Development and Application of Directional Framelets and Complex Multiwavelets

许多复杂的结构是由简单得多的基本结构组成的，例如，音频信号由不同频率的基本正弦组成。因此，可以应用变换将复杂的结构分解成基本的构建块，从而可以分析和研究其特性。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它：称为小波的波状元素。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逼近来提取某些几何结构(如点奇异性)的最优方法。多尺度在自然界中非常常见，在科学上也很有用，例如，2013年诺贝尔化学奖授予三名科学家，以表彰他们对复杂化学体系的多尺度模型的开发。小波和基于小波的方法在解决工业和应用科学中的许多实际问题方面被证明是强大和有效的，并取得了巨大的成功，但许多多维函数/数据具有其他重要的几何结构，如方向(边缘)奇异性，包含了数学建模中的图像和微分方程组的解等函数/数据的大多数关键信息。自傅立叶变换问世以来，寻找有效的数学表示和快速变换来提取不同类型的结构的工作从未停止，在当今数字数据爆炸的信息时代，从数据中快速提取所需的关键信息变得极其关键。在这项拟议的研究中，我们将引入创新的方向多尺度数学表示，具有快速的变换和高效的计算算法，使用所提出的方向复框架和矩阵值多小波方法。Framlet类似于小波，但在其表示中允许冗余。与传统的小波相比，我们提出的方向多尺度表示具有许多有前途的优点和理想的性质。例如，它们具有更好的捕捉多维函数/数据的方向奇异性的能力，具有类似于传统快速小波变换的计算效率的快速框架变换，并且具有许多期望的数学性质，所提出的方法在几何建模、工业问题、计算、成像和信息技术等领域具有潜在的应用前景。我们在这方面的初步实验结果已经显示出比最先进的小波算法在图像去噪问题上的显著进步，这是几乎所有处理数字数据的实际问题的第一步预处理。在这个建议的第一阶段，我们将发展必要的数学理论，在我们提出的使用方向复数框架和矩阵多小波的方向多尺度表示上。在第二阶段，我们计划与业界(例如加拿大的石油/天然气/矿产勘探公司，特别是艾伯塔省的石油/天然气/矿产勘探公司)的学生和研究人员合作，以便将我们开发的定向多尺度表示的数学方法应用于并测试一些实际的工业问题，如地球物理反问题和地震层析成像。这将直接促进加拿大的勘探业，从而促进我们加拿大的经济。拟议的研究还将有助于建立新的数学理论，并为加拿大的未来培养高素质的人才。

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.