复杂区域上的小波框架理论及应用

In recent years, we are experiencing rapid advances in information and computer technology, which contribute greatly to the exponential growth of data. To properly handle, process and analyze such huge and often unstructured data sets, sophisticated mathematical tools and efficient computing methods direly need to be developed. Unlike traditional signals and images, however, these data often have rather complicated structure, which made existing tools and methods in signal and image processing not readily applicable. Therefore, we need to develop brand new mathematical tools, theory and methodology specifically for these new datasets. ..The proposed study aims at developing comprehensive theory and systematic constructions of wavelet frames on generic domains such as surfaces and high dimensional point clouds. The theory and constructions will specifically target some real world problems as well. Furthermore, we will design effective, efficient and stable mathematical models and numerical algorithms for different applications. In theory, we will quantify sparsity of wavelet frames and analyze their approximation properties. To better understand wavelet frame systems on generic domains, we will also consider connections between wavelet frame transforms and differential operators in full depth. In applications, we will conduct comprehensive numerical studies for various applications to test on the models and algorithms developed from the proposed studies.

随着计算机和信息科技的飞速发展，来自各行业的数据呈现爆炸式的增长，如何存储、处理、分析这些海量并且结构复杂的数据是目前全世界的一个科研热点。与传统的数据（比如信号和图像）不同的是，这些新的数据大多结构十分复杂，这使得传统的信号与图像处理的工具并不能直接用来处理这些新型的数据。因此，我们需要为这些全新的、结构复杂的数据量身定做一套新的数学工具，并针对不同实际问题合理的建模。..本项目主要研究的是在与实际问题紧密结合的前提下，讨论小波框架在复杂区域上的构造，并针对实际问题搭建理论体系。另外，我们也会针对不同实际问题设计高效、稳定、低计算代价的模型和数值算法。在理论上，我们将对小波框架进行稀疏性量化和逼近度分析，并挖掘小波框架变换与微分算子的深层联系，从而加深我们对复杂区域上的小波框架的理解。在应用方面，我们将通过大量的数值实验和实际应用问题来检验模型及算法的有效性。

项目主要研究如何非欧式结构数据（点云、图）上的信号处理及其在医学影像、计算机视觉和机器学习中的应用。在欧式结构数据上的信号处理最常用也是最有效的算子是卷积算子，这包括小波变换、微分算子的有限差分离散、卷积神经网络等等。因此，要做好在非欧式结构数据上的信号处理，必须要先给出卷积的定义。本项目基于平行移动定义了2维曲面和3维点云上的卷积算子，设计了相应的深层卷积神经网络，在几何体的分类和分割任务中性能优越。此外，项目还利用基于卷积算子和微分算子之间的深层联系，设计出新的监督学习和半监督学习模型，并在医疗影像、图像识别、反问题计算中得到成功的应用。其中两篇工作发表在机器学习顶级期刊ICML 2018上，目前在谷歌学术上的引用分别为235和182次。