Frame-based Approach for Sparse Representation of Signal Decomposition and Reconstruction

Frame is the corner stone for designing decomposition and reconstruction operations in signal processing. Famous frames include wavelets, curvelets, and Gabor. A celebrated result indicates that if a synthesis frame is chosen for reconstruction, then its canonical dual frame is the analysis frame that performs decomposition, yielding coefficients that minimizes l2-norm of all coefficients of all dual frames. This paper tries to extend this result by investigating whether a dual frame can be constructed so that the corresponding coefficients yield the minimum l1-norm. We show that this mission cannot be achieved for any over-complete frame. However, we present some conditions on a dual frame so that the minimizer of the l1-norm can be derived from the coefficients of the dual frame. We show that, based on frame structure, various applications that seek to optimize sparse analysis and synthesis coefficients are logically equivalent. Finally, we present a general construction method of frames to demonstrate the proposed conditions are feasible. In addition, two design approaches are proposed to derive frames that meet the proposed conditions and yield small non-zero elements of frame coefficients.