CAREER: Partial Differential Equation-based Image Processing with Applications to Radiation Oncology

在过去的十年中，新的非线性偏微分方程（PDEs）被开发用于各种图像处理应用，如降噪、边缘检测、图像分割和恢复。虽然科学界在这一领域的注意力主要集中在创建新的偏微分方程上，但很少有人关注开发近似其解的数值算法。目前使用的少数几种数值算法存在各种各样的问题：它们不够精确，太慢，而且不是无故障的。在这个项目中，研究者为图像处理中的非线性偏微分方程开发了准确、高效和鲁棒的数值算法。研究活动基于研究者在双曲守恒定律领域的广泛工作，包括Hamilton-Jacobi方程的数值方法、高阶非线性偏微分方程的快速算法、稳态解的计算算法、Hamilton-Jacobi方程的数值均匀化和多分辨率分析、非线性扩散方程的分析、约束变形活动轮廓和测地线流的分析。以及图像处理的“非盲”算法。部分研究活动侧重于改进现有算法，以解决放射肿瘤学治疗计划中的特定成像问题。研究人员开发了新的图像处理数学技术，并利用这些技术解决放射肿瘤学成像领域的问题。放射肿瘤学通过向肿瘤提供相对小剂量的辐射来治疗癌症，目的是在不破坏或慢性损害肿瘤生长内部和周围健康组织的情况下消除癌症。CT和MRI扫描被用作三维解剖模型，以确保治疗在几何上符合肿瘤目标。这个过程主要取决于确定肿瘤的位置以及健康器官（为了使这些区域的辐射剂量最小化）。尽管进行了广泛的研究，但现有的图像处理数学工具不适合临床医学应用。CT和MRI扫描的分割仍然是通过手动工具进行的，并且消耗了计划治疗所需时间的一半左右。研究者设计了准确可靠的自动算法，将大大缩短这一时间，并对放射肿瘤学产生重大影响。他将教育活动整合到他的工作中，这些活动证明了应用数学在广泛的科学领域中的重要性。特别强调计算数学在生物学和尖端技术中的应用。计划的教育活动包括初中生、高中生、本科生和研究生。通过鼓励代表性不足的群体学习应用数学并将其作为未来的职业，研究者致力于增加数学科学领域的性别和种族多样性。

In the past decade, new nonlinear partial differentialequations (PDEs) have been developed for various image processingapplications, such as noise reduction, edge detection, imagesegmentation and restoration. While the attention of thescientific community in this area predominantly focused oncreating the new PDEs, very little attention was paid todeveloping numerical algorithms that approximate their solutions.The few numerical algorithms that are currently used suffer froma variety of problems: they are not accurate enough, too slow,and not fault-free. In this project, the investigator developsaccurate, efficient, and robust numerical algorithms fornonlinear PDEs in image processing. The research activities arebased on the investigator's extensive work in the field ofhyperbolic conservation laws, and include numerical methods forthe Hamilton-Jacobi equations, fast algorithms for high-ordernonlinear PDEs, algorithms for computing steady-state solutions,numerical homogenization of Hamilton-Jacobi equations andmulti-resolution analysis, analysis of nonlinear diffusionequations, constrained morphing active contours and geodesicflows, and "non-blind" algorithms for image processing. Aportion of the research activities focuses on improving existingalgorithms in order to solve a specific imaging problem inradiation oncology treatment planning. The investigator develops novel mathematical techniques forimage processing and uses these techniques for solving problemsin the field of radiation oncology imaging. Radiation oncologytreats cancer by delivering relatively small doses of radiationto tumors in order to eliminate cancer without destroying orchronically damaging healthy tissues in and around the growth.CT and MRI scans are used as three-dimensional anatomical modelsto ensure that the treatments conform geometrically to the tumortarget. This process depends critically upon identifying thelocation of the tumor as well as the healthy organs (in order tominimize the dose of radiation in these areas). Despite extendedresearch, the existing mathematical tools for image processingare unsuitable for clinical medical applications. Thesegmentation of the CT and MRI scans is still carried out bymanual tools, and consumes about one-half of the time required toplan the treatments. The investigator designs accurate andreliable automated algorithms that would significantly shortenthis time and have a big impact on radiation oncology. Heintegrates into his work educational activities that demonstratethe importance of applied mathematics in a broad spectrum ofsciences. Special emphasis is given to applications ofcomputational mathematics in biology and cutting-edgetechnologies. The planned educational activities includeprograms for junior-high, high-school, undergraduate, andgraduate students. The investigator works to increase the genderand ethnic diversity in the mathematical sciences by encouragingunder-represented groups to study applied mathematics and chooseit as a future career.