Lucas Kanade Optical Flow, Direction Vector

I am working on optical flow, and based on the lecture notes here and some samples on the Internet, I wrote this Python code.

All code and sample images are there as well. For small displacements of around 4-5 pixels, the direction of vector calculated seems to be fine, but the magnitude of the vector is too small (that's why I had to multiply u,v by 3 before plotting them).

Is this because of the limitation of the algorithm, or error in the code? The lecture note shared above also says that motion needs to be small "u, v are less than 1 pixel", maybe that's w开发者_如何转开发hy. What is the reason for this limitation?

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@belisarius says "LK uses a first order approximation, and so (u,v) should be ideally << 1, if not, higher order terms dominate the behavior and you are toast. ".

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A standard conclusion from the optical flow constraint equation (OFCE, slide 5 of your reference), is that "your motion should be less than a pixel, less higher order terms kill you". While technically true, you can overcome this in practice using larger averaging windows. This requires that you do sane statistics, i.e. not pure least square means, as suggested in the slides. Equally fast computations, and far superior results can be achieved by Tikhonov regularization. This necessitates setting a tuning value(the Tikhonov constant). This can be done as a global constant, or letting it be adjusted to local information in the image (such as the Shi-Tomasi confidence, aka structure tensor determinant).

Note that this does not replace the need for multi-scale approaches in order to deal with larger motions. It may extend the range a bit for what any single scale can deal with.

Implementations, visualizations and code is available in tutorial format here, albeit in Matlab not Python.