Reconstructing Polygons from Scanner Data

Abstract

A range-finding scanner can collect information about the shape of an (unknown) polygonal room in which it is placed. Suppose that a set of scanners returns not only a set of points, but also additional information, such as the normal to the plane when a scan beam detects a wall. We consider the problem of reconstructing the floor plan of a room from different types of scan data. In particular, we present algorithmic and hardness results for reconstructing two-dimensional polygons from point-wall pairs, point-normal pairs, and visibility polygons. The polygons may have restrictions on topology (e.g., to be simply connected) or geometry (e.g., to be orthogonal). We show that this reconstruction problem is NP-hard under most models, but that some restrictive assumptions do allow polynomialtime reconstruction algorithms.

Reconstructing polygons from scanner data

“…✩ Some of these results appeared in preliminary form at the International Symposium on Algorithms and Computation (Biedl et al (2009) [5]). 2.…”

“…If the leftmost data point does not lie on a vertical line, then initialize t to be the x-coordinate of the intersection of these two lines, and set f (t + ε, R, R) = true. 4 If the leftmost data point has a vertical line, then initialize t to be the x-coordinate of that data point, and initialize f (t + ε, R, R) = true if and only if the data point lies below the upper-right and above the lower-right lines on the vertical line {x = t}. All other values f (t + ε, u, ℓ) are initialized to be false.…”

Geometric reconstruction from point-normal data

“…There is little work, however, on reconstructing polyhedra from sparse point-plane or point-normal data, let alone more complex metadata. Biedl et al discuss several polygon reconstruction problems based on point-normal data and related data [12]. Their reconstruction results are limited to two dimensions.…”

Covering points with orthogonally convex polygons

“…Recently, Biedl et al [5] studied reconstructing polygons from scanner data such that the provided points are in the interior of the polygon edges. One of their results is that a monotone orthogonal polygon can be reconstructed from a point set in O (n log n) time, if we know the corresponding edge orientations.…”