CIF: Small: Theory and Algorithms for Scalable Learning of Sparse Representations

在信号和图像处理和成像中的许多应用严重依赖于自然信号或图像的稀疏表示。本研究解决了直接适应数据的改进稀疏表示的发展，而不是通过一般理论考虑先验地固定。这种数据驱动的稀疏结构学习已经得到了广泛的应用。对于高维数据，相对于固定表示的改进尤其显著。本研究旨在克服现有方法的局限性，通过减少计算量以适应大数据问题，提高结果的鲁棒性和可预测性，并发展一种量化预期性能及其影响因素的理论。预计应用于科学和工程的所有领域，包括医疗诊断、多媒体、国防、制造、通信、数据库检索和数据分析。特别是，这项研究利用了PI最近引入的一种新公式，用于数据驱动的“稀疏化变换”学习，这是分析字典的亲戚。图像去噪的初步结果显示，与学习合成字典相比，PSNR略好，但计算量减少了几个数量级。理论预测与样本大小更好的缩放，实验证明鲁棒收敛与初始化无关。本研究的具体目标是：(1)发展学习稀疏化变换的理论和可扩展算法；(2)发展压缩感知中稀疏化变换和信号恢复的联合学习理论及其他逆问题；(3)用实际数据演示关键的大规模应用。这些应用包括：(i)磁共振成像、计算机断层扫描、显微镜和视频中3D和4D数据的去噪、恢复和压缩感知；（二）图像分类与识别。

Many applications in signal and image processing and imaging depend critically on sparse representations for natural signals or images. This research addresses the development of improved sparse representations that are directly adapted to the data, rather than being fixed a priori by general theoretical considerations. Such data-driven learning of sparse structure has been finding broad applications. The improvements over fixed representation are especially significant for high-dimensional data. This research aims to overcome limitations of the current methods by reducing the computation to enable scaling to big data problems, improving robustness and predictability of outcome, and developing a theory quantifying the expected performance and the factors affecting it. Applications are foreseen in all areas of science and engineering, including medical diagnostics, multimedia, defense, manufacturing, communications, database retrieval, and data analytics. In particular, this research leverages a new formulation recently introduced by the PI for data-driven learning of "sparsifying transforms", which are relatives of analysis dictionaries. Initial results in image denoising show slightly better PSNR than with learnt synthesis dictionaries, but at orders of magnitude less computation. Theory predicts better scaling with exemplar size, and experiments demonstrate robust convergence irrespective of initialization. Specific objectives of this research are: (1) develop theory and scalable algorithms for learning sparsifying transforms; (2) develop theory for joint learning of sparsifying transforms and signal recovery in compressed sensing and other inverse problems; and, (3) demonstrate key large-scale applications with real data. These applications include: (i) denoising, restoration, and compressed sensing of 3D and 4D data in magnetic resonance imaging, in computerized tomography, in microscopy, and in video; and (ii) image classification and recognition.