Foundations: the Laplace–Beltrami operator#

Everything in SpectralBrain is built on one object: the spectral decomposition of the Laplace–Beltrami operator (LBO) of a surface or point cloud. Understand this page and the rest of the library is just a matter of which descriptor you read off the same basis.

Intuition#

The LBO is the curved-surface generalization of the familiar Laplacian \(\Delta = \partial_x^2 + \partial_y^2 + \partial_z^2\). On a mesh it measures how much a function at a vertex differs from the average of its neighbors, weighted by the local geometry. Its eigenfunctions are the natural “vibration modes” of the shape — the harmonics of a drum, but for a hippocampus or a cortical surface.

Crucially, the LBO is intrinsic: it depends only on the surface metric, not on how the shape is positioned, rotated, or scaled in space. That is exactly the property that makes spectral descriptors robust to registration and pose, where volume and thickness are not.

The eigenproblem#

We solve the generalized eigenvalue problem

\[ \mathbf{W}\,\boldsymbol{\phi}_i = \lambda_i \,\mathbf{A}\,\boldsymbol{\phi}_i , \qquad 0 = \lambda_0 \le \lambda_1 \le \lambda_2 \le \dots \]

where \(\mathbf{W}\) is the cotangent stiffness matrix, \(\mathbf{A}\) the mass (area) matrix, \(\lambda_i\) the eigenvalues, and \(\boldsymbol{\phi}_i\) the eigenfunctions. The eigenvalues \(\{\lambda_i\}\) are the spectrum; the eigenfunctions \(\{\boldsymbol{\phi}_i\}\) form an orthonormal basis for functions on the surface. Low \(\lambda\) describes coarse, global shape; high \(\lambda\) describes fine, local detail.

In SpectralBrain#

You compute the decomposition once and reuse it for every descriptor:

import spectralbrain as sb

vertices, faces = sb.load_freesurfer_surface("lh.pial")
mesh = sb.BrainMesh(vertices, faces)

decomp = mesh.decompose(k=300)   # -> SpectralDecomposition

decomp.eigenvalues      # (k,)        the spectrum  {λ_i}
decomp.eigenvectors     # (n_vert, k) the basis     {φ_i}

Choosing k (the number of eigenpairs) trades detail for cost: a few hundred modes is plenty for whole-structure descriptors like ShapeDNA, while per-vertex signatures benefit from more.

Where to next#

Global fingerprints & embeddings — read the spectrum directly (ShapeDNA) or embed points with it (GPS).
Heat-diffusion descriptors and Wave / band-pass descriptors — multiscale per-vertex signatures built from \(\{\lambda_i, \boldsymbol{\phi}_i\}\).