CAREER: Geometric Shape Deformation with Applications in Medicine

尽管最近取得了重大进展，但当面对医疗级要求时，3D计算机图形仍然很谦卑，因此医疗插画家经常继续依赖2D手绘。一个基本的挑战是，详细的几何模型和先进的非线性材料增加了计算复杂性，使它们难以应用于实时交互应用。在这项研究中，PI将研究一种基于几何形状变形的替代方法，而不是创建它们的过程。他认为，直观的形状变形可以通过基本几何特性的保证来实现，比如平滑性和注入性（无自交）。关键是要设计出能够快速完成这一任务的算法，同时为用户提供一组小而富有表现力的可调节控件，以确保高效的交互体验；形状变形技术的任务是以一种自然和可预测的方式将这种简洁的、人类可管理的输入控制集合外推到全尺寸的3D变形场中。PI的假设是这个要求可以用基本的几何性质来正式表达。为此，PI将探索直接（封闭形式）和变分方法，因为虽然直接方法在速度方面优于变分方法，但变分方法提供了更强的保证和先进的几何性质。在直接方法方面，PI将开发新的方法来快速混合某些3D变换组（例如，借助新的几何代数结构）。转换混合将辅以高级影响权重，允许用户显式地控制生成的稀疏性。在变分方法方面，PI将研究满足旋转不变性等传统性质但增加了高阶连续性和注入性的变形能；在这里，主要的挑战将是为潜在的优化问题找到有效的数值解。PI认为，通过适当地将变分方法与直接方法相结合，从而将一些变分问题转换为凸优化，从而为高效的凸求解器打开大门，从而证明有可能减轻变分方法固有的计算复杂性。更广泛的影响：形状变形与建筑、计算机辅助设计（CAD）、科学和工程的许多领域以及娱乐业有关。但这个项目的主要动机是医疗应用，灵感来自PI在费城儿童医院的合作者的要求。如果有合适的工具，手绘医学插图的经典领域将演变成3D动画医学地图集，为医学教育树立新的标准。在治疗病理性颅骨畸形（颅缝闭锁）时，形状变形技术可以促进诊断和术前计划，最终有助于临床实践。专家手中的形状建模工具可以帮助降低CT扫描所需的辐射剂量，通过应用新的重建方法，将用户输入与模板模型和精确的表面扫描（通过激光扫描等无辐射方法获得）相结合。PI还将举办研讨会和课程，使医学和工程专业的学生，包括代表性不足的群体的成员聚集在一起，从而促进研究和教育方面的跨学科合作。

In spite of significant recent advances 3D computer graphics are still humbled when confronted with medical-grade requirements, so medical illustrators often continue to rely on 2D hand drawing. A fundamental challenge is that detailed geometric models and advanced nonlinear materials increase computational complexity, making them difficult to apply in real-time interactive applications. In this research, the PI will investigate an alternative approach based on geometric shape deformations rather than the processes which created them. He argues that intuitive shape deformation can be facilitated by guarantees of basic geometric properties such as smoothness and injectivity (no self-intersection). The key is to design algorithms that can do this quickly while providing the user with a small yet expressive set of adjustable controls to ensure an efficient interactive experience; the task of shape deformation techniques is to extrapolate this parsimonious, human manageable set of input controls into a full-scale 3D deformation field in a natural and predictable way. The PI's hypothesis is that this requirement can be formally expressed in terms of basic geometric properties. To this end, the PI will explore both direct (closed-form) and variational methods, because while direct methods excel in speed variational methods offer stronger guarantees and advanced geometric properties. In terms of direct methods, the PI will develop new ways to quickly blend certain groups of 3D transformations (e.g., with the help of new geometric algebraic structures). Transformation blending will be complemented by advanced influence weights that allow the user to explicitly control the resulting sparseness. In terms of variational methods, the PI will study deformation energies satisfying traditional properties such as rotation invariance but augmented with higher-order continuity and injectivity; here, the main challenge will be to find efficient numerical solutions for the underlying optimization problems. The PI believes it will prove possible to mitigate the inherent computational complexity of variational methods by suitably combining them with direct methods so as to cast some of the variational problems as convex optimizations, thereby opening the door to highly efficient convex solvers.Broader Impacts: Shape deformation is relevant to architecture, computer aided design (CAD), and many areas of science and engineering, as well as to the entertainment industry. But this project has primarily been motivated by medical applications, inspired by requests from the PI's collaborators at The Children's Hospital of Philadelphia. Given the right tools, the classical field of hand drawn medical illustration will evolve into 3D animated medical atlases, setting new standards in medical education. Shape deformation techniques could ultimately contribute to clinical praxis, by facilitating diagnosis and pre-operative planning when treating conditions such as pathological skull deformities (craniosynostosis). And shape modeling tools in expert hands could help lower the radiation dose required in CT scanning, by applying new reconstruction methods that combine user input with template models and accurate surface scans (obtained with radiation-free methods such as laser scanning). The PI also will organize seminars and courses that bring together medical and engineering students, including members of underrepresented groups, thereby promoting interdisciplinary collaboration in both research and education.