Two-sided prediction, orthogonalization, and random fields in images

A useful way of finding new signal-processing algorithms is to think of problems like linear prediction in a geometric way-finding prediction coefficients is the same as orthogonalizing the basis of powers of z on the unit circle. This means that powerful concepts from the theory of approximation and orthogonal polynomials, such as three-term recurrences, can be used to find highly efficient algorithms to solve least squares problems. We show how to apply these concepts to different problems than just prediction. In particular, we find optimal linear interpolation coefficients by orthogonalizing a new, different basis of Chebyshev-type functions in order 8 n-squared steps. Interpolation is also known as the non-causal or two-sided prediction problem. It is important not just for linear phase applications, but also in image processing, where the optimum linear interpolator describes a Gauss-Markov random field, as used in image restoration, texture analysis, segmentation and spatial clutter reduction