[Work Log] Hyperprior

December 09, 2013

Training hyperprior over indices vs. observation ordinal. Implemented in wacv_2012/wacv_train_index_prior.

Training is apparently very sensitive to initial point. Is the hessian wrong? Implemented test_mu_deriv() to test hessian using finite differences.

Struggling with modeling offset and rate. Mean is usually zero, but not in this case. Don't really care about initial conditions for now (and struggling with learning it anyway), so just learn smoothing sigma and set offset and rate covariance to large.

Index GP hyperprior doesn't fix index shrinkage issue, which is the root of several problems. Tried another hack that adds an energy term to keep the total length constant.

Still getting unexplained extension of curves. Time to review the literature for ideas.

Reading more on GP Latent Variable Models.

[1] N. D. Lawrence and A. J. Moore, “Hierarchical Gaussian process latent variable models,” presented at the ICML '07: Proceedings of the 24th international conference on Machine learning, 2007.

At once similar and different from what we're doing. In some sense, the index is a latent variable, but we aren't using latent variables as a dimension reduction technique, as is the crux of the GPLVM approach.

Reading up on existing literature dealing with reconstructing inextensible structures.

[1] M. Salzmann and P. Fua, “Linear Local Models for Monocular Reconstruction of Deformable Surfaces,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 33, no. 5, pp. 931–944. [2] M. Salzmann, R. Urtasun, and P. Fua, “Local deformation models for monocular 3D shape recovery,” presented at the Computer Vision and Pattern Recognition, 2008. CVPR 2008. IEEE Conference on, 2008, pp. 1–8. [3] M. Perriollat, R. Hartley, and A. Bartoli, “Monocular Template-based Reconstruction of Inextensible Surfaces,” Int J Comput Vision, vol. 95, no. 2, Nov. 2011. [4] J. Taylor, A. D. Jepson, and K. N. Kutulakos, “Non-rigid structure from locally-rigid motion,” presented at the Computer Vision and Pattern Recognition (CVPR), 2010 IEEE Conference on, 2010, pp. 2761–2768.

All these papers assume we can match keypoints of some patches, which isn't the case in our problem since our curves don't have distinctive texture. However, I can draw inspiration from [1], in which several interesting energy functions are constructed as homogeneous linear equations and with linear constraints. In one formulation, the normal posterior energy is supplemented with extra terms that prevent edge-lengths from changing. In another formulation, the rigidity constraint is replaced with an inequality that prevents extension but disallows contraction, which permits sharp folds.

There are significant differences in our case. First, we don't have a reference structure to compare the model against, so we don't know the reference length of each segment. Second, this assumes known correspondences, which we don't have. But we can use the idea of adding an energy term based on length.

Reference [2] uses GPLVM (Gaussian process latent variable model) to model local deformations. The latent variable formulation allows for nonlinear deformations (e.g. kinks, creases). On the downside, it's nonconvex, so is best suited for tracking only.

Posted by Kyle Simek
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