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形成的，例如，音频信号由具有不同频率的基本正弦波组成。因此，可以应用变换将复杂结构分解为基本构建块，以便可以分析和研究其属性。变换的有效性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，用于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：称为小波的波形元素。具有许多期望的属性，小波与快速变换是已知的最佳提取某些几何结构，如点奇异通过稀疏多尺度近似。多尺度在自然界中非常常见，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在解决工业和应用科学中的许多实际问题上被证明是强大而有效的，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，如方向性（边状）奇异性，包含了数学建模中函数/数据的大多数关键信息，如图像和微分方程的解。由于影响深远的傅立叶变换，搜索有效的数学表示与快速变换提取不同类型的结构永远不会结束，并成为非常重要的，在今天的信息时代与爆炸的数字数据快速提取所寻求的关键信息从数据。在这项研究中，我们将引入创新的方向多尺度数学表示，快速变换和高效的计算算法，使用方向复框架和矩阵值多小波的方法。小框架类似于小波，但在它们的表示中允许冗余。我们提出的方向多尺度表示有许多有前途的优势和传统小波所需的属性。例如，它们具有更好的捕获多维函数/数据中方向奇异性的能力，具有与传统快速小波变换相似的计算效率高的快速小框架变换，以及许多所需的数学性质，在几何建模、工业问题、计算、成像和信息技术等领域具有广泛的应用前景.我们在这个方向上的初步实验结果已经显示出显着的进步，国家的最先进的小波算法的图像去噪问题，这是第一步预处理，在几乎所有的实际问题处理数字数据。在这个建议的第一阶段，我们将开发必要的数学理论，我们提出的方向多尺度表示使用方向复框架和矩阵值多小波。在第二阶段，我们计划与来自行业的学生和研究人员合作（例如，石油/天然气/矿产勘探公司在加拿大，特别是在阿尔伯塔），使我们将应用和测试我们开发的数学方法的方向多尺度表示的一些实际的工业问题，如地球物理逆问题和地震层析成像。这将直接促进加拿大的勘探工业，从而促进我们加拿大的经济。拟议的研究也将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。

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.