A 3D shape descriptor based on spectral analysis of medial axis

The medial axis of a 3D shape is widely known for its ability as a compact and complete shape representation. However, there is still lack of a generative description defined over the medial axis directly which limits its actual application to 3D shape analysis such as shape matching and retrieval. In this paper, we propose a new spectral shape descriptor that directly applies spectral analysis to the medial axis of a 3D shape, which we call the medial axis spectrum for a 3D shape. We develop a newly defined Minkowski–Euclidean ratio inspired by the Minkowski inner product to characterize the geometry of the medial axis of a 3D mesh. We then generalize the Laplace–Beltrami operator to the medial axis, and take the solution to a Laplacian eigenvalue problem defined on it as the medial axis spectrum. The medial axis spectrum is invariant under rigid transformation and isometry of the medial axis, and is robust to shape boundary noise as shown by our experiments. The medial axis spectrum is finally used for 3D shape retrieval, and its superiority over previous work is shown by extensive comparisons.