3 research outputs found
A Unified Definition and Computation of Laplacian Spectral Distances
Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion,
wave kernel distances) are easily defined through a filtering of the Laplacian
eigenpairs. They play a central role in several applications, such as
dimensionality reduction with spectral embeddings, diffusion geometry, image
smoothing, geometric characterisations and embeddings of graphs. Extending the
results recently derived in the discrete
setting~\citep{PATANE-STAR2016,PATANE-CGF2017} to the continuous case, we
propose a novel definition of the Laplacian spectral kernels and distances,
whose approximation requires the solution of a set of inhomogeneous Laplace
equations. Their discrete counterparts are equivalent to a set of sparse,
symmetric, and well-conditioned linear systems, which are efficiently solved
with iterative methods. Finally, we discuss the optimality of the Laplacian
spectrum for the approximation of the spectral kernels, the relation between
the spectral and Green kernels, and the stability of the spectral distances
with respect to the evaluation of the Laplacian spectrum and to multiple
Laplacian eigenvalues
Laplacian Spectral Basis Functions
Representing a signal as a linear combination of a set of basis functions is
central in a wide range of applications, such as approximation, de-noising,
compression, shape correspondence and comparison. In this context, our paper
addresses the main aspects of signal approximation, such as the definition,
computation, and comparison of basis functions on arbitrary 3D shapes. Focusing
on the class of basis functions induced by the Laplace-Beltrami operator and
its spectrum, we introduce the diffusion and Laplacian spectral basis
functions, which are then compared with the harmonic and Laplacian
eigenfunctions. As main properties of these basis functions, which are commonly
used for numerical geometry processing and shape analysis, we discuss the
partition of the unity and non-negativity; the intrinsic definition and
invariance with respect to shape transformations (e.g., translation, rotation,
uniform scaling); the locality, smoothness, and orthogonality; the numerical
stability with respect to the domain discretisation; the computational cost and
storage overhead. Finally, we consider geometric metrics, such as the area,
conformal, and kernel-based norms, for the comparison and characterisation of
the main properties of the Laplacian basis functions
Wavelet-based Heat Kernel Derivatives: Towards Informative Localized Shape Analysis
In this paper, we propose a new construction for the Mexican hat wavelets on
shapes with applications to partial shape matching. Our approach takes its main
inspiration from the well-established methodology of diffusion wavelets. This
novel construction allows us to rapidly compute a multiscale family of Mexican
hat wavelet functions, by approximating the derivative of the heat kernel. We
demonstrate that it leads to a family of functions that inherit many attractive
properties of the heat kernel (e.g., a local support, ability to recover
isometries from a single point, efficient computation). Due to its natural
ability to encode high-frequency details on a shape, the proposed method
reconstructs and transfers -functions more accurately than the
Laplace-Beltrami eigenfunction basis and other related bases. Finally, we apply
our method to the challenging problems of partial and large-scale shape
matching. An extensive comparison to the state-of-the-art shows that it is
comparable in performance, while both simpler and much faster than competing
approaches.Comment: 14 lage