The complexity of understanding line drawings of origami scenes

Abstract

This paper deals with the interpretation of line drawings of Origami scenes (Kanade, 1980), that is scenes obtained by assembling planar panels of negligible thickness, and it addresses the computational complexity of the problem of consistently assigning suitable labels to the segments describing 3D properties as convexity, concavity and occlusion (labeling problem). The main results of the paper are the following: (a) the labeling problem for line drawings of Origami scenes is NP, as for the case of trihedral scenes; (b) the problem remains NP even if the location of the vanishing points in the image plane is given, whereas for trihedral scenes the problem was polynomially solvable; (c) in case the vanishing points are known the labeling problem can be subdivided into two subproblems, the paneling problem and the labeling-a-paneled-line-drawing problem which are both polynomially solvable (there may be, however, exponentially many legal panelings to check against labelability). The approach provides geometrical constraints which help select ‘natural’ interpretations even in the presence of occlusions and in the absence of apparent 3D-symmetries, and which may help highlight the role of geometrical regularities in the design of biologically plausible algorithms for the interpretation of line drawings.