Polynomial Representation of Pictures

In many image processing applications, the discrete values of an image can be embedded in a continuous function. This type of representation can be useful for interpolation, geometrical transformations or special feature extraction. Given a rectangular M x N discrete image (or sub-image), it is shown how to compute a continuous polynomial function that guarantees an exact fit at the considered pixel locations. the polynomial coefficients can be expressed as a linear one-to-one separable transforms of the pixels. The trasform matrices can be computed using a fast recursive algorithm which enables efficient inversion of a Vandermonde matrix. It is also shown that the least square polynomial approximation with M' x N' coefficients, in the separable formulation, involves the inversion of two M' x M' and N' x N' Hankel matrices.