Two-Dimensional $2\times$ Oversampled DFT Modulated Filter Banks and Critically Sampled Modified DFT Modulated Filter Banks

This paper investigates two-dimensional (2D)
$2\times$
oversampled DFT modulated filter banks and 2D critically sampled modified DFT (MDFT) modulated filter banks as well as their design. The structure and perfect reconstruction (PR) condition of 2D
$2\times$
oversampled DFT modulated filter banks are presented in terms of the polyphase decompositions of prototype filters (PFs). In the double-prototype case, the part solutions of the PR condition are parameterized by imposing the 2D two-channel lifting structure on each pair of the polyphase components of analysis and synthesis PFs. Based on the parametric structure, the analysis and synthesis PFs are separately designed by constrained quadratic programs. The obtained filter banks are of structurally PR. Moreover, 2D critically sampled MDFT modulated filter banks are proposed. It is proved that 2D critically sampled PR MDFT modulated filter banks can be rebuilt from 2D
$2\times$
oversampled PR DFT modulated filter banks when the decimation matrices satisfy a permissible condition and the analysis and synthesis PFs are identical and symmetric with respect to the origin. A numerical algorithm is given to design 2D critically sampled PR MDFT modulated filter banks and the obtained filter banks are of numerically PR.