Algebraic Solution for the Visual Hull

Abstract

We introduce an algebraic dual-space method for reconstructing the visual hull of a three-dimensional object from occluding contours observed in 2D images. The method exploits the differential structure of the manifold rather than parallax geometry, and therefore requires no correspondences. We begin by observing that the set of 2D contour tangents determines a surface in a dual space where each point represents a tangent plane to the original surface. The primal and dual surfaces have a symmetric algebra: A point on one is orthogonal to its dual point and tangent basis on the other. Thus the primal surface can be reconstructed if the local dual tangent basis can be estimated. Typically this is impossible because the dual surface is noisy and riddled with tangent singularities due to self-crossings. We identify a directionally-indexed local tangent basis that is well-defined and estimable everywhere on the dual surface. The estimation procedure handles singularities in the dual surface and degeneracies arising from measurement noise. The resulting method has O(N) complexity for N observed contour points and gives asymptotically exact reconstructions of surfaces that are totally observable from occluding contours.