Development and Application of Directional Framelets and Complex Multiwavelets

定向框架和复多小波的开发与应用

• 批准号: RGPIN-2014-05865
• 负责人: Han, Bin
• 金额: $ 2.48万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=652975
• 关键词: Development; Application; Directional; Framelets; Complex

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.

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。**但很多多维函数/数据都有其他重要的几何结构，如定向（类边）奇点，在数学建模中保存着函数/数据的大部分关键信息，如图像和微分方程的解。自傅立叶变换以来，寻找有效的数学表示和快速变换来提取不同类型的结构从未结束，在当今数字数据爆炸的信息时代，从数据中快速提取所寻找的关键信息变得至关重要。在本研究中，我们将采用定向复框架和矩阵值多小波的方法，引入具有快速变换和高效计算算法的创新定向多尺度数学表示。小框架类似于小波，但在它们的表示中允许冗余。与传统小波相比，我们提出的定向多尺度表示具有许多有前途的优点和理想的特性。例如，它们在捕获多维函数/数据中的方向奇异性方面具有更好的能力，具有类似于传统快速小波变换的计算效率高的快速框架变换，并且具有许多理想的数学性质。**提出的方法有许多潜在的应用，包括几何建模、工业问题、计算、成像和信息技术。我们在这个方向上的初步实验结果已经表明，在图像去噪问题上，最先进的小波算法已经取得了重大进展，这是几乎所有处理数字数据的实际问题的第一步预处理。在本建议的第一阶段，我们将利用定向复框架和矩阵值多小波发展我们提出的定向多尺度表示的必要数学理论。在第二阶段，我们计划与来自工业界的学生和研究人员（例如，加拿大的石油/天然气/矿产勘探公司，特别是在阿尔伯塔省）合作，以便我们将应用和测试我们开发的定向多尺度表示的数学方法，以解决一些实际的工业问题，如地球物理反演问题和地震层析成像。这将直接促进加拿大的勘探行业，从而促进加拿大的经济。拟议的研究还将有助于建立新的数学理论和为加拿大的未来培养高素质的人才。