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 R d to a sphere. Here, a sphere is represented by its center x ∈ R d and radius r > 0 . The goal is to minimize the sum ∑ p ∈ P ∣ 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 2 D 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.

Hypersphere Fitting From Noisy Data Using an EM Algorithm

“…These latent vectors are then corrupted by a zero-mean Gaussian noise with covariance matrix σ 2 I d . For each run, the coordinates of the true center and the radius are chosen uniformly in the intervals [5,10] and [1,10]. The EM algorithm is iterative and requires to be initialized properly.…”

“…F ITTING a circle, a sphere or more generally an hypersphere to a noisy point cloud is a recurrent problem in many applications including object tracking [1]- [3], robotics [4]- [6] or image processing and pattern recognition [7]- [9]. Popular methods available in the literature are based on least squares [10]- [14] or maximum likelihood (ML) estimation [15], [16].…”

Deconvolution of spherical data corrupted with unknown noise

“…The statistical estimation of the center and of the radius of the sphere is of interest in various applications such as object tracking, robotics, pattern recognition, see for instance [5], [6], [13], among others, see also [3] and references therein. Several methods have been proposed based on least squares, maximum likelihood, see [11] for a recent likelihood based algorithm, most of them modeling the noise distribution with a Gaussian distribution.…”

Robust Hypersphere Fitting from Noisy Data Using an EM Algorithm

“…Fitting a circle, a sphere or more generally a hypersphere to a noisy point cloud is a recurrent problem in many applications including object tracking [1]- [3], robotics [4]- [6] or image processing and pattern recognition [7]- [9]. This problem was recently investigated in [10] by introducing latent variables defined as affine transformations of random vectors distributed according to von Mises-Fisher distributions.…”