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:<math>I_x u + I_y v+ I_t = 0.</math>
The optimization problem can now be rewritten as
:<math>E = \iint_\Omega \Psi(I_x u + I_y v + I_t) + \alpha \Psi(|\nabla u|) + \alpha \Psi(|\nabla v|) dx dy. </math>
For the choice of <math>\Psi(x) = x^2</math>, this method is the same as the [[Horn-Schunck method]].
<ref name="Horn_1980"/>
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This yields the local loss function, <ref>{{cite conference |last=Lucas |first=Bruce D. |last2=Kanade |first2=Takeo |date=1981-08-24 |title=An iterative image registration technique with an application to stereo vision |url=https://dl.acm.org/doi/10.5555/1623264.1623280 |journal=Proceedings of the 7th International Joint Conference on Artificial intelligence - Volume 2 |series=IJCAI'81 |___location=San Francisco, CA, USA |publisher=Morgan Kaufmann Publishers Inc. |pages=674–679}}</ref>
:<math>
\hat{\boldsymbol{\alpha}} = \arg \min_{\boldsymbol{\alpha}} \sum_{(x, y) \in \mathcal{R}} [ I(x + u_{\boldsymbol{\alpha}}, y + v_{\boldsymbol{\alpha}}, t + 1) - I(x, y, t)]^2 .
</math>
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