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，包括医学应用中的并行磁共振成像和安全识别中的指纹和面部识别。这类图像或信号重建问题通常被建模为大规模优化问题。本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。该项目的研究成果有望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，并有助于提高临床决策的准确性。大四本科生和研究生在这个项目的过程中得到培养。本研究计划旨在发展多尺度接近演算法，以解决影像或讯号处理中出现的最佳化问题。具有实际重要性的信号处理问题，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层次结构，或者允许自己在多尺度分析中被稀疏表示。多尺度分析作为一种方便的计算工具，在表述优化问题时主要用于对底层信号进行稀疏化；它在优化问题的有效算法开发中没有得到充分利用。在本项目中，为了系统地利用所关注的优化问题中存在的层次结构，研究者将综合并结合多尺度分析和邻近算法，以精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.