Functional maps & distances#

The descriptors so far live on a single shape. This page covers the tools that relate two or more shapes: functional maps and the shape difference operator for correspondence, the collection-aware DWKS, and the spectral distance metrics that turn shape comparison into a number.

Functional maps and the shape difference operator#

A functional map represents a correspondence between two surfaces not as a vertex-to-vertex matching but as a small matrix \(\mathbf{C}\) that translates functions from one shape’s spectral basis to the other’s. It is compact, robust, and the natural language for comparing structures of slightly different tessellation.

The shape difference operator (SDO) then localizes where and how two shapes differ, given their functional map — separating area-based from conformal (angle-based) change.

C = sb.compute_functional_map(decomp_a, decomp_b, descriptors_a, descriptors_b)
sdo = sb.shape_difference_operator(decomp_a, decomp_b, C)

DWKS — collection-aware wave signature#

The Derivative WKS extends the wave signature across a collection of shapes related by functional maps, exposing how band-pass geometry varies through the collection rather than within a single surface.

from spectralbrain.spectral.collections import compute_dwks, compute_dwks_collection

Spectral distances#

Once shapes are described, these metrics quantify how far apart they are:

Function	Distance
`wesd()` / `wesd_matrix()`	Weighted Spectral Distance — eigenvalue-based, whole-shape
`shapedna_distance()`	distance between ShapeDNA fingerprints
`biharmonic_distance()`	smooth, globally-aware point-to-point distance
`commute_time_distance()`	expected random-walk commute time between points
`diffusion_distance()`	heat-diffusion distance (with a multiscale variant)
`descriptor_distance()`	generic distance between two descriptor fields

From point-to-point spectral distances you can also assemble a geometric connectome — build_geometric_connectome() — and aggregate it to networks for graph-level analysis.

W = sb.wesd_matrix(decompositions)            # pairwise whole-shape distances
conn = sb.build_geometric_connectome(decomp)  # intrinsic connectome