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[[Image:Opticfloweg.png|thumb|right|400px|The optic flow experienced by a rotating observer (in this case a fly). The direction and magnitude of optic flow at each ___location is represented by the direction and length of each arrow.]]
'''Optical flow''' or '''optic flow''' is the pattern of apparent [[motion (physics)|motion]] of objects, surfaces, and edges in a visual scene caused by the [[relative motion]] between an observer and a scene.<ref>{{Cite book |url={{google books|plainurl=yes|id=CSgOAAAAQAAJ|pg=PA77|text=optical flow}} |title=Thinking in Perspective: Critical Essays in the Study of Thought Processes |last1=Burton |first1=Andrew |last2=Radford |first2=John |publisher=Routledge |year=1978 |isbn=978-0-416-85840-2}}</ref><ref>{{Cite book |url={{google books|plainurl=yes|id=-I_Hazgqx8QC|pg=PA414|text=optical flow}} |title=Electronic Spatial Sensing for the Blind: Contributions from Perception |last1=Warren |first1=David H. |last2=Strelow |first2=Edward R. |publisher=Springer |year=1985 |isbn=978-90-247-2689-9}}</ref> Optical flow can also be defined as the distribution of apparent velocities of movement of brightness pattern in an image.<ref name="Horn_1980">{{Cite journal |last1=Horn |first1=Berthold K.P. |last2=Schunck |first2=Brian G. |date=August 1981 |title=Determining optical flow |url=http://image.diku.dk/imagecanon/material/HornSchunckOptical_Flow.pdf |journal=Artificial Intelligence |language=en |volume=17 |issue=1–3 |pages=185–203 |doi=10.1016/0004-3702(81)90024-2|hdl=1721.1/6337 }}</ref>
The concept of optical flow was introduced by the American psychologist [[James J. Gibson]] in the 1940s to describe the visual stimulus provided to animals moving through the world.<ref>{{Cite book |title=The Perception of the Visual World |last=Gibson |first=J.J. |publisher=Houghton Mifflin |year=1950}}</ref> Gibson stressed the importance of optic flow for [[Affordance|affordance perception]], the ability to discern possibilities for action within the environment. Followers of Gibson and his [[Ecological Psychology|ecological approach to psychology]] have further demonstrated the role of the optical flow stimulus for the perception of movement by the observer in the world; perception of the shape, distance and movement of objects in the world; and the control of [[Animal locomotion|locomotion]].<ref>{{Cite journal |last1=Royden |first1=C. S. |last2=Moore |first2=K. D. |year=2012 |title=Use of speed cues in the detection of moving objects by moving observers |journal=Vision Research |volume=59 |pages=17–24 |doi=10.1016/j.visres.2012.02.006|pmid=22406544 |s2cid=52847487 |doi-access=free }}</ref>
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===Classical Models===
Many classical models use the intuitive assumption of ''brightness constancy''; that even if a point moves between frames, its brightness stays constant.
<ref name="Fortun_Survey_2015">{{cite journal |last1=Fortun |first1=Denis |last2=Bouthemy |first2=Patrick |last3=Kervrann |first3=Charles|title=Optical flow modeling and computation: A survey |journal=Computer Vision and Image Understanding |date=2015-05-01 |volume=134 |pages=1-21 |doi=10.1016/j.cviu.2015.02.008 |url=https://www.sciencedirect.com/science/article/pii/S1077314215000429 |access-date=2024-12-23}}</ref> To formalise this intuitive assumption, consider two consecutive frames from a video sequence, with intensity <math>I(x, y, t)</math>, where <math>(x, y)</math> refer to pixel coordinates and <math>t</math> refers to time. In this case, the brightness constancy constraint is :<math>
I(x, y, t)
</math>
where <math>\mathbf{w}:= (u, v
By itself, the brightness constancy constraint cannot be solved for <math>u</math> and <math>v</math> at each pixel, since there is only one equation and two unknowns.
This is known as the ''[[Motion perception#The aperture problem|aperture problem]]''.
Therefore, additional constraints must be imposed to estimate the flow field.<ref name="Brox_2004">{{cite conference |url=http://link.springer.com/10.1007/978-3-540-24673-2_3 |title=High Accuracy Optical Flow Estimation Based on a Theory for Warping |last1=Brox |first1=Thomas |last2=Bruhn |first2=Andrés |last3=Papenberg |first3=Nils |last4=Weickert |first4=Joachim |date=2004 |publisher=Springer Berlin Heidelberg |book-title=Computer Vision - ECCV 2004 |pages=25-36 |___location=Berlin, Heidelberg |conference=ECCV 2004}}</ref><ref name="Baker_2011">{{cite journal |last1=Baker |first1=Simon |last2=Scharstein |first2=Daniel |last3=Lewis |first3=J. P. |last4=Roth |first4=Stefan |last5=Black |first5=Michael J. |last6=Szeliski |first6=Richard |title=A Database and Evaluation Methodology for Optical Flow |journal=International Journal of Computer Vision |date=1 March 2011 |volume=92 |issue=1 |pages=1–31 |doi=10.1007/s11263-010-0390-2 |url=https://link.springer.com/article/10.1007/s11263-010-0390-2 |access-date=25 Dec 2024 |language=en |issn=1573-1405}}</ref>
==== Regularized Models ====
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)) + \Psi(|\nabla u|) + \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, and <math>\Psi()</math> is a [[loss function|loss function]].
<ref name="Fortun_Survey_2015" /> <ref name="Brox_2004" />
This optimisation problem is difficult to solve owing to its non-linearity.
To address this issue, one can use a ''variational approach'' and linearise the brightness constancy constraint using a first order [[Taylor series]] approximation. Specifically, the brightness constancy constraint is approximated as,
:<math>\frac{\partial I}{\partial x}u+\frac{\partial I}{\partial y}v+\frac{\partial I}{\partial t} = 0.</math>
For convenience, the derivatives of the image, <math>\tfrac{\partial I}{\partial x}</math>, <math>\tfrac{\partial I}{\partial y}</math> and <math>\tfrac{\partial I}{\partial t}</math> are often condensed to become <math>I_x</math>, <math>I_y</math> and <math> I_t</math>.
Doing so, allows one to rewrite the linearised brightness constancy constraint as, <ref name="Baker_2011"/>
:<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) + \Psi(|\nabla u|) + \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>
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 |last3=Weickert |first3=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>
==== Parametric Models ====
Instead of applying the regularization constraint on a point by point basis as per a regularized model...
===Learning Based Models===
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