A new approach to through-the-lens camera control

Abstract

The through-the-lens camera control technique originally proposed by M. Gleicher and A. Witkin [Comput. Graphics 26(2), 1992, 331-340], provides a powerful user interface for the control of the virtual camera in 3D computer graphics and animation. Their technique is based on locally inverting the nonlinear perspective viewing transformation. However, given tn image control points, the Jacobian matrix is derived as a quite complex 2m X 8 matrix; furthermore, the Jacobian matrix always has at least one redundant column since its rank can be 7 at most. For the overconstrained case of m greater than or equal to 4, the Lagrange equation is always singular since its 2m x 2m square matrix has rank 7 at most. All these complications result from removing the constraint q(w)(2), + q(x)(2) + q(y)(2) + q(z)(2) = 1 for unit quaternions (qw, qx, qy, qz) is an element of S-3 which represent the camera rotations. In this paper, we interpret the problem as a target tracking problem and formulate it as a constrained nonlinear inversion problem. The problem is then solved by integrating a tangent vector field defined on the configuration space of the virtual camera. The vector field is given by the least-squares solutions of the Jacobian matrix equations. The row and column weighting scheme for the Jacobian matrix provides a convenient way to control the desired least-squares solutions and the associated vector held. The Lie group structure of the unit quaternion space S-3 enables us to derive a simple 2m x 7 Jacobian matrix, which improves both the computational efficiency and numerical stability of the overall algorithm. For the overconstrained case of m greater than or equal to 4, the Jacobian matrix equation is solved (in the least-squares sense) by using an efficient projection method with O(m) time complexity. (C) 1996 Academic Press, Inc.