Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc.

Laplace

The Laplace–Beltrami operator is involved in many important differential equations, such as the heat equation and the wave equation. It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function f:

${\displaystyle \Delta f:=\operatorname {div} \operatorname {grad} f.}$

Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem):

${\displaystyle \Delta \varphi _{i}+\lambda _{i}\varphi _{i}=0.}$

The solutions are the eigenfunctions ${\displaystyle \varphi _{i}}$ (modes) and corresponding eigenvalues ${\displaystyle \lambda _{i}}$, representing a diverging sequence of positive real numbers. The first eigenvalue is zero for closed domains or when using the Neumann boundary condition. For some shapes, the spectrum can be computed analytically (e.g. rectangle, flat torus, cylinder, disk or sphere). For the sphere, for example, the eigenfunctions are the spherical harmonics.

The most important properties of the eigenvalues and eigenfunctions are that they are isometry invariants. In other words, if the shape is not stretched (e.g. a sheet of paper bent into the third dimension), the spectral values will not change. Bendable objects, like animals, plants and humans, can move into different body postures with only minimal stretching at the joints. The resulting shapes are called near-isometric and can be compared using spectral shape analysis.

Discretizations

Geometric shapes are often represented as 2D curved surfaces, 2D surface meshes (usually triangle meshes) or 3D solid objects (e.g. using voxels or tetrahedra meshes). The Helmholtz equation can be solved for all these cases. If a boundary exists, e.g. a square, or the volume of any 3D geometric shape, boundary conditions need to be specified.

Several discretizations of the Laplace operator exist (see Discrete Laplace operator) for the different types of geometry representations. Many of these operators do not approximate well the underlying continuous operator.

Spectral shape descriptors

Spectral Matching

The spectral decomposition of the graph Laplacian associated with complex shapes (see Discrete Laplace operator) provides eigenfunctions (modes) which are invariant to isometries. Each vertex on the shape could be uniquely represented with a combinations of the eigenmodal values at each point, sometimes called spectral coordinates:

${\displaystyle s(x)=(\varphi _{1}(x),\varphi _{2}(x),\ldots ,\varphi _{N}(x)){\text{ for vertex }}x.}$

Spectral matching consists of establishing the point correspondences by pairing vertices on different shapes that have the most similar spectral coordinates. Early work ^[11][12][13] focused on sparse correspondences for stereoscopy. Computational efficiency now enables dense correspondences on full meshes, for instance between cortical surfaces.^[14] Spectral matching could also be used for complex non-rigid image registration, which is notably difficult when images have very large deformations.^[15] Such image registration methods based on spectral eigenmodal values indeed capture global shape characteristics, and contrast with conventional non-rigid image registration methods which are often based on local shape characteristics (e.g., image gradients).

References