Aligning Points to Lines: Provable Approximations

We suggest a new optimization technique for minimizing the sum ∑ i=1n gi(x)∑i=1ngi(x) of n non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational geometry, combinatorics and convex optimization. As an example application, we provide the first constant-factor approximation algorithms whose running-times are polynomial in n for the fundamental problem of Points-to-Lines alignment: Given n points p 1,\p n p1,..,pn and n lines ℓ_1,\ℓ n ℓ1,..,ℓn on the plane and z>0 z>0, compute the matching π:[n]\to [n] π:[n]→[n] and alignment (rotation matrix R and translation vector t t) that minimize the sum of euclidean distances ∑ i=1n dist (Rp_i-t,ℓ π (i))z∑i=1n dist (Rpi-t,ℓπ(i))z between each point to its corresponding line. This problem is non-trivial even if z=1 and the matching ππ is given. If π is given, our algorithms run in O(n3)O(n3) time, and even near-linear in n using core-sets that support: streaming, dynamic, and distributed parallel computations in poly-logarithmic update time. Generalizations for handling e.g., outliers or pseudo-distances such as M-estimators for the problem are also provided. Experimental results and open source code show that our algorithms improve existing heuristics also in practice. A companion demonstration video in the context of Augmented Reality shows how such algorithms may be used in real-time systems.