Learning Parseval Frames for Sparse Representation with Frame Perspective

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 of the emphasis of signal representation in frame theory toward sparse $l_{1}$-minimization problems. In this paper, we apply frame theory to the sparse representation of signals in which a synthesis dictionary is used for a frame and an analysis dictionary is used for a dual frame. We developed a novel dictionary learning algorithm (called Parseval K-SVD) to learn a tight-frame dictionary. We then leveraged the analysis and synthesis perspectives of signal representation with frames to derive optimization formulations for problems pertaining to image recovery. Our preliminary results demonstrate that the images recovered using this approach are correlated to the frame bounds of dictionaries, thereby demonstrating the importance of using different dictionaries for different applications.