Computation of the Dual Frame: Forward and Backward Greville Formulas

摘要

We study the computation of the dual frame for oversampled filter banks (OFBs) by exploiting Greville's formula, which was derived in 1960 to compute the pseudo inverse of a matrix when a new row is appended. In this paper, we first develop the backward Greville formula to handle the case of row deletion. Based on Greville's formula, we then study the dual frame computation of the Laplacian pyramid. Through the backward Greville formula, we investigate OFBs for robust transmission over erasure channels. The necessary and sufficient conditions for OFBs robust to one erasure channel are derived. A post- filtering structure is also presented to implement the dual frame when the transform coefficients in one subband are completely lost.