Collaborative Research: Multiscale Proximity Algorithms for Optimization Problems Arising from Image/Signal Processing

在各种应用中，需要从有限的可用数据中恢复图像或信号，包括医疗应用中的并行磁共振成像以及安全识别中的指纹和人脸识别。这样的图像或信号重建问题通常被建模为大规模的优化问题。这一研究项目旨在开发更有效的计算算法来解决这些优化问题。预计该项目的成果将在实际应用中产生影响。特别是，正在开发的数值方案有望支持医学成像研究，并帮助提高临床决策的准确性。高年级本科生和研究生都是在这个项目的过程中培养的。该研究项目旨在为图像或信号处理中出现的优化问题开发多尺度邻近算法。具有实际意义的信号处理问题，如不完全数据恢复、压缩感知和矩阵补全，被建模为具有不可微目标函数的优化问题。感兴趣的信号自然具有分层结构，或者允许在多尺度分析中稀疏地表示其自身。然而，多尺度分析作为一种方便的计算工具，主要用于在描述优化问题时对底层信号进行稀疏处理，而在开发高效的优化算法时并未得到充分利用。在这个项目中，为了系统地利用感兴趣的优化问题中存在的层次结构，研究人员将综合和结合多尺度分析和邻近算法，以准确和计算高效的方式解决问题。

Restoring images or signals from limited available data is required in variety of applications, including parallel magnetic resonance imaging in medical applications and fingerprint and face recognition in security identification. Such image or signal reconstruction problems are often modeled as large-scale optimization problems. This research project aims to develop more efficient computational algorithms for solving these optimization problems. Results of this project are anticipated to have an impact in practical applications. In particular, the numerical schemes under development are expected to support medical imaging research and assist in improving the accuracy of clinical decisions. Senior undergraduate and graduate students are trained in the course of this project. This research project aims to develop multiscale proximity algorithms for optimization problems arising in image or signal processing. Signal processing problems of practical importance, such as incomplete data recovery, compressive sensing, and matrix completion, are modeled as optimization problems that have non-differentiable objective functions. A signal of interest naturally has a hierarchical structure or allows itself to be sparsely represented in a multiscale analysis. Multiscale analysis, as a convenient tool for computation, however, is mainly used to sparsify the underlying signal in formulating optimization problems; it is not fully exploited in development of efficient algorithms for optimization problems. In this project, to make systematic use of the hierarchical structure that exists in optimization problems of interest, the investigators will synthesize and combine multiscale analysis and proximity algorithms to solve the problems in an accurate and computationally efficient way.