Analytic and Geometric Methods in Inverse Problems and Imaging

反问题和成像中的解析和几何方法

• 批准号: RGPIN-2016-06329
• 负责人: Nachman, Adrian
• 金额: $ 1.97万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=681765
• 关键词: Analytic; Geometric; Methods; Inverse; Problems

This proposal seeks to answer some fundamental mathematical questions in the fields of Inverse Problems and Image Processing, as well as to contribute specific applications to Medical Imaging. The field of Inverse Problems studies novel methods to obtain images from noninvasive measurements, for next generation imaging modalities. Image Processing treats images obtained from existing modalities and the mathematics involved in denoising, segmenting, registering and extracting information from them.*** Classically, the data in Inverse Problems consists of measurements in the exterior of the object under investigation, or on its boundary. There have been considerable advances by numerous researchers in the systematic study of such problems, but several challenging questions remain open, and new ideas for addressing some of them will be sought. In addition, a new class of inverse problems considers situations where some interior information can be obtained (for instance using Magnetic Resonance Imagers). In joint work with A. Tamasan and A. Timonov, we have found a connection between one such problem arising in electric impedance imaging and the theory of minimal surfaces in non-euclidean geometries (determined by the measured data) and of related weighted least gradient problems. In recent joint work with A. Tamasan and J. Veras, the practical application has lead us to an interesting novel boundary value problem which has not been previously considered in geometric measure theory and will be investigated. We will also seek to obtain corresponding efficient and scalable numerical algorithms for its solution, and apply these to experimental data obtained in collaboration with M.Joy's group at the University of Toronto.******The study of weighted least gradient problems has also led us to several analytic and geometric questions in Image Analysis. In a seminal paper, Tadmor, Nessar and Vese introduced a multiscale decomposition of images based on a sequence of variational problems involving a similar regularization functional; they showed that this can be considered as a nonlinear harmonic decomposition, with successive terms adding finer scale details. In joint work with K. Modin and L. Rondi we have recently extended this approach to registration problems, thus obtaining a (multiplicative) harmonic decomposition of diffeomorphisms. Theoretical and practical implications of this novel decomposition will be investigated. We will also apply the multiscale approach to the inverse problem described above, as a possible tool for handling non-smooth impedances.*** Solution of some of the problems we plan to study is expected to have a significant impact in the development of next generation imaging modalities as well as to the image analysis of clinical data. **

该建议旨在回答反问题和图像处理领域中的一些基本数学问题，以及在医学成像中的具体应用。逆问题领域研究新的方法，以获得图像的非侵入性测量，为下一代成像方式。图像处理从现有模式获得的图像以及去噪，分割，配准和从中提取信息所涉及的数学。 传统上，反问题中的数据由被调查对象外部或其边界上的测量值组成。许多研究人员在系统研究这些问题方面取得了相当大的进展，但仍有几个具有挑战性的问题有待解决，将寻求解决其中一些问题的新想法。此外，一类新的逆问题考虑了可以获得某些内部信息的情况（例如使用磁共振成像仪）。在与A。Tamasan和A. Timonov，我们已经发现了一个这样的问题之间的连接所产生的电阻抗成像和理论的最小表面在非欧几里德几何形状（由测量数据确定）和相关的加权最小梯度问题。在最近与A. Tamasan和J. Veras，实际应用使我们得到了一个有趣的新的边值问题，这是以前在几何测度理论中没有考虑过的，将被研究。我们还将寻求获得相应的有效和可扩展的数值算法来求解它，并将其应用于与多伦多大学的M.Joy小组合作获得的实验数据。加权最小梯度问题的研究也使我们在图像分析中遇到了一些解析和几何问题。在一篇开创性的论文中，Tadmor，Nessar和Vese介绍了一种基于一系列涉及类似正则化泛函的变分问题的图像多尺度分解;他们表明这可以被认为是一种非线性谐波分解，连续项添加更精细的尺度细节。在与K。莫丁和L. Rondi，我们最近扩展了这种方法的注册问题，从而获得（乘法）谐波分解的dronomorphisms。这种新的分解的理论和实践意义将进行调查。我们还将多尺度方法应用于上述逆问题，作为处理非平滑阻抗的可能工具。* 我们计划研究的一些问题的解决方案预计将对下一代成像模式的发展以及临床数据的图像分析产生重大影响。**