It is well known that two-dimensional (2D) filter bank is far removed from a straightforward extension of one-dimensional (1D) filter bank. There are many challenging problems on the theory and design methods for the 2D filter bank. Among these problems, the perfect-reconstruction (PR) theory of the 2D DFT modulated filter bank with arbitrary modulation and decimation matrices remains an unsolved difficulty, which is the focus of this paper. The necessary and sufficient condition for perfect reconstruction (PR) is derived by using the polyphase decomposition of the analysis and synthesis filters, as well as the fast implementation structure of the filter bank. Then, the PR condition in frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF), which is utilized to formulate the design problem into an unconstrained optimization problem. An efficient iterative algorithm is proposed to solve the problem. Numerical examples are included to verify the validity of the PR condition and the effectiveness of the design method. 
Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices