Sphere fitting with applications to machine tracking

Abstract

We suggest a provable and practical approximation algorithm for fitting a set P of n points in Rd to a sphere. Here, a sphere is represented by its center x ε Rd and radius r>0. The goal is to minimize the sum ∑pεP | p-x-r | of distances to the points up to a multiplicative factor of 1± ε, for a given constant ε > 0, over every such r and x. Our main technical result is a data summarization of the input set, called coreset, that approximates the above sum of distances on the original (big) set P for every sphere. Then, an accurate sphere can be extracted quickly via an inefficient exhaustive search from the small coreset. Most articles focus mainly on sphere identification (e.g., circles in 2D image) rather than finding the exact match (in the sense of extent measures), and do not provide approximation guarantees. We implement our algorithm and provide extensive experimental results on both synthetic and real-world data. We then combine our algorithm in a mechanical pressure control system whose main bottleneck is tracking a falling ball. Full open source is also provided.