Accurate and Efficient Computation of Laplacian Spectral Distances and Kernels

• Published in: Computer graphics forum 2017-01, Vol.36 (1), p.184-196
• Main Author: Patane, Giuseppe
• Format: Article
• Language: English
• Subjects: Accuracy; Approximation; Coding; Computational efficiency; Computer Graphics [Computing methodologies]: Shape modelling; Convergence; digital geometry processing; Eigenvalues; Electromagnetic wave filters; Filtering; geometric modelling; Invariance; Kernels; Linear systems; Mathematical analysis; Mathematical models; modelling; Operators; Prolongation; Smoothness; Spectra; Systems analysis; Transformations (mathematics)
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12794

This paper introduces the Laplacian spectral distances, as a function that resembles the usual distance map, but exhibits properties (e.g. smoothness, locality, invariance to shape transformations) that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commute‐time distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero. Instead of applying a truncated spectral approximation or prolongation operators, we propose a computation of Laplacian distances and kernels through the solution of sparse linear systems. Our approach is free of user‐defined parameters, overcomes the evaluation of the Laplacian spectrum and guarantees a higher approximation accuracy than previous work. This paper introduces the Laplacian spectral distances, as a function that resembles the usual distancemap, but exhibits properties (e.g. smoothness, locality, invariance to shape transformations) that make them useful to processing and analysing geometric data. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and reduce to the heat diffusion, wave, biharmonic and commutetime distances for specific filters. In particular, the smoothness of the spectral distances and the encoding of local and global shape properties depend on the convergence of the filtered eigenvalues to zero.