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

合作研究：成像变分模型的小波框架：桥接离散和连续体

• 批准号: 1418772
• 负责人: Moysey Brio
• 金额: $ 17.08万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418772&HistoricalAwards=false
• 关键词: Collaborative; Research; Wavelet; Frames; Variational

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.

从科学开始，视觉观察就扮演着重要的角色。计算机技术的进步使得一些最复杂的数学和科学发展应用于设计和实现在大量处理器上运行的快速算法来处理图像数据成为可能。因此，图像处理和分析技术现在几乎应用于所有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；数字图像已经进入了每个人的生活。数学从一开始就在图像和信号处理中扮演着重要的角色。图像恢复有两种主要的数学方法，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，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、国家研究实验室以及工业界的研究人员受益。对小波帧的几何方面的理解以及与微分算子的联系将有助于计算调和分析和变分技术和数值偏微分方程的社区。该教育计划将把本科生和研究生带到计算数学、计算机视觉和医学成像的研究前沿；加强数学家、工程师、计算机科学家和医生之间的合作。小波帧是函数系统，它提供了特定函数空间（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，这在许多应用中是理想的。虽然小波帧的大多数理论方面已经在文献中得到了很好的理解，但小波帧变换的几何意义仍然普遍未知。事实上，缺乏几何解释是小波框架的主要缺陷之一，它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存在的感兴趣的对象进行几何正则化。本文的主要研究目标是通过研究小波帧变换与各种变分框架内的微分算子的关系，对小波帧变换进行通用的几何解释。基于几何解释，我们提出了一些重要的应用，如图像恢复（去模糊，修复，CT/MR成像等）新的模型和算法。通过理论分析和数值实验，我们将探讨所提出的基于小波框架的模型相对于现有变分模型和微分模型在不同应用中的优势。本文的研究主要集中在：(1)用小波框架变换在一般变分框架内逼近微分算子；(2)利用小波帧进行凸/非凸优化，求解大规模病态逆问题（如图像恢复、盲反卷积）；(3)设计并求解基于小波帧的模型在实际成像中的应用，如低剂量CT图像重建、去除相机抖动引起的模糊等。对小波帧变换几何意义的研究，将从一个全新的角度来解释小波帧及其相关的优化模型。这样的基础研究使我们第一次能够在涉及几何的数据分析任务和求偏微分方程的数值解中充分利用小波帧的独特性质。在各种应用中，小波框架变换相对于标准有限差分近似的实际优势（如反演问题的质量）将在本文提出的研究之后变得更加明显。此外，该项目还将对变分模型的数值求解方法带来新的认识；并回答了一些基本的和重要的问题，从文献中不清楚的变分模型。