Differential techniques for optical flow

Abstract

We show that optical flow, i.e., the apparent motion of the time-varying brightness over the image plane of an imaging device, can be estimated by means of simple differential techniques. Linear algebraic equations for the two components of optical flow at each image location are derived. The coefficients of these equations are combinations of spatial and temporal derivatives of the image brightness. The equations are suggested by an analogy with the theory of deformable bodies and are exactly true for particular classes of motion or elementary deformations. Locally, a generic optical flow can be approximated by using a constant term and a suitable combination of four elementary deformations of the time-varying image brightness, namely, a uniform expansion, a pure rotation, and two orthogonal components of shear. When two of the four equations that correspond to these deformations are satisfied, optical flow can more conveniently be computed by assuming that the spatial gradient of the image brightness is stationary. In this case, it is also possible to evaluate the difference between optical flow and motion field—that is, the two-dimensional vector field that is associated with the true displacement of points on the image plane. Experiments on sequences of real images are reported in which the obtained optical flows are used successfully for the estimate of three-dimensional motion parameters, the detection of flow discontinuities, and the segmentation of the image in different moving objects.