Quasiconvex optimization for robust geometric reconstruction.

Geometric reconstruction problems in computer vision are often solved by minimizing a cost function that combines the reprojection errors in the 2D images. In this paper, we show that, for various geometric reconstruction problems, their reprojection error functions share a common and quasiconvex formulation. Based on the quasiconvexity, we present a novel quasiconvex optimization framework in which the geometric reconstruction problems are formulated as a small number of small-scale convex programs that are ready to solve. Our final reconstruction algorithm is simple and has intuitive geometric interpretation. In contrast to existing local minimization approaches, our algorithm is deterministic and guarantees a predefined accuracy of the minimization result. The quasiconvexity also provides an intuitive method to handle directional uncertainties and outliers in measurements. When there are outliers in the measurements, our method provides a mechanism to locate the global minimum of a robust error function. For large scale problems and when computational resources are constrained, we provide an efficient approximation that gives a good upper bound (but not global minimum) on the reconstruction error. We demonstrate the effectiveness of our algorithm by experiments on both synthetic and real data.