On the recovery of a function on a circular domain

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Pawlak, M
Liao, SX
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We consider the problem of estimating a function f (x, y) on the unit disk {(x, y): x(2) -l- y(2) less than or equal to 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-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.
accuracy, circle orthogonal polynomials, circle problem, circular domain, lattice points, nonparametric estimate, radial functions, rotational invariance, two-dimensional (2-D) functions, Zernike functions, IMAGE-ANALYSIS, ZERNIKE MOMENTS, RECONSTRUCTION, RECOGNITION, REPRESENTATION, POLYNOMIALS
0018-9448; IEEE TRANS INFORM THEORY, OCT 2002, vol. 48, no. 10, p.2736 to 2753.