Global fingerprints & embeddings#

Two descriptors read the spectral basis at the whole-shape level rather than per vertex: ShapeDNA, which is the spectrum itself, and GPS, which embeds every point into a spectral coordinate system.

ShapeDNA — the shape’s fingerprint#

ShapeDNA is simply the (normalized) sequence of LBO eigenvalues:

\[ \text{ShapeDNA} = (\lambda_1, \lambda_2, \dots, \lambda_k). \]

Because the spectrum is intrinsic and isometry-invariant, two surfaces with the same ShapeDNA have, to that truncation, the same intrinsic geometry. It is a compact, fixed-length fingerprint — ideal for comparing whole structures (left vs. right hippocampus, patient vs. normative reference) with a single distance.

sdna = sb.compute_shapedna(decomp)              # (k,) eigenvalue fingerprint
d = sb.shapedna_distance(sdna_a, sdna_b)        # compare two shapes

GPS — the Global Point Signature#

GPS turns each vertex into a point in an infinite-dimensional space whose coordinates are the eigenfunctions scaled by the inverse square-root of the eigenvalues:

\[ \text{GPS}(x) = \left( \frac{\phi_1(x)}{\sqrt{\lambda_1}}, \frac{\phi_2(x)}{\sqrt{\lambda_2}}, \dots \right). \]

In this embedding, intrinsic (geodesic-like) proximity on the surface becomes ordinary Euclidean proximity. It is the bridge from “shape” to “feature vector” for clustering or correspondence.

gps = sb.compute_gps(decomp, n_components=50)   # (n_vert, n_components)