Editing Optical flow (section)

==== Parametric Models ====

Instead of applying the regularization constraint on a point by point basis as per a regularized model, one can group pixels into regions and estimate the motion of these regions.
This is known as a ''parametric model'', since the motion of these regions is [[parameter|parameterized]].
In formulating optical flow estimation in this way, one makes the assumption that the motion field in each region be fully characterised by a set of parameters.
Therefore, the goal of a parametric model is to estimate the motion parameters that minimise a loss function which can be written as,
:<math>
\hat{\boldsymbol{\alpha}} = \arg \min_{\boldsymbol{\alpha}} \sum_{(x, y) \in \mathcal{R}} g(x, y) \rho(x, y, I_1, I_2, u_{\boldsymbol{\alpha}}, v_{\boldsymbol{\alpha}}),
</math>
where <math>{\boldsymbol{\alpha}}</math> is the set of parameters determining the motion in the region <math>\mathcal{R}</math>, <math>\rho()</math> is data cost term, <math>g()</math> is a weighting function that determines the influence of pixel <math>(x, y)</math> on the total cost, and <math>I_1</math> and <math>I_2</math> are frames 1 and 2 from a pair of consecutive frames.<ref name="Fortun_Survey_2015" />

The simplest parametric model is the [[Lucas-Kanade method]]. This uses rectangular regions and parameterises the motion as purely translational. The Lucas-Kanade method uses the original brightness constancy constrain as the data cost term and selects <math>g(x, y) = 1</math>.
This yields the local loss function,
:<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)| .
</math>
Other possible local loss functions include the negative normalized [[cross-correlation]] between the two frames.<ref>{{cite conference |last1=Lucas |first1=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>