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M.A. Snyder, "On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 13, no. 11, pp. 11051114, November, 1991.  
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@article{ 10.1109/34.103272, author = {M.A. Snyder}, title = {On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {13}, number = {11}, issn = {01628828}, year = {1991}, pages = {11051114}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.103272}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Pattern Analysis and Machine Intelligence TI  On the Mathematical Foundations of Smoothness Constraints for the Determination of Optical Flow and for Surface Reconstruction IS  11 SN  01628828 SP1105 EP1114 EPD  11051114 A1  M.A. Snyder, PY  1991 KW  computer vision; gradientbased methods; smoothness constraints; optical flow; surface reconstruction; minimization technique; greylevel image intensity function; positive definite; Cartesian coordinate system; weight matrix; computer vision; minimisation VL  13 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
Gradientbased 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 greylevel 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 bestknown 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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