The polynomial phase difference operator for parametric modeling of 2-D nonhomogeneous signals

A fundamental problem in two-dimensional signal processing is the modeling and analysis of nonhomogeneous two-dimensional (2-D) signals. For example, in almost any image taken by a camera, perspective exists, and hence the acquired 2-D signal is nonhomogeneous, even if the original scene was homogeneous. Conventional approaches to the problems of perspective and camera orientation estimation usually involve local analysis of the image, by means of edge detection algorithms. Parametric models, when used in image processing, generally assume the observed image to be homogeneous, or piecewise homogeneous. We consider a parametric model which is nonhomogeneous, and attempts to perform global (or at least, less localized) image analysis. We study a model consisting of a sine (or cosine) of a polynomial function of the image coordinates. For practical reasons it is more convenient to work with a complex valued model in which the sinusoidal function is replaced by a complex exponential. In applications where the 2-D signal is real, it can be converted subject to some restrictive conditions, into complex form through the Hilbert transform. We consider 2-D signals which can be represented by a constant amplitude complex exponential whose phase is a polynomial function of the coordinates.