What question did this study set out to answer?

The aim is to introduce Harmonic Shape Transform as a topology-independent framework for shape correspondence using scalar fields.

May 10, 2026Open Access

Harmonic Shape Transform: A Scalar-Field Framework for Topology-Independent 3D Shape Mapping and Universal Spectral Initialization

Key Points

The aim is to introduce Harmonic Shape Transform as a topology-independent framework for shape correspondence using scalar fields.
Introduced HST for 3D shape mapping using normalized scalar fields and harmonic notes.
Tested on the FAUST dataset with 99 pairs and 6890 vertices, showcasing performance metrics.
Provided GPU-acceleration for significant speed improvements over existing methods such as ZoomOut and Functional Maps.
Mean geodesic error of HST Note was 0.129 with a computation time of 0.805s per pair, which is 53× faster than ZoomOut.
HST initialized ZoomOut improved performance by 42.3%, while Functional Maps improved by 52.5%.
HST Dual Note achieved a mean geodesic error of 0.120 with 61 out of 99 wins, demonstrating enhanced accuracy.

Abstract

This paper introduces the Harmonic Shape Transform (HST), a general framework for mapping one shape onto another using normalized scalar fields defined intrinsically on each domain. The core idea is that every 3D shape possesses a harmonic note — a normalized Laplace-Beltrami eigenfunction that encodes its internal proportional structure. HST constructs mappings by aligning these normalized level sets across shapes, requiring no matching topology, mesh connectivity, or explicit point alignment. Key results on the full FAUST dataset (99 pairs, 6890 vertices, fully deterministic, zero failures): - HST Note: mean geodesic error 0.129, time 0.805s/pair (53× faster than ZoomOut)- HST as initializer improves ZoomOut by 42.3% (0/99 random wins)- HST as initializer improves Functional Maps by 52.5% (1/99 random wins)- HST Dual Note: mean geodesic error 0.120 (+7.1% over Single, 61/99 wins)- GPU acceleration (RTX 4070): ZoomOut 6.5× faster, FMaps 15× faster- Full pipeline: 142 min → 13 min (11× GPU speedup)- CPU and GPU produce identical results — hardware-independent- Volumetric HST: surface + interior (tetrahedral Laplacian) + exterior (SDF) — 1.70s per shape The central finding is that the harmonic note acts as a universal geometric predictor: a single normalized eigenfunction systematically improves any spectral shape correspondence method as initialization, regardless of the specific algorithm. Tested independently on ZoomOut (Melzi et al. 2019) and Functional Maps (Ovsjanikov et al. 2012). Additional contributions include HST Dual Note (two eigenfunctions for symmetry disambiguation), volumetric extension via tetrahedral Laplacian and SDF, GPU-accelerated Blender addons, correspondence visualization, and texture transfer tools. This is the first publicly documented GPU-accelerated spectral shape correspondence pipeline combining volumetric Laplace-Beltrami operators with HST initialization — including the first GPU implementation of Functional Maps and ZoomOut refinement. Full code, raw CSV data, and Blender addons:https://github.com/sel8888/harmonic-shape-transform-2026-koncepthttps://orcid.org/0009-0003-9680-3333

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Cite This Study

Krahulík Pavel (Thu,) studied this question.

synapsesocial.com/papers/6a002126c8f74e3340f9bfec https://doi.org/https://doi.org/10.5281/zenodo.20089996

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