Compressed sensing is a powerful tool in applications such as magnetic
resonance imaging (MRI). It enables accurate recovery of images from highly
undersampled measurements by exploiting the sparsity of the images or image
patches in a transform domain or dictionary. In this work, we focus on blind
compressed sensing (BCS), where the underlying sparse signal model is a priori
unknown, and propose a framework to simultaneously reconstruct the underlying
image as well as the unknown model from highly undersampled measurements.
Specifically, our model is that the patches of the underlying image(s) are
approximately sparse in a transform domain. We also extend this model to a
union of transforms model that better captures the diversity of features in
natural images. The proposed block coordinate descent type algorithms for blind
compressed sensing are highly efficient, and are guaranteed to converge to at
least the partial global and partial local minimizers of the highly non-convex
BCS problems. Our numerical experiments show that the proposed framework
usually leads to better quality of image reconstructions in MRI compared to
several recent image reconstruction methods. Importantly, the learning of a
union of sparsifying transforms leads to better image reconstructions than a
single adaptive transform.

本研究提出了一种同时重建基础图像和未知模型的盲压缩感知框架，用于利用稀疏变换或字典中图像 / 图像补丁的稀疏性从高度欠采样的测量中精确恢复图像。我们将这个模型扩展到了一个转换融合模型，以更好地捕捉自然图像中的特征。与其他最近的图像重建方法相比，我们的数值实验表明，提出的框架通常会导致 MRI 图像重建的质量更好。

盲压缩感知的稀疏变换联合学习模型基于数据驱动

Data-Driven Learning of a Union of Sparsifying Transforms Model for  Blind Compressed Sensing

The fundamental principle underlying compressed sensing is that a signal,
which is sparse under some basis representation, can be recovered from a small
number of linear measurements. However, prior knowledge of the sparsity basis
is essential for the recovery process. This work introduces the concept of
blind compressed sensing, which avoids the need to know the sparsity basis in
both the sampling and the recovery process. We suggest three possible
constraints on the sparsity basis that can be added to the problem in order to
make its solution unique. For each constraint we prove conditions for
uniqueness, and suggest a simple method to retrieve the solution. Under the
uniqueness conditions, and as long as the signals are sparse enough, we
demonstrate through simulations that without knowing the sparsity basis our
methods can achieve results similar to those of standard compressed sensing,
which relay on prior knowledge of the sparsity basis. This offers a general
sampling and reconstruction system that fits all sparse signals, regardless of
the sparsity basis, under the conditions and constraints presented in this
work.

介绍了盲压缩感知的概念，提出了三种可能的基稀疏性约束，并证明了唯一性条件及提出了检索解决方案的简洁方法。在唯一性条件下，只要信号足够稀疏，我们的方法可以通过模拟实现类似于依赖于稀疏基先验知识的标准压缩感知的结果，这提供了一个适用于所有稀疏信号的通用采样和重构系统，无需了解稀疏基，只需满足本文中提出的条件和约束即可。