The usefulness is illustrated of two-dimensional (2-D) Gaussian-Markov random field (MRF) models for coding of textures. The MRF models used are noncausal; the mean of observationy(s) at positionsis written as a linear weighted sum of neighboring observations surroundingsin all directions. The method of least squares is used to obtain estimates of the model parameters. The model is then used with periodic boundary conditions to regenerate the original texture. Results obtained indicate that this method could be used to compress textures with low bit rates.

A transform coding scheme for closed image boundaries on a plane is described. The given boundary is approximated by a series of straight line segments. Depending on the shape, the boundary is represented by the (x-y) coordinates of the endpoints of the line segments or by the magnitude of the successive radii vectors that are equispaced in angle around the given boundary. Due to the circularity present in the data, the discrete Fourier transform is used to exactly decorrelate the finite boundary data. By fitting a Gaussian circular autoregressive model to represent the boundary data, estimates of the variances of the Fourier coefficients are obtained. Using the variances of the Fourier coefficients and the MAX quantizer, the coding scheme is implemented. The scheme is illustrated by an example.