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It should be quite doable to modify the Gunnar Farneb├Ąck algorithm implemented in OpenCV to find flows along a fixed axis. The equation that finds the displacement from the calculated polynomials can be changed to solve along any given dimension.

Please have a look at the paper for reference: http://www.diva-portal.org/smash/get/diva2:273847/FULLTEXT01.pdf

Sadly, the OpenCV implementation has virtually no comments, thus I'm not able to identify the corresponding lines in the code quickly.