On the mathematical foundations of smoothness constraints for thedetermination of optical flow and for surface reconstruction
Snyder, M.A.
Dept. of Comput. & Inf. Sci., Massachusetts Univ., Amherst, MA;
This paper appears in: Pattern Analysis and Machine Intelligence, IEEE Transactions on
Publication Date: Nov 1991
Volume: 13,
Issue: 11
On page(s): 1105-1114
ISSN: 0162-8828
References Cited: 20
CODEN: ITPIDJ
INSPEC Accession Number: 4093253
Digital Object Identifier: 10.1109/34.103272
Current Version Published: 2002-08-06
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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