Compactly supported directional wavelet frames and their applications

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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.