Fundamentals of Nonparametric Bayesian Line Detection Anne C. van Rossum 1,2,3(B ) , Hai Xiang Lin 1,2,3 , Johan Dubbeldam 1,2,3 , and H. Jaap van den Herik 1,2,3 1 Distributed Organisms B.V., Rotterdam, The Netherlands anne@dobots.nl, {h.x.lin,j.l.a.dubbeldam}@tudelft.nl, h.j.vandenherik@law.leidenuniv.nl 2 Delft University of Technology, Delft, The Netherlands 3 Leiden University, Leiden, The Netherlands Abstract. Line detection is a fundamental problem in the world of com- puter vision. Many sophisticated methods have been proposed for per- forming inference over multiple lines; however, they are quite ad-hoc. Our fully Bayesian model extends a linear Bayesian regression model to an infinite mixture model and uses a Dirichlet Process as a prior. Gibbs sampling over non-unique parameters as well as over clusters is performed to fit lines of a fixed length, a variety of orientations, and a variable number of data points. Bayesian inference over data is optimal given a model and noise definition. Initial computer experiments show promising results with respect to clustering performance indicators such as the Rand Index, the Adjusted Rand Index, the Mirvin metric, and the Hubert metric. In future work, this mathematical foundation can be used to extend the algorithms to inference over multiple line segments and multiple volumetric objects. Keywords: Bayesian nonparametrics · Line detection 1 Introduction The task of line detection in point clouds has hitherto been performed through rather ad-hoc methods. Two of the most familiar methods have been RANSAC and Hough. The RANSAC [3] method iteratively tests a hypothesis for line parameters. A set of points is selected out of all existing points and a line is fitted through them. Points that are not in this set, but fit the same line (according to a predefined loss function), are added to this set. If the fit, considering all points, is not sufficiently adequate, this process is reiterate till a predefined performance level is obtained. The Hough transform [15] is not an iterative method, but it deterministically maps all points in the image space to curves in the so-called Hough space. The Hough space has typically two dimensions: slope and intercept. A line is characterized in Hough space by a curves intersecting at the same point. Each point defines a unique slope and intercept. By rasterizing the Hough space c  Springer International Publishing AG 2017 A. Fred et al. (Eds.): ICPRAM 2016, LNCS 10163, pp. 175–193, 2017. DOI: 10.1007/978-3-319-53375-9 10