What question did this study set out to answer?

The research aims to develop a framework for mapping shapes without needing topological matching, using harmonic properties.

May 14, 2026Open Access

Harmonic Shape Transform: A Scalar-Field Framework for Topology-Independent 3D/2D Shape Mapping and Universal Spectral Initialization

Key Points

The research aims to develop a framework for mapping shapes without needing topological matching, using harmonic properties.
Introduced the Harmonic Shape Transform (HST) framework for 3D and 2D shape mapping.
Utilized normalized Laplace-Beltrami eigenfunctions to encode intrinsic structures of shapes.
Tested on various shape datasets including organic and manufactured geometries.
Achieved a mean geodesic error of 0.110 at 1.25s/pair on GPU, an improvement from 0.129.
Demonstrated clean harmonic gradients in 2D photographs of organic matter with a geodesic error of 0.132.
Showed that HST works effectively for organic shapes but not for chaotic CAD geometries.

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. What's New in This Version Major New Results **1. HST Pipeline v6 — Best-of-All**Mean geodesic error **0.110** (down from 0.129 standalone) at 1.25s/pair on GPU.Pipeline returns minimum result across all methods per pair. **2. Production Implementation**C++ CPU + GPU (RTX 4070) — bit-identical results across Python, C++ CPU, and GPUon all 99 FAUST pairs. FMaps 17.9× faster than Python CPU. **3. Organic vs. Manufactured Shapes**HST tested on ModelNet10 CAD geometry — geo error 0.35–0.58.Confirms that HST is specifically suited to shapes found in nature. First-of-Its-Kind Discovery **4. 2D HST on Photographs**First experimental demonstration that Laplace-Beltrami eigenfunctionsproduce clean, globally smooth harmonic gradients on 2D photographsof living organic matter (human faces and bodies) — identical incharacter to 3D mesh results. - Human face: geo error **0.132** (best pair, similar expressions)- Human body photograph: smooth global gradient, no training required- CAD geometry: chaotic eigenfunctions — does not work- Never demonstrated before in spectral geometry or image processing **The boundary between organic and artificial geometryis now measurable through spectral geometry.** Paper Updates- New Section 7: Extension to 2D Image Domain- Updated abstract with 2D discovery- Updated conclusion with philosophical implications- New figures: 2D harmonic notes on photographs- 12 pages total (was 11) Full code, raw CSV data, and Blender addons:https://github.com/sel8888/harmonic-shape-transform-2026-koncepthttps://orcid.org/0009-0003-9680-3333

Harmonic Shape Transform: A Scalar-Field Framework for Topology-Independent 3D/2D Shape Mapping and Universal Spectral Initialization

Key Points

Abstract

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