Heat-diffusion descriptors#

This family describes local geometry through heat flow: drop a unit of heat at a point and watch how much stays put over time. Slow time scales see broad, global shape; fast time scales see fine, local detail.

HKS — the Heat Kernel Signature#

The HKS records the amount of heat remaining at a point \(x\) after time \(t\), summed over the spectrum:

\[ \text{HKS}(x, t) = \sum_{i} e^{-\lambda_i t}\, \phi_i(x)^2 . \]

Sampling a range of \(t\) gives each vertex a multiscale signature: small \(t\) emphasizes curvature and ridges, large \(t\) emphasizes global position within the structure. The HKS is intrinsic, isometry-invariant, and stable under noise.

hks = sb.compute_hks(decomp, t_values=[1, 10, 100])   # (n_vert, n_t)

SI-HKS — Scale-Invariant HKS#

The plain HKS is invariant to isometry but not to scale: a larger copy of the same shape diffuses heat differently. SI-HKS removes the global scale factor (via a logarithmic time sampling and a Fourier transform of the signature), so two structures that differ only in size match.

si_hks = sb.compute_si_hks(decomp)

Anisotropic HKS#

Standard heat flow is isotropic — it spreads equally in all directions. The anisotropic variant biases diffusion along principal curvature directions, sharpening sensitivity to directional features such as sulcal banks or the long axis of the hippocampus. It is built on an anisotropic_laplacian() rather than the isotropic LBO.

from spectralbrain.spectral.anisotropic import compute_anisotropic_hks
a_hks = compute_anisotropic_hks(mesh, ...)