Multidimensional filter bank signal reconstruction from multichannel acquisition

We study the theory and algorithms of an optimal use of multidimensional signal reconstruction from multichannel acquisition by using a filter bank setup. Suppose that we have an N-channel convolution system, referred to as N analysis filters, in M dimensions. Instead of taking all the data and applying multichannel deconvolution, we first reduce the collected data set by an integer M × M uniform sampling matrix D, and then search for a synthesis polyphase matrix which could perfectly reconstruct any input discrete signal. First, we determine the existence of perfect reconstruction (PR) systems for a given set of finite-impulse response (FIR) analysis filters. Second, we present an efficient algorithm to find a sampling matrix with maximum sampling rate and to find a FIR PR synthesis polyphase matrix for a given set of FIR analysis filters. Finally, once a particular FIR PR synthesis polyphase matrix is found, we can characterize all FIR PR synthesis matrices, and then find an optimal one according to design criteria including robust reconstruction in the presence of noise.