In this paper, 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.
Presenting a thorough overview of the theoretical foundations of non-parametric system identification for nonlinear block-oriented systems, this book shows that non-parametric regression can be successfully applied to system identification, and it highlights the achievements in doing so. With emphasis on Hammerstein, Wiener systems, and their multidimensional extensions, the authors show how to identify nonlinear subsystems and their characteristics when limited information exists. Algorithms using trigonometric, Legendre, Laguerre, and Hermite series are investigated, and the kernel algorithm, its semirecursive versions, and fully recursive modifications are covered. The theories of modern non-parametric regression, approximation, and orthogonal expansions, along with new approaches to system identification (including semiparametric identification), are provided. Detailed information about all tools used is provided in the appendices. This book is for researchers and practitioners in systems theory, signal processing, and communications and will appeal to researchers in fields like mechanics, economics, and biology, where experimental data are used to obtain models of systems.
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.
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