# Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces

arXiv: https://arxiv.org/abs/1610.04531 · HTML: https://mlsanity.com/paper/1610.04531/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2016.

## Built on by (most cited first, 12 of 12)

- [Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework](https://mlsanity.com/paper/2204.04238/) (2022)
- [4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data](https://mlsanity.com/paper/2101.09403/) (2021)
- [A Riemannian Framework for Analysis of Human Body Surface](https://mlsanity.com/paper/2108.11449/) (2021)
- [Dynamic Neural Surfaces for Elastic 4D Shape Representation and Analysis](https://mlsanity.com/paper/2503.03132/) (2025)
- [Elastic Shape Analysis of Tree-like 3D Objects using Extended SRVF Representation](https://mlsanity.com/paper/2110.08693/) (2021)
- [Shape-Graph Matching Network (SGM-net): Registration for Statistical Shape Analysis](https://mlsanity.com/paper/2308.06869/) (2023)
- [Procrustes Analysis with Deformations: A Closed-Form Solution by Eigenvalue Decomposition](https://mlsanity.com/paper/2206.14528/) (2022)
- [A function space perspective on stochastic shape evolution](https://mlsanity.com/paper/2302.05382/) (2023)
- [PaNDaS: Learnable Deformation Modeling with Localized Control](https://mlsanity.com/paper/2412.02306/) (2024)
- [TreeSRNF: Square-Root Normal Fields for Generative Modelling of the Geometric and Structural Variability in Tree-like 3D Objects](https://mlsanity.com/paper/2607.13456/) (2026)
- [Elastic Shape Analysis of Brain Structures for Predictive Modeling of PTSD](https://mlsanity.com/paper/2105.11547/) (2021)
- [NeuralSRNF: Neural Square Root Normal Fields for the Statistical Shape Analysis and Generation of Nonrigid 3D and 4D Objects](https://mlsanity.com/paper/2609.27728/) (2026)

## Abstract (arXiv)

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides comprehensive frameworks for simultaneous registration, deformation, and comparison of shapes. These methods achieve computational efficiency using certain square-root representations that transform invariant elastic metrics into Euclidean metrics, allowing for applications of standard algorithms and statistical tools. For analyzing shapes of embeddings of mathbbS2 in ℝ3, Jermyn et al. introduced square-root normal fields (SRNFs) that transformed an elastic metric, with desirable invariant properties, into the mathbbL2 metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is to invert solutions (deformations, averages, modes of variations, etc) computed in the SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNFs maps and their inverses, we take a numerical approach and derive an efficient multiresolution algorithm, based on solving an optimization problem in the surface space, that estimates surfaces corresponding to given SRNFs. This solution is found effective, even for complex shapes, e.g. human bodies and animals, that undergo significant deformations including bending and stretching. Specifically, we use this inversion for computing elastic shape deformations, transferring deformations, summarizing shapes, and for finding modes of variability in a given collection, while simultaneously registering the surfaces. We demonstrate the proposed algorithms using a statistical analysis of human body shapes, classification of generic surfaces and analysis of brain structures.