Generalized Hermite polynomials for image reconstruction from zero crossing contours

Generalized Hermite polynomials in two variables are employed for the reconstruction of images from a knowledge of their zero crossing contours. The problem of reconstruction of signals as functions of two variables is not a mere extension of that of a single variable. This is a consequence of the fact that the spatial and spectral characteristics of two-variable functions are quite distinct from what one can expect from their separate projections on to the coordinate axes. One of the results of the paper is that we cannot guarantee uniqueness in reconstruction unless we impose certain constraints on, for instance, space-bandwidth products/ratios in the xω_x, yω _y directions, of the unknown image. Further, a distinguishing feature of the proposed approach is that the standard assumption of bandlimitedness is not invoked. The proposed framework is believed to provide a more unified procedure for signal reconstruction (of uni- and multi-dimensional signals) from partial information than most of the results of the literature. For lack of space, only the main analytical and computational results are presented