2D, 3D, and 4D Shape Analysis

Statistical analysis of the shape of 2D and 3D objects as well as modeling the spatio-temporal variability in longitudinal 3D shape (also known as 4D shape) data are an important problem in mathematics, computer vision, and computer graphics. It has a wide range of applications in biology, medicine, 3D modeling, and simulation. Our group has been undertaking research in this important area for more than a decade. This page summarizes our contributions to this important field of research. It also contains links to our publications, codes we developed over the years, and data, which hopefully will benefit the readers and the community.

Note that the codes provided here are for research purposes only:

• You are free to use them for research purposes, just make sure you cite the papers and give credit to the authors.

• If you plan to use any of these resources for commercial purposes, please contact the corresponding authors.

Also, the codes are written for research purposes and thus will likely contain bugs and issues. Unfortunately, we cannot provide support on how to use them or how to fix potential bugs (due to the lack of resources and time). However, if you find a bug and manage to fix it, we will be glad to hear from you.

Source Codes

• Square Root Velocity Functions (SRVF): Elastic analysis of curves, whether they are in 2D, 3D, or nD [Project Page]. You may also be interested in the Matlab library for elastic functional data analysis [Project Page] maintained by Derek Tucker.

• Spherical parameterization of Genus-0 surfaces [Download the Code].

• Elastic registration and geodesics computation between parameterized surfaces. The code can handle genus-0, and thus spherically parameterized, surfaces and disk-like surfaces [Code - Coming Soon].

• Square-Root Normal Fields (SRNFs): This repository provides the code for the numerical inversion of SRNF maps. It also shows how to compute geodesics between registered surfaces, and how to transfer deformations (using geodesic shooting in the SRNF space) [Download the Code].

• Atlas4D: Spatio-temporal registration and statistical analysis of 4D surfaces (dynamic surfaces), also known as longitudinal 3D shape data, i.e., 3D shapes that evolve over time due to (abnormal) growth, actions, etc. [Code - Coming Soon]

People

Many people have contributed to this work over the years. It all started with planar curves (by Anuj Srivastava) and the work of Sebastian Kurtek on surfaces.

• Hamid Laga (Murdoch University, Australia).

• Sebastian Kurtek (The Ohio State University, US).

• Ian H. Jermyn (Durham University, UK).

• Anuj Srivastava (Florida State University, US)

• Mohammed Bennamoun (University of Western Australia, Australia)

• Eric Klassen (Florida State University, US)

• Qian Xie

• Marcel Padilla (TU Berlin, Germany)

References

4D Surfaces (Longitudinal 3D shape data)

• Hamid Laga, Marcel Padilla, Ian H. Jermyn, Sebastian Kurtek, Mohammed Bennamoun, and Anuj Srivastava. "4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data." Accepted for publication in IEEE transactions on pattern analysis and machine intelligence (2022). [Link to the Arxiv version]

Surfaces

• Hamid Laga, Qian Xie, Ian H. Jermyn, and Anuj Srivastava. "Numerical inversion of SRNF maps for elastic shape analysis of genus-zero surfaces." IEEE transactions on pattern analysis and machine intelligence 39, no. 12 (2017): 2451-2464.

• Jermyn, Ian H., Sebastian Kurtek, Hamid Laga, and Anuj Srivastava. "Elastic shape analysis of three-dimensional objects." Synthesis Lectures on Computer Vision 12, no. 1 (2017): 1-185.

• Sebastian Kurtek, Anuj Srivastava, Eric Klassen, and Hamid Laga. "Landmark‐guided elastic shape analysis of spherically‐parameterized surfaces." In Computer graphics forum, vol. 32, no. 2pt4, pp. 429-438. Oxford, UK: Blackwell Publishing Ltd, 2013.

• Jermyn, Ian H., Sebastian Kurtek, Eric Klassen, and Anuj Srivastava. "Elastic shape matching of parameterized surfaces using square root normal fields." In European conference on computer vision, pp. 804-817. Springer, Berlin, Heidelberg, 2012.

• Kurtek, Sebastian, Eric Klassen, John C. Gore, Zhaohua Ding, and Anuj Srivastava. "Elastic geodesic paths in shape space of parameterized surfaces." IEEE transactions on pattern analysis and machine intelligence 34, no. 9 (2011): 1717-1730.

• Kurtek, Sebastian, Eric Klassen, Zhaohua Ding, and Anuj Srivastava. "A novel Riemannian framework for shape analysis of 3D objects." In 2010 IEEE computer society conference on computer vision and pattern recognition, pp. 1625-1632. IEEE, 2010.

Curves in 2D, 3D, and nD