Low Rank Pure Quaternion Approximation for Pure Quaternion Matrices

：丁维洋
：2021-07-22 15:20
：厦大海韵园实验楼105报告厅

报告人：丁维洋（复旦大学）

时  间：2021年7月22日下午15:20

地  点：厦大海韵园实验楼105报告厅

内容摘要:

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains real component which is not useful for the representation of a color image. The main contribution of this talk is to find an optimal rank-r pure quaternion matrix approximation for a pure quaternion matrix. Our idea is to use a projection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating projections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the projection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a non-trivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the projection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices.

个人简介：

丁维洋博士分别于2011年和2016年在复旦大学获学士和博士学位，随后在香港理工大学作博士后研究，2017年9月至2020年11月在香港浸会大学任研究助理教授，其后于2020年11月加入复旦大学类脑智能科学与技术研究院，担任青年研究员。丁博士的主要研究兴趣包括结构矩阵、张量计算和优化及其在脑与类脑科学、数据分析、信号处理等领域的应用。丁博士已出版学术专著1本，发表高质量期刊论文15篇（包括SIAM系列，JSC，LAA等），其中有3篇是ESI高被引论文。目前，他主持国家自然科学基金委的青年科学基金项目以及一项上海“求索杰出青年”计划项目。

联系人：杜魁教授