3D Shape Reconstruction from 2D Landmarks: A Convex Formulation

Nice ! As you know, I am a big fan of phase transitions especially when they are used to show what can and cannot be done. Today the authors, our new map makers, went about solving a problem in a convex fashion and they then featured the limit of their algorithm in a field that is not really known to care too much about these things.   And yes, if there are too few landmarks and too few coefficient, your brain the solver can't figure out the pose. I wonder how this transposes to optical illusions.  
3D Shape Reconstruction from 2D Landmarks: A Convex Formulation by Xiaowei Zhou, Spyridon Leonardos, Xiaoyan Hu, Kostas Daniilidis

We investigate the problem of reconstructing the 3D shape of an object, given a set of landmarks in a single image. To alleviate the reconstruction ambiguity, a widely-used approach is to confine the unknown 3D shape within a shape space built upon existing shapes. While this approach has proven to be successful in various applications, a challenging issue remains, i.e. the joint estimation of shape parameters and camera-pose parameters requires to solve a nonconvex optimization problem. The existing methods often adopt an alternating minimization scheme to locally update the parameters, and consequently the solution is sensitive to initialization. In this paper, we propose a convex formulation to address this issue and develop an efficient algorithm to solve the proposed convex program. We demonstrate the exact recovery property of the proposed method, its merits compared to the alternative methods, and the applicability in human pose, car and face reconstruction. 

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