Symmetric extension for lifted filter banks and obstructions to reversible implementation

Symmetric pre-extension is a standard approach to boundary handling for finite-length input vectors with linear phase filter banks. It works with both conventional linear implementations and the so-called reversible, or integer-to-integer, implementations of odd-length linear phase (whole-sample symmetric) filter banks. In comparison, significant difficulties arise when using symmetric pre-extension on reversible filter banks with even-length (half-sample symmetric) linear phase filters. An alternative approach is presented using lifting step extension, in which boundary extensions are performed in each step of a lifting factorization, avoiding some of these difficulties while preserving reversibility and retaining the nonexpansive property of symmetric pre-extension. Another alternative that is capable of preserving both reversibility and subband symmetry for half-sample symmetric filter banks is developed based on ideas from the theory of lattice vector quantization. The practical ramifications of this work are illustrated by describing its influence on the specification of filter bank algorithms in Part 2 of the ISO/IEC JPEG 2000 image coding standard.