National Institute on Aging, National Institutes of Health, Baltimore, Maryland, U.S.A.
NMR Biomed. 2020 Dec;33(12):e4315. doi: 10.1002/nbm.4315. Epub 2020 Aug 16.
There has been a tremendous increase in applications of the inverse problem framework to parameter estimation in magnetic resonance. Attempting to capture both the basics of this formalism and modern developments would require an article of inordinate length. Therefore, in the following, we provide basic material as a practical introduction to the topic and an entree to the literature. First, we describe the formulation of linear and nonlinear inverse problems, with an emphasis on signal equations arising in magnetic resonance. We then describe the Fredholm equation of the first kind as a paradigm for these problems. This is followed by much more detailed considerations for determining solutions in the linear case, including central concepts such as condition number, regularization, and stability. Solution methods for nonlinear inverse problems are described next, followed by a treatment of their stability and regularization. Finally, we provide an introduction to compressed sensing, with signal reconstruction formulated as the solution to an inverse problem, making use of much of the previous material. Throughout, the emphasis is on outlines of the theory and on numerical examples, rather than on mathematical rigor and completeness.
磁共振参数估计中反问题框架的应用有了巨大的增长。尝试捕捉这个形式主义和现代发展的基础需要一篇冗长的文章。因此，在下面，我们提供基本的材料作为这个主题的实用介绍和文献的入口。首先，我们描述线性和非线性反问题的公式，重点是磁共振中出现的信号方程。然后，我们将第一类弗雷德霍姆方程描述为这些问题的范例。接下来是更详细地考虑线性情况下的解，包括条件数、正则化和稳定性等核心概念。接下来描述非线性反问题的解方法，然后是它们的稳定性和正则化的处理。最后，我们介绍压缩感知，将信号重建表述为反问题的解，利用前面的大部分材料。在整个过程中，重点是理论的概述和数值例子，而不是数学的严谨性和完整性。
