# 4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data

arXiv: https://arxiv.org/abs/2101.09403 · HTML: https://mlsanity.com/paper/2101.09403/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 57 references that are in the map.
- First posted on arXiv 23 Jan 2021.

## Builds on (most cited first, 12 of 12)

- [Expressive Body Capture: 3D Hands, Face, and Body from a Single Image](https://mlsanity.com/paper/1904.05866/) (2019)
- [3D Morphable Face Models -- Past, Present and Future](https://mlsanity.com/paper/1909.01815/) (2019)
- [Generating 3D faces using Convolutional Mesh Autoencoders](https://mlsanity.com/paper/1807.10267/) (2018)
- [Capture, Learning, and Synthesis of 3D Speaking Styles](https://mlsanity.com/paper/1905.03079/) (2019)
- [Three-D Safari: Learning to Estimate Zebra Pose, Shape, and Texture from Images "In the Wild"](https://mlsanity.com/paper/1908.07201/) (2019)
- [A Survey on Deep Learning Techniques for Stereo-based Depth Estimation](https://mlsanity.com/paper/2006.02535/) (2020)
- [Image-based 3D Object Reconstruction: State-of-the-Art and Trends in the Deep Learning Era](https://mlsanity.com/paper/1906.06543/) (2019)
- [Articulation-aware Canonical Surface Mapping](https://mlsanity.com/paper/2004.00614/) (2020)
- [Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces](https://mlsanity.com/paper/1610.04531/) (2016)
- [A Survey on Non-rigid 3D Shape Analysis](https://mlsanity.com/paper/1812.10111/) (2018)
- [A numerical framework for elastic surface matching, comparison, and interpolation](https://mlsanity.com/paper/2006.11652/) (2020)
- [Statistical Analysis and Modeling of the Geometry and Topology of Plant Roots](https://mlsanity.com/paper/1905.06812/) (2019)

## Built on by (most cited first, 8 of 8)

- [Elastic shape analysis of surfaces with second-order Sobolev metrics: a comprehensive numerical framework](https://mlsanity.com/paper/2204.04238/) (2022)
- [BaRe-ESA: A Riemannian Framework for Unregistered Human Body Shapes](https://mlsanity.com/paper/2211.13185/) (2022)
- [Basis restricted elastic shape analysis on the space of unregistered surfaces](https://mlsanity.com/paper/2311.04382/) (2023)
- [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)
- [A Riemannian Approach for Spatiotemporal Analysis and Generation of 4D Tree-shaped Structures](https://mlsanity.com/paper/2408.12443/) (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)
- [Hallo4D: Multi-Modal Hallucination Mitigation for Consistent Spatio-Temporal Generation](https://mlsanity.com/paper/2607.12752/) (2026)

## Abstract (arXiv)

We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, also called 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields where the L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a Euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries; and (4) the synthesis of random 4D surfaces.