Subspace Procrustes Analysis
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Procrustes analysis (PA) has been a popular technique to align and build 2-D statistical models of shapes. Given a set of 2-D shapes PA is applied to remove rigid transformations. Later, a non-rigid 2-D model is computed by modeling the residual (e.g., PCA). Although PA has been widely used, it has several limitations for modeling 2-D shapes: occluded landmarks and missing data can result in local minima solutions, and there is no guarantee that the 2-D shapes provide a uniform sampling of the 3-D space of rotations for the object. To address previous issues, this paper proposes subspace PA (SPA). Given several instances of a 3-D object, SPA computes the mean and a 2-D subspace that can model rigid and non-rigid deformations of the 3-D object. We propose a discrete (DSPA) and continuous (CSPA) formulation for SPA, assuming that 3-D samples of an object are provided. DSPA extends the traditional PA, and produces unbiased 2-D models by uniformly sampling different views of the 3-D object. CSPA provides a continuous approach to uniformly sample the space of 3-D rotations, being more efficient in space and time. We illustrate the benefits of SPA in two different applications. First, SPA is used to learn 2-D face and body models from 3-D datasets. Experiments on the FaceWarehouse and CMU motion capture (MoCap) datasets show the benefits of our 2-D models against the state-of-the-art PA approaches and conventional 3-D models. Second, SPA learns an unbiased 2-D model from CMU MoCap dataset and it is used to estimate the human pose on the Leeds Sports dataset.
KeywordsProcrustes analysis Learning 2D shape models Functional subspace learning
This work is partly supported by the Spanish Ministry of Science and Innovation (Projects TIN2012-38416-C03-01, TIN2015-66951-C2-1-R, TIN2013-43478-P), Project 2014 SGR 1219, SUR, Departament d’Economia i Coneixement, and Comissionat per a Universitats i Recerca del Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya.
- Andriluka, M., Roth, S., & Schiele, B. (2009). Pictorial structures revisited: People detection and articulated pose estimation. In IEEE computer vision and pattern recognition (CVPR), pp. 1014–1021.Google Scholar
- Brand, M. (2001). Morphable 3d models from video. In IEEE computer vision and pattern recognition (CVPR), Vol. 2, pp. II–456.Google Scholar
- Cao, C., Weng, Y., Zhou, S., Tong, Y., & Zhou, K. (2013). Facewarehouse: A 3d facial expression database for visual computing. IEEE Transactions on Visualization and Computer Graphics, 1(1), 99.Google Scholar
- Carnegie mellon motion capture database. http://mocap.cs.cmu.edu
- Cootes, T. F. & Taylor, C. J. (2004). Statistical models of appearance for computer vision.Google Scholar
- Goodall, C. (1991). Procrustes methods in the statistical analysis of shape. Journal of the Royal Statistical Society. Series B (Methodological), pp. 285–339Google Scholar
- Johnson, S., & Everingham, M. (2010). Clustered pose and nonlinear appearance models for human pose estimation. In Proceedings of the British machine vision conference. doi: 10.5244/C.24.12.
- Kokkinos, I. & Yuille, A. (2007). Unsupervised learning of object deformation models. In IEEE international conference on computer vision (ICCV), pp. 1–8Google Scholar
- Korte, B., Lovász, L., & Schrader, R. (1991). Greedoids. 1991. Algorithms and Combinatorics. Berlin: SpringerGoogle Scholar
- Li, H., Huang, X., & He, L. (2013). Object matching using a locally affine invariant and linear programming techniques. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(2), 411–424.Google Scholar
- Marques, M., Stosić, M., & Costeira, J. (2009). Subspace matching: Unique solution to point matching with geometric constraints. In IEEE international conference on computer vision (ICCV), pp. 1288–1294.Google Scholar
- Minka, T. P. (2000). Old and new matrix algebra useful for statistics. http://research.microsoft.com/~minka/papers/matrix/.
- Naimark, M. A. (1964). Linear representatives of the Lorentz group (translated from Russian). New York: Macmillan.Google Scholar
- Park, D. & Ramanan, D. (2011). N-best maximal decoders for part models. In IEEE international conference on computer vision (ICCV), pp. 2627–2634Google Scholar
- Perez-Sala, X., De la Torre, F., Igual, L., Escalera, S., & Angulo, C. (2014). Subspace procrustes analysis. In ECCV Workshop on ChaLearn Looking at People.Google Scholar
- Pishchulin, L., Andriluka, M., Gehler, P., & Schiele, B. (2013a). Poselet conditioned pictorial structures. In IEEE conference on computer vision and pattern recognition (CVPR), pp. 588–595Google Scholar
- Pishchulin, L., Andriluka, M., Gehler, P., & Schiele, B. (2013b). Strong appearance and expressive spatial models for human pose estimation. In IEEE international conference on computer vision (ICCV), pp. 3487–3494.Google Scholar
- Pizarro, D., & Bartoli, A. (2011). Global optimization for optimal generalized procrustes analysis. In IEEE conference on computer vision and pattern recognition CVPR, pp. 2409–2415.Google Scholar
- Roig, G., Boix, X., & De la Torre, F. (2009). Optimal feature selection for subspace image matching. In IEEE international conference on computer vision workshops, pp. 200–205.Google Scholar
- Yang, F., Shechtman, E., Wang, J., Bourdev, L., & Metaxas, D. (2012). Face morphing using 3d-aware appearance optimization. In Graphics Interface (pp. 93–99). Canadian Information Processing SocietyGoogle Scholar
- Zhou, F., Brandt, J., & Lin, Z. (2013). Exemplar-based graph matching for robust facial landmark localization. In IEEE international conference on computer vision (ICCV), pp. 1025–1032.Google Scholar