Polynomial wavelets on a disc

A set of 2-D orthogonal functions defined on the unit disc B(0,1), called Zernike circle polynomials, is used in the analysis of optical systems by expanding optical wavefront functions as series of these functions. These circle polynomials form an orthonormal basis for the space of square integrable functions on the unit disc, L^2(B(0,1)), and can thus be used to effectively represent signals on circular domains. Wavelets have turned out to be very efficient in image compression but most wavelet analysis done in higher dimensions is suited for functions on a rectangular domain. In this talk, we shall address the problem of constructing wavelets for subspaces of L^2(B(0,1)). The primary motivation is efficient representation of 2-D signals on circular domains, such as images of corneal surfaces, by using wavelets constructed from circle polynomials.