Property of circular convolution for subband image coding

Classification of Nonexpansive Symmetric Extension Transforms for Multirate Filter Banks

“…The topic of M-channel SET's has subsequently been treated in more detail in [21,3,49,4,8,13,14,47,36]. Bamberger et al [3,49,4] also considered M-channel nonuniformly downsampled filter banks.…”

“…Nishikawa et al [47,36] consider two different types of decimation, termed "Decimation I," which corresponds to (2.1), and "Decimation II," which amounts to (2.1) preceded by a phase shift of M/2 samples (only for M even). Since any such phase shift can be incorporated into the phase of the analysis filter, h, this paper will only use the downsampler (2.1) and will classify SETs in terms of the group delays of the analysis filters.…”

“…While this has not yet been studied using real data, the concept is appealing insofar as it adheres to the philosophy of viewing cascaded subband coders (e.g., discrete wavelet transforms) as quantitative approximations of ideal analog multiresolution decompositions. Readers interested in transforms based on antisymmetric extensions are directed to [47,36]. In the present paper, antisymmetric extensions will be used exclusively for reconstructing antisymmetric subbands during synthesis.…”

Symmetric-extension-compatible reversible integer-to-integer wavelet transforms

“…In the case of conventional wavelet transforms, a popular technique for handing finite-length signals in a nonexpansive manner is symmetric extension. The use of symmetric extension was first proposed by Smith and Eddins [12], and subsequently received considerable attention in the literature (e.g., [11,[18][19][20][21][22][23][24]). With this technique, a signal is extended so that it is both symmetric and periodic.…”

“…where x, y ∈ Z and α is a real constant. If α ∈ Z, (19) is not invertible (i.e., one cannot always uniquely solve for x in terms of y). For any α ∈ R, (20) and 21…”