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: 1418737
• 负责人: Weiyu Xu
• 金额: $ 12.92万
• 依托单位: University of Iowa
• 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 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418737&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.

从科学诞生之初，目视观察就一直扮演着重要的角色。计算机技术的进步使得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成为可能。因此，图像处理和分析技术现在几乎应用于所有的自然科学和技术学科，从计算机科学、电子工程到生物和医学科学；数字图像已经走进了每个人的生活。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。这个项目的主要研究目标是通过将前一种方法与后一种方法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、国家研究实验室以及工业界的研究人员受益。理解小波框架的几何方面以及与微分算子的联系将有助于计算调和分析的社区以及变分技术和数值偏微分方程组的社区。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研究的前沿，并加强数学家、工程师、计算机科学家和医学博士之间的合作。小波框架是一种函数系统，它提供了生活在特定函数空间(如L2(Rn))中的函数的线性表示。与经典的(双)正交小波基相比，这种表示通常是冗余的，这在许多应用中是可取的。虽然小波框架的大多数理论方面在文献中已经被很好地理解了，但小波框架变换的几何意义仍然是未知的。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的感兴趣对象进行几何正则化。这一建议的主要研究目标是通过研究小波框架变换与各种变分框架内的微分算子之间的关系，发展对小波框架变换的一般几何解释。基于几何解释，我们提出了一些新的模型和算法，用于图像恢复(去模糊、修复、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基于小波框架的模型相对于现有的变分模型和微分模型在不同应用中的优势。主要研究内容包括：(1)在一般变分框架下利用小波框架对微分算子的逼近；(2)利用小波框架进行凸/非凸优化，解决大规模的不适定反问题(如图像恢复、盲反卷积等)；(3)设计和求解基于小波框架的成像模型，如低剂量CT图像重建、去除相机抖动引起的模糊等。研究小波框架变换的几何意义将从一个全新的角度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几何数据分析任务中充分利用小波框架的独特性质，并找到偏微分方程组的数值解。在所提出的研究之后，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用优势(如反问题的恢复质量)将变得更加明显。此外，该项目还将给求解变分模型的数值方法带来新的认识，并回答了变分模型的一些基本和重要的问题，这些问题在文献中还不清楚。