This paper revisits the problem of continuous-time structure from motion, and introduces a number of extensions that improve convergence and efficiency. The formulation with a @math -continuous spline for the trajectory naturally incorporates inertial measurements, as derivatives of the sought trajectory. We analyse the behaviour of split interpolation on @math and on @math , and a joint interpolation on @math , and show that the latter implicitly couples the direction of translation and rotation. Such an assumption can make good sense for a camera mounted on a robot arm, but not for hand-held or body-mounted cameras. Our experiments show that split interpolation on @math and on @math is preferable over @math interpolation in all tested cases. Finally, we investigate the problem of landmark reprojection on rolling shutter cameras, and show that the tested reprojection methods give similar quality, while their computational load varies by a factor of 2.
This paper proposes a new class of unit quaternion curves in 3 . A general method is developed that transforms a curve in 3 (defined as a weighted sum of basis functions) into its unit quaternion analogue in 3 . Applying the method to well-known spline curves (such as B´ ezier, Hermite, and B-spline curves), we are able to construct various unit quaternion curves which share many important differential properties with their original curves. Many of our naive common beliefs in geometry break down even in the simple non-Euclidean space 3 or 3 . For example, the de Casteljau type construction of cubic B-spline quaternion curves does not preserve 2 -continuity [10]. Through the use of decomposition into simple primitive quaternion curves, our quaternion curves preserve most of the algebraic and differential properties of the original spline curves.