Shape analysis of objects in images is a critical area of research, and several approaches, including those that utilize elastic Riemannian metrics, have been proposed. While elastic techniques for shape analysis of curves are pretty advanced, the corresponding results for higher-dimensional objects (surfaces and disks) are less developed. This paper studies shapes of solid planar objects that are embeddings of a compact domain—a unit square or a unit disk—in \({\mathbb {R}}^2\). Specifically, it introduces a mathematical representation of objects using tensor fields and uses a re-parametrization-invariant Riemannian metric on these tensor fields to analyze object shapes elastically. The essential contribution here is developing an efficient numerical technique to map tensor fields back to the object space, allowing one to approximate geodesic paths in these objects’ shape spaces. Finally, the paper extends this framework to reach landmark-driven registration and improve geodesic computations. The paper illustrates this framework using several simulated and natural objects.

图像中物体的形状分析是一个关键的研究领域，已经提出了几种方法，包括那些利用弹性黎曼度量的方法。虽然用于曲线形状分析的弹性技术非常先进，但对于高维对象（曲面和圆盘）的相应结果却不太发达。本文研究了实体平面物体的形状，这些物体是紧致域（单位正方形或单位圆盘）在\({\mathbb {R}}^2\)中的嵌入. 具体来说，它使用张量场引入了对象的数学表示，并在这些张量场上使用重新参数化不变的黎曼度量来弹性地分析对象形状。这里的主要贡献是开发一种有效的数值技术，将张量场映射回对象空间，允许人们在这些对象的形状空间中近似测地线路径。最后，本文扩展了该框架以达到地标驱动的配准并改进测地线计算。该论文使用几个模拟和自然对象说明了这个框架。