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We consider the analysis operator and synthesis dictionary learning problemsbased on the the $\ell_1$ regularized sparse representation model. We revealthe internal relations between the $\ell_1$-based analysis model and synthesismodel. We then introduce an approach to learn both analysis operator andsynthesis dictionary simultaneously by using a unified framework of bi-leveloptimization. Our aim is to learn a meaningful operator (dictionary) such thatthe minimum energy solution of the analysis (synthesis)-prior based model is asclose as possible to the ground-truth. We solve the bi-level optimizationproblem using the implicit differentiation technique. Moreover, we demonstratethe effectiveness of our leaning approach by applying the learned analysisoperator (dictionary) to the image denoising task and comparing its performancewith state-of-the-art methods. Under this unified framework, we can compare theperformance of the two types of priors.

Learning $\ell_1$-based analysis and synthesis sparsity priors using bi-level optimization