Biharmonic density estimate: a scale-space descriptor for 3-D deformable surfaces

The wide variability in deformable three-dimensional (3-D) shapes calls for the formulation of a multiscale surface signature for effective characterization and analysis of the underlying 3-D intrinsic geometry. To this end, a novel intrinsic geometric scale-space descriptor for 3-D deformable surfaces, termed as the biharmonic density estimate (BDE), is proposed. The BDE, derived from the biharmonic distance measure, is shown to provide an intrinsic geometric scale-space signature for multiscale surface feature-based representation of deformable 3-D shapes that is both effective and useful for practical applications. The proposed BDE signature provides a theoretical framework for the concept of intrinsic geometric scale space, resulting in a highly descriptive characterization of both the local surface structure and the global metric of the underlying 3-D shape. The compactness and robustness of the BDE are experimentally demonstrated on two standard benchmark datasets. The applications of the BDE in the detection of key components on a deformable 3-D surface and determination of sparse point correspondences between two deformable 3-D shapes are also demonstrated.