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One can combine both of these constraints to formulate estimating optical flow as an [[Optimization problem|optimization problem]], where the goal is to minimize the cost function of the form,
:<math>E = \iint_\Omega \Psi(I(x + u, y + v, t + 1) - I(x, y, t)) + \alpha \Psi(|\nabla u|) + \alpha \Psi(|\nabla v|) dx dy, </math>
where <math>\Omega</math> is the extent of the images <math>I(x, y)</math>, <math>\nabla</math> is the gradient operator, <math>\alpha</math> is a constant, and <math>\Psi()</math> is a [[loss function]].<ref name="Fortun_Survey_2015" /><ref name="Brox_2004" />
This optimisation problem is difficult to solve owing to its non-linearity.
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:<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" />
Of course, other choices of cost function have been used such as <math>\Psi(x) = \sqrt{x^2 + \epsilon^2}</math>, which is a differentiable variant of the [[Taxicab geometry |<math>L^1</math> norm]].<ref name="Fortun_Survey_2015" /><ref>{{cite conference |url=https://ieeexplore.ieee.org/abstract/document/5539939 |title=Secrets of optical flow estimation and their principles |last1=Sun |first1=Deqing |last2=Roth |first2=Stefan |last3=Black |first3="Micahel J." |date=2010 |publisher=IEEE |book-title=2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition |pages= 2432-2439 |___location=San Francisco, CA, USA |conference=2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition}}</ref>
To solve the aforementioned optimization problem, one can use the [[Euler-Lagrange equations]] to provide a system of partial differential equations for each point in <math>I(x, y, t)</math>. In the simplest case of using <math>\Psi(x) = x^2</math>, these equations are,
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