Aligning Points to Lines: Provable Approximations

We suggest a new optimization technique for minimizing the sum ∑ni=1gi(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 p1,…,pn and n lines ℓ1,…,ℓn on the plane and z>0 , compute the matching π:[n]→[n] and alignment (rotation matrix R and translation vector t ) that minimize the sum of euclidean distances ∑ni=1dist(Rpi−t,ℓπ(i))z between each point to its corresponding line. This problem is non-trivial even if z=1 and the matching π is given. If π is given, our algorithms run in 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.