identified by a two-step semiparametric approach. The impulse response function of the linear part is identified via the nonlinear least-squares approach with the system nonlinearity estimated by a pilot nonparametric kernel regression estimate. The obtained estimate of the linear part is then used to form a nonparametric kernel estimate of the nonlinear element of the Wiener system. The proposed method permits recovery of a wide class of nonlinearities which need not be invertible. As a result, the proposed algorithm is computationally very efficient since it does not require a numerical procedure to calculate the inverse of the estimate. Furthermore, our approach allows non-Gaussian input signals and the presence of additive measurement noise. However, only linear systems with a finite memory are admissible. The conditions for the convergence of the proposed estimates are given. Computer simulations are included to verify the basic theory

Fourier and Hermite series estimates of regression functions

SummaryIn the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X1,Y1),…, (Xn, Yn) of a pair (X, Y) of random variables. We examine an estimate of a type

Distribution-free consistency of a nonparametric kernel regression estimate and classification

{{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} \mathord{\left/ {\vphantom {{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}