Spectral

Spectral decomposition of a 3D shape. More...

Enumerations
enum	Euclid::SpecOp { Euclid::SpecOp::laplace_beltrami, Euclid::SpecOp::graph_laplacian }
	The operator to use for spectral decomposition. More...
enum	Euclid::SpecDecomp { Euclid::SpecDecomp::symmetric, Euclid::SpecDecomp::generalized }
	The method to use for spectral decomposition. More...

Functions
template<typename Mesh , typename DerivedA , typename DerivedB >
unsigned	Euclid::spectrum (const Mesh &mesh, unsigned k, Eigen::MatrixBase< DerivedA > &lambdas, Eigen::MatrixBase< DerivedB > &phis, SpecOp op=SpecOp::laplace_beltrami, SpecDecomp decomp=SpecDecomp::symmetric, unsigned max_iter=1000, double tolerance=1e-10)
	Spectral decomposition of a mesh. More...

Detailed Description

This package contains several methods to conduct spectral decomposition of a given shape by computing the eigenvalues and eigenvectors of the Laplacian matrix. It provides the basis for various spectral shape processing algorithms.

References

[1] Zhang, H., Van Kaick, O., Dyer, R. Spectral Mesh Processing. Computer Graphics Forum, 2010.

[2] Vallet, B., Levy, B. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum, 2008.

Enumeration Type Documentation

enum Euclid::SpecDecomp

strong

Enumerator
symmetric	Solving the symmetric Laplacian matrix. Let \(L=D^{-1/2}SD^{-1/2}\), then solving the eigenvalue problem of this symmetric matrix \(LX=\lambda X\).
generalized	Solving the generalized eigenvalue problem. Let \(L=D^{-1}S\), then solving the generalized eigenvalue problem \(SX=\lambda DX\).

enum Euclid::SpecOp

strong

Enumerator
laplace_beltrami	The Laplace-Beltrami operator.
graph_laplacian	The graph Laplacian operator.

Function Documentation

template<typename Mesh , typename DerivedA , typename DerivedB >

unsigned Euclid::spectrum	(	const Mesh &	mesh,
		unsigned	k,
		Eigen::MatrixBase< DerivedA > &	lambdas,
		Eigen::MatrixBase< DerivedB > &	phis,
		SpecOp	op = SpecOp::laplace_beltrami,
		SpecDecomp	decomp = SpecDecomp::symmetric,
		unsigned	max_iter = 1000,
		double	tolerance = 1e-10
	)

Parameters

mesh	The input mesh.
k	The number of eigenvalues to compute. Note that the actual size of the spectrum might be smaller than k when the computation doesn't converge.
lambdas	The output eigenvalues, sorted in ascending order.
phis	The output eigenfunctions corresponding to the eigenvalues.
op	The Laplace operator to use.
decomp	The eigen system to solve.
max_iter	The maximum number of iterations for eigen decomposition.
tolerance	The tolerance of accuracy loss in eigen decomposition.

Returns

The number of converged eigenvalues.

See also

SpecOp

SpecDecomp