Abstract
We present a closed-form solution to the problem of recovering the 3D shape of a non-rigid inelastic surface from 3D-to-2D correspondences. This lets us detect and reconstruct such a surface by matching individual images against a reference configuration, which is in contrast to all existing approaches that require initial shape estimates and track deformations from image to image.
We represent the surface as a mesh, and write the constraints provided by the correspondences as a linear system whose solution we express as a weighted sum of eigenvectors. Obtaining the weights then amounts to solving a set of quadratic equations accounting for inextensibility constraints between neighboring mesh vertices. Since available closed-form solutions to quadratic systems fail when there are too many variables, we reduce the number of unknowns by expressing the deformations as a linear combination of modes. The overall closed-form solution then becomes tractable even for complex deformations that require many modes.
This work was supported in part by the Swiss National Science Foundation and in part by the European Commission under the IST-project 034307 DYVINE (Dynamic Visual Networks).
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References
Metaxas, D., Terzopoulos, D.: Constrained deformable superquadrics and nonrigid motion tracking. PAMI 15, 580–591 (1993)
Cohen, L., Cohen, I.: Deformable models for 3-d medical images using finite elements and balloons. In: CVPR, pp. 592–598 (1992)
Pentland, A.: Automatic extraction of deformable part models. IJCV 4, 107–126 (1990)
Blanz, V., Vetter, T.: A Morphable Model for The Synthesis of 3–D Faces. ACM SIGGRAPH, 187–194 (1999)
Matthews, I., Baker, S.: Active Appearance Models Revisited. IJCV 60, 135–164 (2004)
Torresani, L., Hertzmann, A., Bregler, C.: Learning non-rigid 3d shape from 2d motion. In: NIPS (2003)
Salzmann, M., Pilet, J., Ilić, S., Fua, P.: Surface Deformation Models for Non-Rigid 3–D Shape Recovery. PAMI 29, 1481–1487 (2007)
White, R., Forsyth, D.: Combining cues: Shape from shading and texture. In: CVPR (2006)
Moreno-Noguer, F., Lepetit, V., Fua, P.: Accurate Non-Iterative O(n) Solution to the PnP Problem. In: ICCV (2007)
Courtois, N., Klimov, A., Patarin, J., Shamir, A.: Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807. Springer, Heidelberg (2000)
Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active Contour Models. IJCV 1, 321–331 (1988)
Terzopoulos, D., Witkin, A., Kass, M.: Symmetry-seeking Models and 3D Object Reconstruction. IJCV 1, 211–221 (1987)
Bartoli, A., Zisserman, A.: Direct Estimation of Non-Rigid Registration. In: BMVC (2004)
Gay-Bellile, V., Bartoli, A., Sayd, P.: Direct estimation of non-rigid registrations with image-base self-occlusion reasoning. In: ICCV (2007)
Pilet, J., Lepetit, V., Fua, P.: Real-Time Non-Rigid Surface Detection. In: CVPR (2005)
Zhu, J., Lyu, M.R.: Progressive finit newton approach to real-time nonrigid surface detection. In: ICCV (2007)
McInerney, T., Terzopoulos, D.: A Finite Element Model for 3D Shape Reconstruction and Nonrigid Motion Tracking. In: ICCV (1993)
Delingette, H., Hebert, M., Ikeuchi, K.: Deformable surfaces: A free-form shape representation. Geometric Methods in Computer Vision (1991)
Perriollat, M., Hartley, R., Bartoli, A.: Monocular Template-based Reconstruction of Inextensible Surfaces. In: BMVC (2008)
Cootes, T., Edwards, G., Taylor, C.: Active Appearance Models. In: Burkhardt, H., Neumann, B. (eds.) ECCV 1998. LNCS, vol. 1407. Springer, Heidelberg (1998)
Llado, X., Bue, A.D., Agapito, L.: Non-rigid 3D Factorization for Projective Reconstruction. In: BMVC (2005)
Salzmann, M., Lepetit, V., Fua, P.: Deformable Surface Tracking Ambiguities. In: CVPR (2007)
Salzmann, M., Hartley, R., Fua, P.: Convex Optimization for Deformable Surface 3–D Tracking. In: ICCV (2007)
Quan, L., Lan, Z.: Linear N-Point Camera Pose Determination. PAMI 21, 774–780 (1999)
Fiore, P.D.: Efficient linear solution of exterior orientation. PAMI 23, 140–148 (2001)
Ansar, A., Daniilidis, K.: Linear pose estimation from points or lines. PAMI 25, 578–589 (2003)
Lowe, D.: Distinctive Image Features from Scale-Invariant Keypoints. IJCV 20, 91–110 (2004)
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Salzmann, M., Moreno-Noguer, F., Lepetit, V., Fua, P. (2008). Closed-Form Solution to Non-rigid 3D Surface Registration. In: Forsyth, D., Torr, P., Zisserman, A. (eds) Computer Vision – ECCV 2008. ECCV 2008. Lecture Notes in Computer Science, vol 5305. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88693-8_43
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