Compactly supported directional wavelet frames and their applications

小波变换将复杂的对象/功能分解为基本的构建块（称为小波），以便可以通过小波系数分析和理解对象。小波理论以其稀疏的多尺度表示和快速的计算算法，在工业和应用科学领域有着广泛的应用，如JPEG 2000图像压缩标准、生物医学信号/数据处理、大数据小波网络等。 在当今大数据爆炸的信息时代，具有快速转换的有效表示系统用于从数据中提取关键结构和信息是非常重要的。框架小波是小波的一种推广，具有冗余性和灵活性等特点。数据的关键结构往往在于它们的低维奇异性，如图像中的边缘。方向标架被证明比小波更有效地捕捉这种奇异性，例如，小框架在噪声去除方面明显优于小波。为了大大推进小波理论及其应用，本提案的目的是开发创新的方向多尺度表示系统，使用compensation支持的多变量方向framelets旨在超越国家的最先进的方法在应用中，如图像处理和数据分析。此外，许多数据是在有界域中给出的，例如，图像和微分方程的解。这就要求表示系统使用复杂的支持元素来正确处理域边界，实现良好的空间定位，并具有快速算法。 为了实现我们的目标，我们的两个主要相关的方法是发展的理论和探索的应用程序的紧支持的方向复杂紧框架和多元拟紧框架。首先，我们将发展必要的数学理论和构造方法，我们提出的方向复杂的紧框架和准紧框架。接下来，与学生合作，我们计划开发他们的基础计算算法，以便我们和其他研究人员可以将我们的算法应用于图像处理和许多其他问题。最后，我们将与来自行业的学生和研究人员（例如，加拿大的石油/矿产勘探公司，如Quartic.ai Inc.）从而将我们所发展的关于方向复紧标架和多元拟紧标架的数学方法和计算算法应用于地球物理反问题、地震层析成像和数据分析等实际工业问题。这将直接促进加拿大的工业，从而促进我们加拿大的经济。拟议的研究也将大大有助于建立新的数学理论和培养高素质的人才，为加拿大的未来。

A wavelet transform decomposes a complicated object/function into elementary building blocks (called wavelets) so that the object can be analyzed and understood through wavelet coefficients. Featured by sparse multiscale representation and fast computational algorithms, wavelet theory, which is a hot and fast-growing interdisciplinary area, has numerous applications in industry and applied sciences with great success such as JPEG 2000 standard for image compression, biomedical signal/data processing, and wavelet networks for big data. Effective representation systems with fast transforms for extracting key structures and information from data are extremely crucial in today's information era with explosion of big data. Framelets generalize wavelets with the extra much desired features of redundancy and flexibility. Key structures of data often lie in their low dimensional singularities such as edges in images. Directional framelets are proven to be much more effective to capture such singularities than wavelets, e.g., framelets significantly outperform wavelets for noise removal. To significantly advance wavelet theory and its applications, the objective of this proposal is to develop innovative directional multiscale representation systems using compactly supported multivariate directional framelets intended to outperform the state-of-the-art methods in applications such as image processing and data analysis. Also, many data are given in bounded domains, e.g., images and solutions of differential equations. This demands for representation systems using compactly supported elements to properly handle domain boundaries, to achieve good spatial localization, and to have fast algorithms. To achieve our objective of this proposal, our two main correlated approaches are to develop the theories and explore applications of compactly supported directional complex tight framelets and multivariate quasi-tight framelets. First, we will develop the necessary mathematical theory and construction methods on our proposed directional complex tight framelets and quasi-tight framelets. Next, working with students, we plan to develop their underlying computational algorithms so that we and other researchers can apply our algorithms for image processing and many other problems. Finally, we will team up with students and researchers from industry (e.g., oil/mineral exploration companies in Canada such as Quartic.ai Inc.) so that we will apply and test our developed mathematical methods and computational algorithms on directional complex tight framelets and multivariate quasi-tight framelets to some practical industrial problems such as geophysical inverse problems, seismic tomography and data analysis. This will directly contribute to industry in Canada and consequently to our Canadian economy. The proposed research will also significantly contribute in establishing new mathematical theory and in training highly qualified personnel for the future of Canada.