  We consider the analysis operator and synthesis dictionary learning problems
based on the the $\ell_1$ regularized sparse representation model. We reveal
the internal relations between the $\ell_1$-based analysis model and synthesis
model. We then introduce an approach to learn both analysis operator and
synthesis dictionary simultaneously by using a unified framework of bi-level
optimization. Our aim is to learn a meaningful operator (dictionary) such that
the minimum energy solution of the analysis (synthesis)-prior based model is as
close as possible to the ground-truth. We solve the bi-level optimization
problem using the implicit differentiation technique. Moreover, we demonstrate
the effectiveness of our leaning approach by applying the learned analysis
operator (dictionary) to the image denoising task and comparing its performance
with state-of-the-art methods. Under this unified framework, we can compare the
performance of the two types of priors.
