Polynomial approximation of images

A transform method for compressing digital data on a two-dimensional lattice, based on polynomial approximation is studied. The least square polynomial approximation in the sense that the “distance” between the original and the reconstructed image is used. The mathematical results establish a basis for a useful implementation for image compression, and transmission having a specific tolerable distortion. Properties of the set of matrix-compressors which minimizes the distance, the matrix-transform, and the minimal error are explicitly found. A necessary and sufficient condition for “distortion-free compression” (i.e., lossless-compression) is found. The tradeoff between the image distortion and the transmitted information is discussed. Experimental results address practical issues of quantization, bit allocation, pyramidal decomposition, and error distribution.