  This paper addresses a new learning algorithm for the recently introduced
co-sparse analysis model. First, we give new insights into the co-sparse
analysis model by establishing connections to filter-based MRF models, such as
the Field of Experts (FoE) model of Roth and Black. For training, we introduce
a technique called bi-level optimization to learn the analysis operators.
Compared to existing analysis operator learning approaches, our training
procedure has the advantage that it is unconstrained with respect to the
analysis operator. We investigate the effect of different aspects of the
co-sparse analysis model and show that the sparsity promoting function (also
called penalty function) is the most important factor in the model. In order to
demonstrate the effectiveness of our training approach, we apply our trained
models to various classical image restoration problems. Numerical experiments
show that our trained models clearly outperform existing analysis operator
learning approaches and are on par with state-of-the-art image denoising
algorithms. Our approach develops a framework that is intuitive to understand
and easy to implement.
