On the image recovery on a circular domain from noisy data

We consider the problem of estimating a function f(x, y) on the unit disk {(x, y): x²+y² ≤ 1}, given 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 and noise resistant reconstruction method. For radial functions, the orthogonal set has a particularly simple form being related to the classical Legendre polynomials. We give a detailed statistical accuracy analysis of the proposed estimate of f(x, y) in the sense of the L₂ 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.