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面部识别。这样的图像或信号重建问题通常被建模为大规模优化问题。 本研究计画旨在发展更有效率的演算法来解决这些最佳化问题。预计该项目的结果将在实际应用中产生影响。特别是，正在开发的数值计划，预计将支持医学成像研究，并协助提高临床决策的准确性。在本项目的过程中，对高年级本科生和研究生进行培训。本研究计划旨在发展多尺度邻近演算法，以解决影像或讯号处理中的最佳化问题。信号处理问题的实际意义，如不完整的数据恢复，压缩感知，矩阵完成，建模为优化问题，具有不可微的目标函数。感兴趣的信号自然具有层次结构，或者允许其本身在多尺度分析中稀疏地表示。多尺度分析，作为一种方便的计算工具，然而，主要是用来稀疏的基础信号在制定优化问题，它没有充分利用在开发有效的算法优化问题。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，研究人员将综合和联合收割机多尺度分析和邻近算法，以准确和计算效率高的方式解决问题。

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.