Content deleted Content added
→Parametric Models: Started the parametric model section in earnest |
Added more information about variational methods and parametric models. |
||
Line 36:
Perhaps the most natural approach to addressing the aperture problem is to apply a smoothness constraint or a ''regularization constraint'' to the flow field.
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|loss function]].
<ref name="Fortun_Survey_2015" /> <ref name="Brox_2004" />
Line 47:
:<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
For the choice of <math>\Psi(x) = x^2</math>, this method is the same as the [[Horn-Schunck method]].
<ref name="Horn_1980"/>
Line 53:
<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,
:<math> I_x(I_xu+I_yv+I_t) - \alpha \Delta u = 0,</math>
:<math> I_y(I_xu+I_yv+I_t) - \alpha \Delta v = 0,</math>
where <math>\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} </math> denotes the [[Laplace operator]].
Since the image data is made up of discrete pixels, these equations are discretised.
Doing so yields a system of linear equations which can be solved for <math>(u, v)</math> at each pixel, using an iterative scheme such as [[Gauss-Seidel]]. <ref name="Horn_1980" />
An alternate approach is to discretize the optimisation problem and then perform a search of the possible <math>(u, v)</math> values without linearising it. <ref name="Steinbrucker_2009">{{cite conference |url=https://ieeexplore.ieee.org/document/5459364 |title=Large Displacement Optical Flow Computation without Warping |last1=Steinbr¨ucker |first1=Frank |last2=Pock |first2=Thomas |last3=Cremers |first3=Daniel |last4=Weickert |first4=Joachim |date=2009 |publisher=IEEE |book-title=2009 IEEE 12th International Conference on Computer Vision |pages=1609-1614 |conference=2009 IEEE 12th International Conference on Computer Vision}}</ref>
This search is often performed using [[Max-flow min-cut theorem]] algorithms, linear programming or [[belief propagation]] methods.
==== Parametric Models ====
Line 68 ⟶ 76:
<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, <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)]^.
</math>
===Learning Based Models===
| |||