On the complexity of labeling perspective projections of polyhedral scenes

Abstract

This paper investigates the computational time complexity of the labeling problem for line drawings of trihedral scenes. It is shown that the class of problems having polynomial complexity is larger than the simple case of line drawings of Legoland scenes (Kirousis and Papadimitriou, 1988). Once the location of the vanishing points in the image plane is known, the labeling problem can be solved in time O( Nn) where N is the number of segments and n is the number of vanishing points. The vanishing points can be given a priori, otherwise can, in many cases, be detected by standard techniques from the line drawing itself. The NP-completeness of the labeling problem for line drawings of trihedral scenes (Kirousis and Papadimitriou, 1988) is then due to the lack of knowledge about the vanishing points. which is equivalent to the knowledge of the possible directions for the edges. These results help draw a more accurate boundary between the problems in the interpretation of line drawings that are polynomially solvable and those that are NP-complete.