Scale Invariant Geometry for Nonrigid Shapes

Local scale variations within the same species are common in nature. The shape matching puzzle poses fascinating questions, like how should we measure the discrepancy between a small dog with large ears and a large one with small ears? are there similar geometric structures that are common to an elephant and a giraffe? what is the morphometric similarity between a blue whale and a dolphin? Existing tools that attempt to quantify the resemblance between surfaces which are insensitive to deformations in size are limited to either scale invariant local descriptors, or global normalization methods. Here, we propose novel tools for shape exploration by introducing a scale invariant metric for surfaces. The geometric measures we consider can be used for nonrigid shape analysis, it could help in generating local invariant features, produce scale invariant geodesics, embed one surface into another while being robust to changes in local and global size, and assist in the computational study of intrinsic symmetries where size does not matter.