## On the recovery of a function on a circular domain

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##### Date

2002-10-31T18:59:06Z

##### Authors

Pawlak, M

Liao, SX

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##### Abstract

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.

##### Description

##### Keywords

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

##### Citation

0018-9448; IEEE TRANS INFORM THEORY, OCT 2002, vol. 48, no. 10, p.2736 to 2753.