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）用于解决漫射光学层析成像（DOT）逆问题。在DOT中实现SCR的最新算法有两种：带稀疏性约束的广义条件梯度法（gggm - sc）和带稀疏性约束的广义半光滑牛顿法（GSNM-SC）。对于线性和一些非线性问题，gggm - sc的理论和数值结果都证明了它的有效性。gggm - sc，也被导出用于线性问题，最小化给定的泛函，不一定是凸的，关于给定基的稀疏性约束。在本项目中，DOT逆问题的计算方面将研究：(i)使用gggm - sc开发DOT重建算法，（ii）设计计算自适应基（如有限元）的策略，以充分利用稀疏性约束思想，（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）对所提出的GSNM-SC和gggm - sc在DOT中进行收敛分析。该跨学科研究项目旨在将教育与研究结合起来，促进国际学生交流。该项目的教育活动将包括：(i)为学生提供国际研究经验，（ii）克莱姆森大学和德国不来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）让高中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进行成像的方法。例如，生物组织的光学断层扫描在乳腺癌的早期检测方面具有潜在的应用。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，无创。DOT成像技术在物理学、生物学和医学领域也显示出巨大的潜力。然而，尽管它有很大的潜力，但由于图像重建的不稳定性导致任何生成的图像模糊，它尚未在商业或医学上取得成功。为了克服这些困难，我们将研究新的图像重建数学技术。研究应用将从生物医学成像中的癌症检测到遥感中的地雷探测，再到海洋中的物体成像。研究成果也有望为中子输运、大气科学中的输运、光热光谱和显微镜、激光泵浦探针、漫射光子密度波和新的断层成像技术（如光学、电子和热成像）以及生物医学诊断等领域的科学知识做出贡献。

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.