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I was reading about the Horn & Schunck method and they define the smoothness error as (Robot Control 1988 (SYROCO'88): Selected Papers from the 2nd IFAC Symposium p. 348):

$E_{sm} = \iint{\left(u_x^2+u_y^2+v_x^2+v_y^2\right)}dx\:dy$

, where $(u,v)$ is the optical flow velocity vector and the indexes are shorthand for the $x$ and $y$ derivatives.

But since $u$ is already in the $x$ direction, shouldn't $u_y$ be zero? Same for $v_x$.

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