High-Order Differential Geometry of Curves for Multiview Reconstruction and Matching

The relationship between the orientation and curvature of projected curves and the orientation and curvature of the underlying space curve has been previously established. This has allowed a disambiguation of correspondences in two views and a transfer of these properties to a third view for confirmation. We propose that a higher-order intrinsic differential geometry attribute, namely, curvature derivative, is necessary to account for the range of variation of space curves and their projections. We derive relationships between curvature derivative in a projected view, and curvature derivative and torsion of the underlying space curve. Regardless of the point, tangent, and curvature, any pair of curvature derivatives are possible correspondences, but most would lead to very high torsion and curvature derivatives. We propose that the minimization of third order derivatives of the reconstruction, which combines torsion and curvature derivative of the space curve, regularizes the process of finding the correct correspondences.