Motion Recovery from Image Sequences: Discrete Viewpoint vs. Differential Viewpoint

The aim of this paper is to explore intrinsic geometric methods of recovering the three dimensional motion of a moving camera from a sequence of images. Generic similarities between the discrete approach and the differential approach are revealed through a parallel development of their analogous motion estimation theories. We begin with a brief review of the (discrete) essential matrix approach, showing how to recover the 3D displacement from image correspondences. The space of normalized essential matrices is characterized geometrically: the unit tangent bundle of the rotation group is a double covering of the space of normalized essential matrices. This characterization naturally explains the geometry of the possible number of 3D displacements which can be obtained from the essential matrix. Second, a differential version of the essential matrix constraint previously explored by [19,20] is presented. We then present the precise characterization of the space of differential essential matrices, which gives rise to a novel eigenvector-decomposition-based 3D velocity estimation algorithm from the optical flow measurements. This algorithm gives a unique solution to the motion estimation problem and serves as a differential counterpart of the SVD-based 3D displacement estimation algorithm from the discrete case. Finally, simulation results are presented evaluating the performance of our algorithm in terms of bias and sensitivity of the estimates with respect to the noise in optical flow measurements.