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Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework.

ABSTRACT: This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

Supplementary information

The online version contains supplementary material available at 10.1007/s11263-022-01743-0.

SUBMITTER: Hartman E

PROVIDER: S-EPMC10102155 | biostudies-literature | 2023

REPOSITORIES: biostudies-literature

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Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework.

Hartman Emmanuel E Sukurdeep Yashil Y Klassen Eric E Charon Nicolas N Bauer Martin M

International journal of computer vision 20230121 5

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on s ...[more]

PMID: 37069835

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Project description:BackgroundProtein structure comparison play important role in in silico functional prediction of a new protein. It is also used for understanding the evolutionary relationships among proteins. A variety of methods have been proposed in literature for comparing protein structures but they have their own limitations in terms of accuracy and complexity with respect to computational time and space. There is a need to improve the computational complexity in comparison/alignment of proteins through incorporation of important biological and structural properties in the existing techniques.ResultsAn efficient algorithm has been developed for comparing protein structures using elastic shape analysis in which the sequence of 3D coordinates atoms of protein structures supplemented by additional auxiliary information from side-chain properties are incorporated. The protein structure is represented by a special function called square-root velocity function. Furthermore, singular value decomposition and dynamic programming have been employed for optimal rotation and optimal matching of the proteins, respectively. Also, geodesic distance has been calculated and used as the dissimilarity score between two protein structures. The performance of the developed algorithm is tested and found to be more efficient, i.e., running time reduced by 80-90 % without compromising accuracy of comparison when compared with the existing methods. Source codes for different functions have been developed in R. Also, user friendly web-based application called ProtSComp has been developed using above algorithm for comparing protein 3D structures and is accessible free.ConclusionsThe methodology and algorithm developed in this study is taking considerably less computational time without loss of accuracy (Table 2). The proposed algorithm is considering different criteria of representing protein structures using 3D coordinates of atoms and inclusion of residue wise molecular properties as auxiliary information.

Dataset Information

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework.

Supplementary information

Publications

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework.

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