Reconstructing Convex Sets from Support Line Measurements

This paper proposes algorithms for reconstructing convex sets given noisy support line measurements. We begin by observing that a set of measured support lines may not be consistent with any set in the plane. We then develop a theory of consistent support lines which serves as a basis for reconstruction algorithms that take the form of constrained optimization algorithms. The formal statement of the problem and constraints reveals a rich geometry which allows us to include prior information about object position and boundary smoothness. The algorithms, which use explicit noise models and prior knowledge, are based on ML and MAP estimation principles, and are implemented using efficient linear and quadratic programming codes. Experimental results are presented. This research sets the stage for a more general approach to the incorporation of prior information concerning and the estimation of object shape.