Multiresolution analysis for optimal binary filters

The performance of a designed digital filter is measured by the sum of the errors of the optimal filter and the estimation error. Viewing an image at a high resolution results in optimal filters having smaller errors than at lower resolutions; however, higher resolutions bring increased estimation e rror. Hence, choosing an appropriate resolution for filter design is import ant. The present paper provides expressions for both the error of the optim al filter and the design error for estimating optimal filters in a pyramida l multiresolution framework. The analysis is facilitated by a general chara cterization of suitable sequences of resolution-constraint mappings. The er ror expressions are generated from resolution to resolution in a telescopin g manner. To take advantage of data at all resolutions, one can use a hybri d multiresolution design to arrive at a multiresolution filter. A sequence of filters is designed using data at increasing resolutions, each filter se rves as a prior filter for the next, and the last filter is taken as the de signed filter. The value of the multiresolution filter at a given observati on is based on the highest resolution at which conditioning by the observat ion is considered significant.