Collaborative Research: Wavelet Frames for Variational Models in Imaging: Bridging Discrete and Continuum

从科学诞生之初，视觉观察就一直扮演着重要的角色。计算机技术的进步使得有可能将数学和科学中的一些最复杂的发展应用于在大量处理器上运行以处理图像数据的快速算法的设计和实现。因此，图像处理和分析技术现在应用于几乎所有的自然科学和技术学科，从计算机科学和电子工程到生物学和医学科学;数字图像已经进入每个人的生活。数学从一开始就在图像和信号处理中扮演着重要的角色。图像复原的数学方法主要有两种，即小波紧框架方法和微分/变分方法。该项目的主要研究目标是通过将前者与后者联系起来来研究前者的几何方面。它将产生新的数学模型和数值算法，使学术界、国家研究实验室和工业界的研究人员受益。小波框架的几何方面的理解和与微分算子的连接将有助于社区的计算调和分析和社区的变分技术和数值偏微分方程。该教育计划将使本科生和研究生进入计算数学、计算机视觉和医学成像的前沿研究领域，并加强数学家、工程师、计算机科学家和医生之间的合作。小波框架是函数系统，提供了某些函数空间（如L2（Rn））中函数的线性表示。与经典的（双）正交小波基相比，这种表示通常是冗余的，这在许多应用中是期望的。虽然小波框架的大多数理论方面已经很好地理解在文献中，小波框架变换的几何意义仍然是普遍未知的。事实上，缺乏几何解释是小波框架的主要缺陷之一，这阻碍了小波框架在一些重要的数据分析问题中的应用，这些问题需要对数据中的感兴趣对象进行几何正则化。该建议的主要研究目标是开发一个通用的几何解释小波框架变换，通过研究其与微分算子在各种变分框架内的关系。几何解释的基础上，我们提出了新的模型和算法的几个重要的应用，如图像恢复（去模糊，修补，CT/MR成像等）。通过理论分析和数值实验，我们将探索所提出的小波框架为基础的模型比现有的变分和微分模型的不同应用的优势。拟开展的研究将集中在：（1）在一般变分框架下用小波框架变换逼近微分算子;（2）求解大规模不适定反问题（例如，图像恢复，盲反卷积）通过凸/非凸优化的小波框架;（3）设计和解决基于小波框架的模型在现实世界中的成像应用，如低剂量CT图像重建，消除由相机抖动引起的模糊等。小波框架变换的几何意义的研究将从一个全新的角度解释小波框架及其相关的优化模型。这种基础性的研究使我们能够，第一次，充分利用小波框架的独特性质，在几何涉及的数据分析任务，并找到偏微分方程的数值解。小波框架变换在各种应用中相对于标准有限差分近似的实际优势（如反问题的恢复质量）将在所提出的研究之后变得更加明显。此外，本项目还将为求解变分模型的数值方法带来新的理解;并回答文献中不清楚的变分模型的一些基本和重要问题。

From the beginning of science, visual observations have been playing important roles. Advances in computer technology have made it possible to apply some of the most sophisticated developments in mathematics and the sciences to the design and implementation of fast algorithms running on a large number of processors to process image data. As a result, image processing and analysis techniques are now applied to virtually all natural sciences and technical disciplines ranging from computer sciences and electronic engineering to biology and medical sciences; and digital images have come into everyone's life. Mathematics has been playing an important role in image and signal processing from the very beginning. There are two major mathematical approaches for image restoration, namely, wavelet tight frame approaches and differential/variational approaches. The main research objective of this project is to investigate geometric aspects of the former approach by connecting it with the latter. It will give rise to new mathematical models and numerical algorithms that benefit researchers in academia, national research laboratories, as well as in industry. The understandings of the geometric aspects of the wavelet frames and the connections with differential operators will contribute to both the community of computational harmonic analysis and the community of variational techniques and numerical PDEs. The education plan will bring undergraduate and graduate students to the frontiers of research in computational mathematics, computer vision and medical imaging; and strengthen the collaborations among mathematicians, engineers, computer scientists and medical doctors.Wavelet frames are systems of functions that provide linear representations of functions living in certain function spaces such as L2(Rn). In contrast to the classic (bi)orthogonal wavelet bases, such representations are generally redundant which is desirable in many applications. Although most theoretical aspects of wavelet frames have already been well understood in the literature, geometric meanings of the wavelet frame transform are still generally unknown. In fact, the lack of geometric interpretations is one of the major flaws of wavelet frames that prohibits the applications of wavelet frames in some important problems of data analysis that require geometric regularization of the objects-of-interest reside in the data. The main research objective of this proposal is to develop a generic geometric interpretation to the wavelet frame transform, by studying its relations with differential operators within various variational frameworks. Based on the geometric interpretation, we propose new models and algorithms for several important applications such image restoration (deblurring, inpainting, CT/MR imaging, etc.). Through both theoretical analysis and numerical experiments, we will explore the advantages of the proposed wavelet frame based models over the existing variational and differential models for different applications. The proposed research will focus on: (1) the approximation of the differential operators by the wavelet frame transform within general variational frameworks; (2) solving large-scaled ill-posed inverse problems (e.g., image restoration, blind deconvolution) through convex/nonconvex optimizations using wavelet frames; (3) designing and solving wavelet frame based models in real-world applications in imaging such as low-dose CT image reconstruction, removing blurs caused by camera shaking, etc. The study of the geometric meanings of the wavelet frame transform will interpret wavelet frames and their associated optimization models from a whole new angle. Such fundamental study enables us, for the very first time, to fully utilize the unique properties of wavelet frames in geometry-involved data analysis tasks and finding numerical solutions of PDEs. The practical advantages (such as the quality of restoration for inverse problems) of wavelet frame transform over standard finite difference approximations in various applications will become more evident after the proposed studies. Furthermore, this project will also bring new understandings to numerical methods solving variational models; and answers some fundamental and important questions of variational models that are unclear from the literature.