On the mathematical foundations of smoothness constraints for the determination of optical flow and for surface reconstruction

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Abstract
Gradient-based approaches to the computation of optical flow often use a minimization technique incorporating a smoothness constraint on the optical flow field. The author derives the most general form of such a smoothness constraint that is quadratic in first derivatives of the grey-level image intensity function based on three simple assumptions about the smoothness constraint: (1) it must be expressed in a form that is independent of the choice of Cartesian coordinate system in the image: (2) it must be positive definite; and (3) it must not couple different component of the optical flow. It is shown that there are essentially only four such constraints; any smoothness constraint satisfying (1), (2), or (3) must be a linear combination of these four, possibly multiplied by certain quantities invariant under a change in the Cartesian coordinate system. Beginning with the three assumptions mentioned above, the author mathematically demonstrates that all best-known smoothness constraints appearing in the literature are special cases of this general form, and, in particular, that the 'weight matrix' introduced by H.H. Nagel is essentially (modulo invariant quantities) the only physically plausible such constraint.

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