Spectral descriptors, in general#

The Foundations page ended with a single object: the spectral decomposition \(\{\lambda_i, \boldsymbol{\phi}_i\}\) of a shape’s Laplace–Beltrami operator. A spectral descriptor is anything you read off that decomposition to characterize the shape. Every descriptor in SpectralBrain — ShapeDNA, HKS, WKS, BKS, the wavelets, GPS — is a particular way of reading the same spectrum. This page is about what that reading is, where it comes from, what it actually means geometrically, and how a list of eigenvalues and eigenfunctions ends up encoding morphometry.

What a descriptor is#

A descriptor turns the spectral basis into a number, a vector, or a field that you can compare, cluster, or correlate with a clinical variable. They come in a few flavors:

Global descriptors summarize the whole shape as a single vector — the spectrum itself (ShapeDNA). One vector per surface.
Point signatures assign a feature vector to every vertex — a field over the surface (HKS, WKS, BKS/IBKS, the wavelet descriptors).
Embeddings re-coordinate the surface into a space where intrinsic geometry becomes ordinary geometry (GPS).
Cross-shape operators relate two or more shapes (functional maps, the shape difference operator, the spectral distances).

Different as they look, they are all instances of one idea: applying a function to the Laplace–Beltrami operator and looking at the result.

The common recipe: filtering the spectrum#

The Laplace–Beltrami operator \(\Delta\) is self-adjoint, so the spectral theorem lets us apply any scalar function \(g\) to it through its eigenpairs:

\[ g(\Delta) \;=\; \sum_i g(\lambda_i)\, \boldsymbol{\phi}_i \boldsymbol{\phi}_i^{\top}. \]

This functional calculus is the engine behind every descriptor. The choice of \(g\) is the only thing that changes from one descriptor family to the next — it is a spectral filter that decides which frequencies of the shape you listen to.

The prototype: the heat kernel#

The cleanest derivation starts with the heat equation on the surface. Let heat diffuse from an initial distribution \(u_0\):

\[ \frac{\partial u}{\partial t} = -\Delta u, \qquad\Longrightarrow\qquad u(t) = e^{-t\Delta}\,u_0 . \]

The operator \(e^{-t\Delta}\) is the heat operator; its kernel \(k_t(x, y)\) tells you how much heat travels from \(y\) to \(x\) in time \(t\). Expanded in the spectral basis,

\[ k_t(x, y) = \sum_i e^{-\lambda_i t}\, \phi_i(x)\,\phi_i(y). \]

Two readings of this single kernel give the two most important descriptors:

Its diagonal — how much heat stays put at each point — is the Heat Kernel Signature:

\[ \mathrm{HKS}(x, t) = k_t(x, x) = \sum_i e^{-\lambda_i t}\, \phi_i(x)^2 . \]
Its trace — integrated over the whole surface, since \(\int_M \phi_i^2 = 1\) — is a global quantity, the heat trace:

\[ Z(t) = \int_M k_t(x,x)\,dx = \sum_i e^{-\lambda_i t}, \]

which is a smooth re-encoding of the spectrum, i.e. of ShapeDNA.

So the global fingerprint (ShapeDNA, via the trace) and the local signature (HKS, via the diagonal) are the same object seen at two granularities. That is the template for the whole library.

The general point signature#

Replace the heat filter \(g(\lambda)=e^{-\lambda t}\) with any other filter \(f\) and take the diagonal, and you get the general point signature:

\[ S_f(x) = \sum_i f(\lambda_i)\, \phi_i(x)^2 . \]

The descriptor families are simply different choices of \(f\):

Filter \(f(\lambda)\)	Behavior	Descriptor
\(e^{-\lambda t}\), varying \(t\)	low-pass	HKS
Gaussian in \(\log\lambda\), varying center	band-pass	WKS
alternative spectral kernels	re-weighted	BKS / IBKS
a bank of dilated band-pass kernels	multi-resolution	SGW / ASMWD
identity on \(\{\lambda_i\}\)	the raw spectrum	ShapeDNA

Seeing the families this way demystifies the zoo: choosing a descriptor is choosing a spectral filter, and the trade-offs (multiscale vs. frequency-sharp, isotropic vs. anisotropic) are trade-offs between filters.

Note

Why \(\phi_i(x)^2\) rather than \(\phi_i(x)\)? Eigenfunctions are defined only up to sign (and up to rotation within a degenerate eigenspace), so the raw value \(\phi_i(x)\) is not a well-defined feature of the shape. The square removes the sign ambiguity and, by analogy with quantum mechanics, reads as the energy density of mode \(i\) at point \(x\).

Interpretation: the harmonics of a shape#

The eigenfunctions are the standing waves of the surface — the patterns it would vibrate in if it were a drumhead. The eigenvalue \(\lambda_i\) is the squared frequency of mode \(i\): small \(\lambda\) means a slow, smooth, global undulation; large \(\lambda\) means a fast, wiggly, local one. The zeros of an eigenfunction (its nodal lines) partition the surface the way the nodal lines of a vibrating plate do.

This is the single most useful mental model. A spectral filter that keeps small \(\lambda\) (a low-pass filter) describes a point by its place in the global shape; a filter that keeps large \(\lambda\) describes it by its local geometry. Sweeping the filter — over time \(t\) in the HKS, over energy in the WKS — is sweeping from local to global, which is exactly why these signatures are multiscale.

What they actually mean — and their limits#

Two properties give spectral descriptors their power:

Intrinsic. They depend only on the surface metric, not on the embedding in space. Translate, rotate, or reflect the shape and the descriptors do not move. There is no registration step and no pose confound.
Isometry-invariant. Bend the surface without stretching it and the descriptors are unchanged, because \(\Delta\) itself is unchanged.

The natural question — how much of the shape does the spectrum capture? — is the famous one Mark Kac posed in 1966: “Can one hear the shape of a drum?” The answer is a precise almost. The spectrum determines a great deal (area, topology, total curvature — see below), but not everything: there exist isospectral, non-isometric shapes — geometrically different surfaces with identical spectra. So ShapeDNA is an extremely informative fingerprint, not a perfect one.

This limitation is exactly why point signatures and embeddings exist. Where the global spectrum can confuse two distinct shapes, the spatial pattern of \(\phi_i(x)^2\) across the surface — what HKS, WKS, and the wavelets encode — adds the localized information the bare spectrum throws away.

How they encode morphometry#

The bridge from “a list of eigenvalues” to “a measurement of shape” is the small-time behavior of the heat trace. For a closed surface, the Minakshisundaram–Pleijel expansion gives

\[ Z(t) = \sum_i e^{-\lambda_i t} \;\sim\; \frac{\mathrm{Area}(M)}{4\pi t} \;+\; \frac{\chi(M)}{6} \;+\; O(t) \qquad (t \to 0^+), \]

where \(\chi(M)\) is the Euler characteristic. Read term by term, this is a morphometric statement hiding inside the spectrum:

The leading term is proportional to surface area — equivalently, Weyl’s law \(N(\lambda) \sim \tfrac{\mathrm{Area}}{4\pi}\,\lambda\) says the density of eigenvalues counts area. The spectrum measures size.
The constant term is the Euler characteristic, and by the Gauss–Bonnet theorem \(\int_M K\,dA = 2\pi\chi(M)\), so this term is the total Gaussian curvature and the topology of the structure. The spectrum measures curvature and connectivity.

So even the global descriptor already carries area, integrated curvature, and topology. The point signatures then localize this: the high-frequency content of \(S_f(x)\) concentrates where curvature is high — protrusions, ridges, the necks between hippocampal subfields, the banks of a sulcus — so the field \(x \mapsto S_f(x)\) is a curvature- and scale-sensitive map of the structure.

Contrast this with classical morphometry. Volume and thickness compress a structure to one or two scalars and inherit the registration and voxel-size sensitivities of the segmentation. Spectral descriptors instead provide a pose-free, parameterization-robust characterization of the full geometry — they see the difference between two hippocampi of equal volume but different shape, which is precisely the regime where conditions like mesial temporal lobe epilepsy leave their signature, and where left/right asymmetries become measurable.

From the continuous theory to a mesh#

In practice the surface is a triangle mesh, and the operator is discretized with the cotangent stiffness matrix \(\mathbf{W}\) and the mass matrix \(\mathbf{A}\) (the Foundations page). Two practical consequences follow from the theory above:

Truncation is band-limiting. You compute only the first \(k\) eigenpairs, so every descriptor is implicitly low-passed at mode \(k\). Choosing \(k\) sets the finest geometric scale a descriptor can resolve.
Scale matters — sometimes. Because the leading heat-trace term scales with area, raw descriptors respond to overall size. When you want shape independent of size, use the scale-normalized variants (e.g. SI-HKS) or normalize the spectrum, exactly as shapedna_distance does internally.

import spectralbrain as sb

decomp = sb.BrainMesh(vertices, faces).decompose(k=300)

# Same decomposition, different spectral filters → different descriptors:
sdna = sb.compute_shapedna(decomp)                  # the spectrum (global)
hks  = sb.compute_hks(decomp, t_values=[1, 10, 100])  # low-pass field
wks  = sb.compute_wks(decomp, n_energies=100)         # band-pass field

The family map#

Everything in the Learn section is now one statement: each family is a choice within this single spectral framework.

The spectrum, read directly → ShapeDNA & GPS.
The diagonal, low-pass → heat descriptors.
The diagonal, band-pass → wave descriptors.
Alternative spectral kernels → Bates–Kornfeld.
A bank of band-pass filters → spectral wavelets.
The operator compared across shapes → functional maps & distances.