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We study the problem of 3D shape reconstruction from 2D landmarks extractedin a single image. We adopt the 3D deformable shape model and formulate thereconstruction as a joint optimization of the camera pose and the linear shapeparameters. Our first contribution is to apply Lasserre's hierarchy of convexSums-of-Squares (SOS) relaxations to solve the shape reconstruction problem andshow that the SOS relaxation of minimum order 2 empirically solves the originalnon-convex problem exactly. Our second contribution is to exploit the structureof the polynomial in the objective function and find a reduced set of basismonomials for the SOS relaxation that significantly decreases the size of theresulting semidefinite program (SDP) without compromising its accuracy. Thesetwo contributions, to the best of our knowledge, lead to the first certifiablyoptimal solver for 3D shape reconstruction, that we name Shape*. Our thirdcontribution is to add an outlier rejection layer to Shape* using a truncatedleast squares (TLS) robust cost function and leveraging graduated non-convexityto solve TLS without initialization. The result is a robust reconstructionalgorithm, named Shape#, that tolerates a large amount of outlier measurements.We evaluate the performance of Shape* and Shape# in both simulated and realexperiments, showing that Shape* outperforms local optimization and previousconvex relaxation techniques, while Shape# achieves state-of-the-artperformance and is robust against 70% outliers in the FG3DCar dataset.

In Perfect Shape: Certifiably Optimal 3D Shape Reconstruction from 2D Landmarks