On Orthogonal Transforms of Images Using Paraunitary Filter Banks

Abstract

The use of a paraunitary filter bank (PUFB) for image processing requires a special treatment at image boundaries, to ensure perfect reconstruction (PR) of these regions, and periodic or symmetric extensions are commonly assumed. We reduced the analysis to a one-dimensional signal assuming separable processing. Unlike infinite-length signals, PR PUFBs applied to finite-length signals will not necessarily lead to an orthogonal system. For quantization/processing of the subbands, artifacts at the image boundaries can appear due to artificial discontinuities at borders, lack of orthogonality of the effective boundary filter banks, or improper reconstruction procedure. We will explore linear-phase and nonlinear-phase PUFBs and methods to obtain orthogonality from the boundary filter banks. We will show that for symmetric extensions, orthogonality is only possible for special PUFBs based on linear-phase filters. Using time-varying boundary filter banks, we will discuss a procedure that explores all degrees of freedom of the border filters in a method essentially independent of signal extensions, allowing us to design optimal boundary filter banks, while maintaining fast implementation algorithms.