Development and Application of Directional Framelets and Complex Multiwavelets

许多复杂的结构是由更简单的基本结构构成的，例如，音频信号是由不同频率的基本正弦波组成的。因此，可以应用转换将复杂的结构分解为基本的构建块，以便对其属性进行分析和研究。转换的有效性在很大程度上取决于其基本构建块的属性。小波分析是这样一种数学方法，通过将函数或数字数据表示为基本构建块的线性组合来转换函数或数字数据：称为小波的类波元素。具有快速变换的小波具有许多理想的性质，被认为是通过稀疏多尺度近似提取某些几何结构（如点奇点）的最佳选择。多尺度在自然界中非常普遍，在科学中也很有用，例如，2013年诺贝尔化学奖授予了三位科学家，以表彰他们为复杂化学系统开发了多尺度模型。小波和基于小波的方法被证明在解决工业和应用科学中的许多实际问题方面是强大而有效的，并取得了巨大的成功。

Many complicated structures are formed by much simpler elementary ones, e.g., audio signals are composed of basic sinusoids with different frequencies. Thus, a transform can be applied to break a complicated structure into basic building blocks so that its properties can be analyzed and studied. The effectiveness of a transform largely lies in the properties of its basic building blocks. Wavelet analysis is such a mathematical method for transforming a function or digital data by representing it as a linear combination of basic building blocks: wave-like elements called wavelets. Having many desired properties, wavelets with fast transforms are known to be optimal for extracting certain geometric structures such as point singularities through sparse multiscale approximation. Multiscale is very common in nature and useful in science, e.g., the Nobel Prize in Chemistry 2013 was awarded to three scientists for the development of multiscale models for complex chemical systems. Wavelets and wavelet-based methods are proven to be powerful and effective in solving many practical problems in industry and applied sciences with great success. But a lot of multidimensional functions/data have other important geometric structures such as directional (edge-like) singularities, holding most key information of functions/data such as images and solutions of differential equations in mathematical modeling. Since the far-reaching Fourier transform, the search for effective mathematical representations with fast transforms for extracting different types of structures never ends and becomes extremely crucial in today's information era with explosion of digital data for quickly extracting the sought key information from data. In this proposed research, we will introduce innovative directional multiscale mathematical representations, with fast transforms and efficient computational algorithms, using the proposed approaches of directional complex framelets and matrix-valued multiwavelets. Framelets are similar to wavelets but allow redundancy in their representations. Our proposed directional multiscale representations have many promising advantages and desired properties over the traditional wavelets. For example, they have much better ability in capturing directional singularities in multidimensional functions/data, have a computationally efficient fast framelet transform similar to the traditional fast wavelet transform, and enjoy many desired mathematical properties. The proposed methods have many potential applications, ranging from geometric modeling, industry problems, computing, imaging, and information technologies. Our initial experimental results in this direction have already shown significant advance over the state-of-the-art wavelet algorithms for the image denoising problem, which is the first-step pre-processing in almost all practical problems dealing with digital data. At the first stage of this proposal, we will develop the necessary mathematical theory on our proposed directional multiscale representations using directional complex framelets and matrix-valued multiwavelets. At the second stage we plan to team up with students and researchers from industry (e.g., oil/gas/mineral exploration companies in Canada, in particular, in Alberta) so that we will apply and test our developed mathematical methods on directional multiscale representations to some practical industrial problems such as geophysical inverse problems and seismic tomography. This will directly contribute to exploration industry in Canada and consequently to our Canadian economy. The proposed research will also contribute in establishing new mathematical theory and in training highly qualified personnel for the future of Canada.