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

Laga, Hamid; Xie, Qian; Jermyn, Ian H.; Srivastava, Anuj

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


Authors

Hamid Laga

Qian Xie

Anuj Srivastava



Abstract

Recent developments in elastic shape analysis (ESA) are motivated by the fact that it provides a comprehensive framework 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 the application of standard algorithms and statistical tools. For analyzing shapes of embeddings of S2 in R3, Jermyn et al. [1] introduced square-root normal fields (SRNFs), which transform an elastic metric, with desirable invariant properties, into the L2 metric. These SRNFs are essentially surface normals scaled by square-roots of infinitesimal area elements. A critical need in shape analysis is a method for inverting solutions (deformations, averages, modes of variations, etc.) computed in SRNF space, back to the original surface space for visualizations and inferences. Due to the lack of theory for understanding SRNF 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 to be effective even for complex shapes that undergo significant deformations including bending and stretching, e.g. human bodies and animals. 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.

Citation

Laga, H., Xie, Q., Jermyn, I. H., & Srivastava, A. (2017). Numerical Inversion of SRNF Maps for Elastic Shape Analysis of Genus-Zero Surfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(12), 2451-2464. https://doi.org/10.1109/tpami.2016.2647596

Journal Article Type Article
Acceptance Date Dec 18, 2016
Online Publication Date Jan 5, 2017
Publication Date Dec 1, 2017
Deposit Date Feb 8, 2017
Publicly Available Date Mar 28, 2024
Journal IEEE Transactions on Pattern Analysis and Machine Intelligence
Print ISSN 0162-8828
Electronic ISSN 1939-3539
Publisher Institute of Electrical and Electronics Engineers
Peer Reviewed Peer Reviewed
Volume 39
Issue 12
Pages 2451-2464
DOI https://doi.org/10.1109/tpami.2016.2647596
Related Public URLs https://arxiv.org/abs/1610.04531

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