Scale-invariance in local heat kernel descriptors without scale selection and normalization

Today, only a small fraction of Internet repositories of geometric data is accessible through text search. Fast growth of these repositories makes content-based retrieval one of the next grand challenges in search and organization of such information. Particularly difficult is the problem of \emph{shape retrieval}, as geometric shapes manifest a vast variability due to different scale, orientation, non-rigid deformations, missing data, and also appear in a variety of different formats and representations. One of the biggest challenges in non-rigid shape retrieval and comparison is the design of a shape descriptor that would maintain invariance under a wide class of transformations the shape can undergo. Recently, heat kernel signature was introduced as an intrinsic local shape descriptor based on diffusion scale-space analysis. In this paper, we develop a scale-invariant version of the heat kernel descriptor. Our construction is based on a logarithmically sampled scale-space in which shape scaling corresponds, up to a multiplicative constant, to a translation. This translation is undone using the magnitude of the Fourier transform. The proposed scale-invariant local descriptors can be used in the bag-of-features framework for shape retrieval in the presence of transformations such as isometric deformations, missing data, topological noise, and global and local scaling. We get significant performance improvement over state-of-the-art algorithms on recently established non-rigid shape retrieval benchmarks.