Equiareal Shape-from-Template
Authors
Casillas Pérez, David; Pizarro Pérez, Daniel; Fuentes Jiménez, David; Mazo Quintas, Manuel Ramón; Bartoli, AdrienIdentifiers
Permanent link (URI): http://hdl.handle.net/10017/60350DOI: 10.1007/s10851-018-0862-5
ISSN: 0924-9907
Publisher
Springer
Date
2018-11-26Funders
Agencia Estatal de Investigación
Universidad de Alcalá
European Commission
Bibliographic citation
Casillas Pérez, D. [et al.], 2019, "Equiareal Shape-from-Template", Journal of Mathematical Imaging and Vision, vol. 61, pp. 607–626.
Project
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/TIN2016-80939-R/ES/RECONSTRUCCION DE OBJETOS DEFORMABLES A PARTIR DE IMAGENES Y SUS APLICACIONES A LA REALIDAD AUMENTADA EN CIRUGIA MINIMAMENTE INVASIVA/
info:eu-repo/grantAgreement/UAH//CCGP2017%2FEXP-048
info:eu-repo/grantAgreement/EC/FP7/307483/EU/DEFORMABLE MULTIPLE-VIEW GEOMETRY AND 3D RECONSTRUCTION, WITH APPLICATION TO MINIMALLY INVASIVE SURGERY/FLEXABLE
Document type
info:eu-repo/semantics/article
Version
info:eu-repo/semantics/acceptedVersion
Publisher's version
https://doi.org/10.1007/s10851-018-0862-5Rights
Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
© 2018 Springer
Access rights
info:eu-repo/semantics/openAccess
Abstract
This paper studies the 3D reconstruction of a deformable surface from a single image and a reference surface, known as the template. This problem is known as Shape-from-Template and has been recently shown to be well-posed for isometric deformations, for which the surface bends without altering geodesics. This pa- per studies the case of equiareal deformations. They are elastic deformations where the local area is pre- served and thus include isometry as a special case. Elas- tic deformations have been studied before in Shape- from-Template, yet no theoretical results were given on the existence or uniqueness of solutions. The equiareal model is much more widely applicable than the isomet- ric and the conformal models. This paper introduces Monge?s theory, widely used for studying the solutions of non-linear first-order PDEs to the field of 3D recon- struction. It uses this theory to establish a theoretical framework for equiareal Shape-from-Template and an- swers the important question of whether it is possible to reconstruct a surface exactly which a much weaker prior than isometry. Indead, we prove that equiareal Shape-from-Template has a maximum of two local so- lutions sufficiently near an initial curve that lies on the surface. In addition we propose a convex reconstruction algorithm that can recover the multiple solutions. Our algorithm uses standard numerical tools for ODEs. We use the perspective camera model and give reconstruc- tion results with both synthetic and real examples.
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