Compactly supported directional wavelet frames and their applications

小波变换将复杂的对象/函数分解为基本构建块（称为小波），以便可以通过小波系数来分析和理解对象。小波理论以其稀疏多尺度表示和快速计算算法为特点，是一个热门且快速发展的交叉学科领域，在工业和应用科学领域有着广泛的应用，并取得了巨大成功，例如用于图像压缩的JPEG 2000标准、生物医学信号/数据处理和用于大数据的小波网络。 在当今大数据爆炸的信息时代，具有快速转换功能的有效表示系统从数据中提取关键结构和信息至关重要。小框架概括了小波，具有额外的冗余和灵活性特征。数据的关键结构通常在于其低维奇点，例如图像中的边缘。事实证明，定向框架比小波更能有效地捕获此类奇点，例如，框架在噪声消除方面明显优于小波。为了显着推进小波理论及其应用，该提案的目标是使用紧凑支持的多元定向框架来开发创新的定向多尺度表示系统，旨在超越图像处理和数据分析等应用中最先进的方法。此外，许多数据都是在有界域中给出的，例如图像和微分方程的解。这要求表示系统使用紧凑支持的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。 为了实现本提案的目标，我们的两个主要相关方法是发展紧支撑定向复杂紧框架和多元准紧框架的理论和探索应用。首先，我们将对我们提出的定向复杂紧框架和准紧框架开发必要的数学理论和构造方法。接下来，我们计划与学生合作开发他们的底层计算算法，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，我们将与工业界的学生和研究人员（例如加拿大的石油/矿产勘探公司，如 Quartic.ai Inc.）合作，将我们开发的数学方法和计算算法应用于定向复杂紧框架和多元准紧框架，并测试一些实际的工业问题，如地球物理反演问题、地震层析成像和数据分析。这将直接为加拿大的工业做出贡献，从而为加拿大经济做出贡献。拟议的研究还将为建立新的数学理论和为加拿大的未来培养高素质人才做出重大贡献。

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.