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.

本文提出了一种新的学习算法来应对近期提出的共稀疏分析模型，使用双层优化技术来学习分析算子，研究了共稀疏分析模型的不同方面，并将训练模型应用于多个经典的图像恢复问题。

分析算子学习的深入洞见：从基于补丁的稀疏模型到高阶 MRFs

Insights into analysis operator learning: From patch-based sparse models  to higher-order MRFs

High-resolution depth maps can be inferred from low-resolution depth
measurements and an additional high-resolution intensity image of the same
scene. To that end, we introduce a bimodal co-sparse analysis model, which is
able to capture the interdependency of registered intensity and depth
information. This model is based on the assumption that the co-supports of
corresponding bimodal image structures are aligned when computed by a suitable
pair of analysis operators. No analytic form of such operators exist and we
propose a method for learning them from a set of registered training signals.
This learning process is done offline and returns a bimodal analysis operator
that is universally applicable to natural scenes. We use this to exploit the
bimodal co-sparse analysis model as a prior for solving inverse problems, which
leads to an efficient algorithm for depth map super-resolution.

通过学习联合分析运算符的方法，提出了一种双模共稀疏分析模型，将强度图像和深度信息结合起来用于高分辨率深度图像的超分辨率重建。