A fast algorithm for Toeplitz-block-Toeplitz linear systems

ABSTRACT A Toeplitz-block-Toeplitz (TBT) matrix is block Toeplitz with Toeplitz blocks. TBT systems of equations arise in 2D interpolation, 2-D linear prediction and 2-D least-squares deconvolution problems. Although the doubly Toeplitz structure should be exploitable in a fast algorithm, existing fast algorithms only exploit the block Toeplitz structure, not the Toeplitz structure of the blocks. Iterative algorithms can employ the 2-D FFT, but usually take thousands of iterations to converge. We develop a new fast algorithm that assumes a smoothness constraint (described in the text) on the matrix entries. For an TBT matrix with M Toeplitz blocks along each edge, the algorithm requires only operations to solve an linear system of equations; parallel computing on 2M processors can be performed on the algorithm as given. Two examples show the operation and performance of the algorithm.