Frame-Based Sparse Analysis and Synthesis Signal Representations and Parseval K-SVD

Frames are the foundation of the linear operators used in the decomposition and reconstruction of signals, such as the discrete Fourier transform, Gabor, wavelets, and curvelet transforms. The emergence of sparse representation models has shifted the emphasis in frame theory toward sparse-norm minimization problems. In this paper, we apply frame theory to the sparse representation of signals, wherein a frame is used for synthesis dictionary and a dual frame is used for analysis dictionary. Our objective was to formulate a novel dual-frame design in which the sparse vector obtained through the decomposition of any signal is also the sparse solution representing signals based on a reconstruction frame. Our findings demonstrate that this type of dual frame cannot be constructed for overcomplete frames, thereby precluding the use of any linear analysis operator to derive the sparse synthesis coefficient for signal representation. We also developed a novel dictionary learning algorithm (called Parseval K-SVD) to learn a tight-frame dictionary with normalized atoms. We then dealt with the problem of signal representation using frames from the analysis and synthesis perspective to derive optimization formulations for problems pertaining to image recovery. Our preliminary results demonstrate that the images recovered using this approach are correlated with the frame bounds of dictionaries, thereby demonstrating the importance of using different dictionaries for different applications.