HeatAtlas: Surface Textures as Learnable Atlases of Local Heat Kernels

Heat Kernel Textures (HKTex) represent surface appearance with learnable anisotropic heat kernels, but evaluate each kernel through a global spectral construction that requires fifty Laplace–Beltrami eigendecompositions per mesh and a resident basis of shape [50, V, 256]. We show that this evaluator is mismatched to the learned primitive: trained HKTex responses are local and lie in the short-time regime of the heat equation, where the Minakshisundaram–Pleijel expansion is the natural evaluator and depends only on local geometry. HeatAtlas replaces the global eigensystem by radius-bounded local charts from batched mesh unfolding, on which the anisotropic heat kernel becomes a closed-form geodesic Gaussian with a first-order curvature correction. We then make these charts the representation itself: each kernel is a local chart whose non-negative heat response gives a normalised overlap weight and whose appearance is a local function $c_i(u,v)$, instantiated with intrinsic Gabor content in a parallel-transported frame. Separating where a primitive contributes from what it contains removes the spectral mode ceiling and all global Laplacian preprocessing, and supports learnable per-kernel scale, directional high-frequency appearance, local barriers and edits, planar, disconnected, non-manifold and million-vertex inputs, and transfer across discretisations without retraining. On the 32-object structural-failure set, HeatAtlas reaches 32.29 dB against 19.16 dB for HKTex and 26.82 dB for the strongest baseline, a hash grid; across five meshes of 50k–374k vertices, initialisation is 99–206× faster and peak memory 20–45× lower at equal quality; and at matched storage, intrinsic Gabor content improves view PSNR over constant charts by up to +4.0 dB on periodic textures, and on real high-frequency assets exceeds the hash grid by +1.9 dB (LPIPS 0.044 → 0.027).