  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.
