#### Technical Report USC-SIPI-127

“A Spatio-Temporal Approach to Motion Understanding”

by Min Shao and Rama Chellappa

July 1988

Algorithms for the interpretation of optical flow are difficult to design due to the nonlinearity of constraint equations and the high dimensionality of the parameter space. Here we show that when two velocity fields from the same moving object are given, the rotational component of the motion parameters can be eliminated from the difference velocity field. Thus the translational component, or the focus of expansion (FOE) can be robustly found by solving a set of linear equations. This in turn facilitates closed-form solutions for the rotational component and environment depth. This approach can be applied to multi-object motion segmentation using the Hough transform. If a dense sequence of images is available, then the structure for the environment and the 3-D motion parameters can be recovered directly at every image point from the given velocity filed. In this approach both spatial and temporal information are used in a uniform way. The structure-from-motion (SFM) problem is then reduced to solving a quadratic equation. If the optical flow field is not available, the SFM problem based directly on the first-order derivatives of the image brightness is underdetermined. However, by exploiting the image brightness constancy constraint in both spatial and temporal domains we show that, given the first and second order spatio-temporal derivatives of image brightness, the SFM problem becomes overdetermined.

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