Determining Optical Flow for Large Motions Using Parametric Models in a Hierarchical Framework

The problem of computing optical ow (image velocity) for large image motions has been addressed by a number of researchers, including Bergen et al. 5], who proposed using a number of parametric models in a hierarchical (pyramid) scheme to compute large motions using image intensity derivatives computed from \warped" image patches at each pyramid level. We have implemented a modiied version of their algorithm for four parametric models (0 th order, i.e. assuming constant image velocity in a local neighbourhood, 1 st order, i.e. assuming at most an aane transformation of image velocity in a local neighbourhood, and 2 nd order, i.e. assuming image velocities are computed on either planar and curved environmental surfaces). While the description of Bergen et al.'s algo-rithm's is less detailed than desirable in some places, we followed it as faithfully as we could (with only minor modiications). We present two novel ideas in this paper. First, we present a quantitative analysis of the error obtained at each level in the pyramid (using Fleet's angle error measure 7]) and second, we propose four ways of thresholding on the optical ow eld at each level in the pyramid, to avoid projecting bad image velocities down the pyramid. We investigate the use of condition numbers, determinants and eigenvalues of the least squares integration matrix or of Gaussian curvature of the local (warped) image patches as thresholds. We show that as velocity is propagated down the pyramid using our thresholding scheme it becomes more accurate for controlled optical ow eld densities.