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 features extraction. Given a rectangular M × 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 polynomials coefficients can be expressed as a linear one-to-one separable transform of the pixels. The transform 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′ × N′ coefficients, in the separable formulation, involves the inversion of two M′ × M′ and N′ × N′ Hankel matrices.