Efficiently estimating projective transformations

Abstract

The estimation of the parameters of a projective transformation that relates the coordinates of two image planes is a standard problem that arises in image and video mosaicking, virtual video, and problems in computer vision. This problem is often posed as a least squares minimization problem based on a finite set of noisy point samples of the underlying transformation. While in some special cases this problem can be solved using a linear approximation, in general, it results in an 8-dimensional nonquadratic minimization problem that is solved numerically using an 'off-the-shelf' procedure such as the Levenberg-Marquardt algorithm. We show that the general least squares problem for estimating a projective transformation can be analytically reduced to a 2-dimensional nonquadratic minimization problem. Moreover, we provide both analytical and experimental evidence that the minimization of this function is computationally attractive. We propose a particular algorithm that is a combination of a projection and an approximate Gauss-Newton scheme, and experimentally verify that this algorithm efficiently solves the least squares problem.