Multiview Differential Geometry of Curves

The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available in certain applications and result in unstructured point cloud reconstructions. General image curves provide a complementary feature when keypoints are scarce, and result in 3D curve geometry, but face challenges not addressed by the usual projective geometry of points and algebraic curves. We address these challenges by laying the theoretical foundations of a framework based on the differential geometry of general curves, including stationary curves, occluding contours, and non-rigid curves, aiming at stereo correspondence, camera estimation (including calibration, pose, and multiview epipolar geometry), and 3D reconstruction given measured image curves. By gathering previous results into a cohesive theory, novel results were made possible, yielding three contributions. First, we derive the differential geometry of an image curve (tangent, curvature, curvature derivative) from that of the underlying space curve (tangent, curvature, curvature derivative, torsion). Second, we derive the differential geometry of a space curve from that of two corresponding image curves. Third, the differential motion of an image curve is derived from camera motion and the differential geometry and motion of the space curve. The availability of such a theory enables novel curve-based multiview reconstruction and camera estimation systems to augment existing point-based approaches. This theory has been used to reconstruct a “3D curve sketch”, to determine camera pose from local curve geometry, and tracking; other developments are underway.