Generalized Hermite Polynomials for Image Reconstruction from Zero Crossing Contours

Generalized Hermite polynomials in two variables are employed f o r 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 f rom what one can expect from their separate projections o n 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 x w z , ?) w y directions, of the unknown image. A r t h e r , a distinguishing feature of the proposed approach i s that the standard assumption of bandlimitedness is not invoked. T h e proposed framework is believed to provide a more unified procedure for signal reconstruction (of unaand multi-dimensional signals) f rom partial information than most of the results of the literature. For lack of space, only the m a i n analytical and computational results are presented. Indexing Terms: Hermite polynomials, Image representation, Image reconstruction, Scale Space Analysis, Zerocrossings