Inverse imaging problems that are ill-posed can be encountered across multiple domains of science and technology, ranging from medical diagnosis to astronomical studies. To reconstruct images from incomplete and distorted data, it is necessary to create algorithms that can take into account both, the physical mechanisms responsible for generating these measurements and the intrinsic characteristics of the images being analyzed. In this work, the sparse representation of images is reviewed, which is a realistic, compact and effective generative model for natural images inspired by the visual system of mammals. It enables us to address ill-posed linear inverse problems by training the model on a vast collection of images. Moreover, we extend the application of sparse coding to solve the non-linear and ill-posed problem in microwave tomography imaging, which could lead to a significant improvement of the state-of-the-arts algorithms.

翻译：逆成像问题通常是不适定的，常见于从医学诊断到天文研究等多个科学与技术领域。为了从不完整和失真的数据中重建图像，需要设计算法，既能考虑生成这些测量值的物理机制，又能兼顾被分析图像的内在特征。本文回顾了图像的稀疏表示，这是一种受哺乳动物视觉系统启发的、针对自然图像的现实、紧凑且有效的生成模型。通过在大规模图像集合上训练该模型，我们能够解决不适定的线性逆问题。此外，我们将稀疏编码的应用扩展到解决微波层析成像中的非线性不适定问题，这可能显著提升现有最先进算法的性能。