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 error. Hence, choosing an appropriate resolution for filter design is important. The present paper provides expressions for both the error of the optimal filter and the design error for estimating optimal filters in a pyramidal multiresolution framework. The analysis is facilitated by a general characterization of suitable sequences of resolution-constraint mappings. The error expressions are generated from resolution to resolution in a telescoping manner. To take advantage of data at all resolutions, one can use a hybrid multiresolution design to arrive at a multiresolution filter. A sequence of filters is designed using data at increasing resolutions, each filter serves as a prior filter for the next, and the last filter is taken as the designed filter. The value of the multiresolution filter at a given observation is based on the highest resolution at which conditioning by the observation is considered significant.