  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 in frame theory toward sparse
l1-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 sought 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 over-complete frames, thereby precluding
the use of any linear analysis operator in driving the sparse synthesis
coefficient for signal representation. Nonetheless, the best approximation to
the sparse synthesis solution can be derived from the analysis coefficient
using the canonical dual frame. In this study, 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.
