Computation of crowded geodesics on the universal Teichmueller space for planar shape matching in computer vision

量化两个形状之间的相似性是计算机视觉中的一个中心问题。通过将平面形状空间定义为泛Teichmueller空间的子集，并为其配备Weil-Petersson度量，实现了平面形状空间上的一种距离度量。这产生了形状上的比例和平移不变的度量，并且在两个形状端点之间具有唯一的测地线流。这项提议的工作开发了稳健的计算方法，用于计算该空间上形状之间的公制距离和测地线。主要的困难在于涉及“拥挤”形状的计算，即那些具有细长、弯曲或延长突起的形状。这种形状阻碍了有限精度的计算，因为直接算法受到严重的舍入误差的影响。这项提案的主要目的是开发算法方法来解决舍入误差和相关问题：拉链保角映射算法将得到增强，以生成针对拥挤形状的精确保形映射。测地线上的速度场表示将被重写为抵抗舍入误差的形式。测地线方程将被转换为利用前述速度场变换的表达式，并且可以有效地在拥挤的形状之间流动。该项目的最后阶段将演示精确的测地线流动和拥挤形状之间的距离计算。在这个项目下开发的方法可以应用于科学计算中的几个相关问题：通过保角映射求解不规则几何上的微分方程，与病态粒子系统的保守积分方法，以及移动网格核近似。该项目的工作可以促进在科学和计算机视觉问题中的广泛应用：自动对象识别(例如投射物体识别)，轮廓分类(动物物种的确定)，医学成像(使用MRI诊断痴呆症和相关疾病)，以及人工智能(视觉识别和解释)。所有计算交付成果(计算机代码、示例模拟、文档)都将公开提供。通过让学生参与相关的研究任务，这个项目将有助于未来工程师、数学家和计算机科学家的教育发展。

Quantifying the (dis)similarity between two shapes is a central problem in computer vision. One distance metric on the space of planar shapes is realized by identifying this space as a subset of the Universal Teichmueller Space, and equipping it with the Weil-Petersson metric. This results in a metric that is scale- and translation-invariant on shapes, and has unique geodesic flow between two shape endpoints. The work of this proposal develops robust computational methods for the computation of metric distances and geodesics between shapes on this space. The major difficulty lies in computations involving "crowded" shapes, i.e., those with elongated, winding, or extended protrusions. Such shapes stymie finite-precision computations because direct algorithms suffer from severe roundoff error. The major thrusts of this proposal develop algorithmic methodologies to address roundoff error and related issues: The Zipper conformal mapping algorithm will be augmented to produce accurate conformal maps for crowded shapes. The velocity field representation on a geodesic will be rewritten into a form that is resistant to roundoff error. The geodesic equation will be transformed into a expression that takes advantage of the aforementioned velocity field transformation, and can effectively flow between crowded shapes. The final phase of this project will demonstrate accurate geodesic flow and distance computations between crowded shapes. The methods developed under this project can be applied to several related problems in scientific computing: solutions to differential equations on irregular geometries through conformal mapping, conservative integration methods with ill-conditioned particle systems, and moving-mesh kernel approximations.The work of this project can contribute to far-reaching applications in scientific and computer vision problems: automated object recognition (e.g. projectile identification), outline classification (determination of an animal's species), medical imaging (usage of MRI to diagnose dementia and related diseases), and artificial intelligence (visual recognition and interpretation) to name a few. All computational deliverables (computer code, example simulations, documentation) will be made publicly available. Through the engagement of students in related research tasks, this project will contribute to the educational development of future engineers, mathematicians, and computer scientists.