Image Reconstruction In Diffuse Optical Tomography With Sparsity Constraints

在这个项目中，PI将研究稀疏约束正则化（SCR）来解决漫射光学层析成像（DOT）逆问题。两个最近的算法可用于SCR在DOT中的实施：稀疏约束的广义条件梯度法（GCGM-SC）和稀疏约束的广义半光滑牛顿法（GSNM-SC）。GCGM-SC的有效性已被证明与线性和一些非线性问题的理论和数值结果。GCGM-SC，这也是推导出的线性问题，最小化一个给定的功能，不一定是凸的，稀疏约束相对于一个给定的基础。在这个项目中，DOT逆问题的计算方面将被研究以（i）使用GCGM-SC开发DOT的重建算法，（ii）设计用于计算自适应基（例如有限元）的策略以充分利用稀疏约束思想，（iii）将GSNM-SC扩展到非线性问题并将GSNM-SC与Tikhonov正则化进行比较，以及（iv）在DOT中执行所提出的GSNM-SC和GCGM-SC的收敛性分析。跨学科研究项目旨在整合教育和研究，促进学生的国际交流。该项目的教育活动将包括：（i）学生的国际研究经验，（ii）克莱姆森大学和德国的不莱梅大学之间的学生交流，（iii）将大部分研究活动纳入教学活动，以帮助学生将课程材料与研究联系起来，（iv）高中学生接触应用数学研究，以及（v）吸引代表性不足的群体追求应用数学。漫射光学层析成像（DOT）是一种利用近红外和可见光对高散射介质成像的方法。例如，生物组织的光学层析成像对于乳腺癌的早期检测具有潜在的应用。DOT的一个主要优点是，与X射线乳房X线摄影相比，它更便宜且无创。DOT成像技术也显示出作为物理学、生物学和医学发现的工具的巨大潜力。然而，尽管其具有巨大的潜力，但由于图像重建的不稳定性导致任何所得图像的模糊，因此其在商业或医学上尚未成功。为了克服这些困难，我们将研究新的数学技术的图像重建。研究应用将有所不同，从生物医学成像中的癌症检测到遥感中的地雷检测，再到海洋中的物体成像。研究成果还将有助于增进以下方面的科学知识：中子输运、大气科学中的输运、光热光谱学和显微镜学、激光泵浦探针、扩散光子密度波和新的断层摄影技术，如光学、电子和热成像以及生物医学诊断。

In this project, the PI will investigate Sparsity Constrained Regularization (SCR) for solving the Diffuse Optical Tomography (DOT) inverse problem. Two recent algorithms are available for the implementation of SCR in DOT: the Generalized Conditional Gradient Method with Sparsity Constraint (GCGM-SC) and the Generalized Semi-smooth Newton's Method with Sparsity Constraint (GSNM-SC). The efficacy of GCGM-SC has been demonstrated with both theoretical and numerical results for linear and some non-linear problems. GCGM-SC, which has also been derived for linear problems, minimizes a given functional, not necessarily convex, with sparsity constraint with respect to a given basis. In this project, the computational aspect of the DOT inverse problem will be investigated to (i) develop a reconstruction algorithm for DOT using GCGM-SC, (ii) devise a strategy for computing adaptive basis such as finite elements to take full advantage of the sparsity constraint idea, (iii) extend GSNM-SC to nonlinear problems and compare GSNM-SC to Tikhonov regularization, and (iv) perform convergence analysis of the proposed GSNM-SC and GCGM-SC in DOT. The interdisciplinary research project seeks to integrate education and research and foster an international exchange of students. The educational activities in the project will include: (i) international research experience for students, (ii) student exchange between Clemson University and the University of Bremen, Germany, (iii) the incorporation of much of the research activities into teaching activities to help students bridge course materials with research, (iv) exposure of high school students to applied mathematics research, and (v) attraction of underrepresented groups to pursue appliedmathematics.Diffuse Optical Tomography (DOT) is a method for imaging a highly scattering medium using near infrared and visible light. For example, optical tomography of biological tissue has potential applications for the early detection of breast cancer. One major advantage of DOT is that it is less expensive and non-invasive as compared with x-ray mammography. DOT imaging technology also shows great promise as a tool for initiating discoveries in physics, biology, and medicine. However, despite its great potential, it has yet to be commercially or medically successful as the instability in image reconstruction results in the blurring of any resulting image. To overcome these difficulties, we will investigate novel mathematical techniques for image reconstruction. Research applications will vary from cancer detection in biomedical imaging to land mine detection in remote sensing to imaging objects in the ocean. Research results are also expected to contribute to scientific knowledge in neutron transport, transport in atmospheric science, photothermal spectroscopies and microscopies, laser pump probes, diffuse photon density waves and new tomography technologies, such as optical, electronic and thermal imaging, and biomedical diagnostics.