See also Moving Image Coding, Compression: Using Vector Fields, Flow Fields.

See also Motion and Video Coding: General.

*Beauchemin, S.S.[Steven S.]*,
*Barron, J.L.[John L.]*,

**The Computation of Optical-Flow**,

*Surveys(27)*, No. 3, September 1995, pp. 433-467.

DOI Link
*Survey, Optic Flow*.
BibRef
**9509**

*Gibson, J.J.*,

**The Perception of the Visual World**,

Boston:
*Houghton Mifflin*1955. ??
BibRef
**5500**
*Book*Basic perception book where optical flow is formally introduced.
BibRef

*Gibson, J.J.*,

**Optical Motion and Transformations as Stimuli for Visual Perceptions**,

*PsychR(64)*, No. 5, 1957, pp. 288-295.
BibRef
**5700**

*Gibson, J.J.*,

**What Gives Rise to the Perception of Motion?**,

*PsychR(75)*, No. 4, 1968, pp. 335-346.
BibRef
**6800**

*Limb, J.O.*, and
*Murphy, J.A.*,

**Estimating Velocity of Moving Images in Television Signals**,

*CGIP(4)*, No. 4, December 1975, pp. 311-327.

Elsevier DOI
BibRef
**7512**

And:

**Measuring the Speed of Moving Objects from Television Signals**,

*Commun(23)*, No. 4, April 1975, pp. 474-478.
Early gradient based method for computation directly from image
measurements. The basic results here are that velocity estimates
work only for single moving objects.
Essentially the subpixel interpolation of the correlation peak in
the matchpoint neighborhood.

See also Source-Receiver Encoding of Television Signals.
BibRef

*Cafforio, C.*, and
*Rocca, F.*,

**Methods for Measuring Small Displacements of Television Images**,

*IT(22)*, No. 5 September, 1976, pp. 573-579.
BibRef
**7609**

*Cafforio, C.*,
*Rocca, F.*,

**Tracking Moving Objects in Television Images**,

*SP(1)*, 1979, pp. 133-140.
BibRef
**7900**

*Cafforio, C.*,

**Remarks on the Differential Method for the Estimation of
Movement in Television Images**,

*SP(4)*, 1982, pp. 45-52.
BibRef
**8200**

*Cafforio, C.*, and
*Rocca, F.*,

**The Differential Method for Image Motion Estimation**,

*ISPDSA83*(104-124).
BibRef
**8300**

*Horn, B.K.P.*, and
*Schunck, B.G.*,

**Determining Optical Flow**,

*AI(17)*, No. 1-3, August 1981, pp. 185-203.

Elsevier DOI
BibRef
**8108**

Earlier:
*DARPA81*(144-156).
BibRef

And:
*MIT AI Memo*-572, April 1980.

WWW Link.
*Optical Flow*.
The standard reference for original optical flow equation
computations. The exact formulations are not quite right since
they only work in special cases.
BibRef

*Schunck, B.G.*,
*Horn, B.K.P.*,

**Constraints on Optical Flow Computation**,

*PRIP81*(205-210).
BibRef
**8100**

*Horn, B.K.P.[Berthold K.P.]*,
*Schunck, B.G.*,

**Determining Optical Flow: A Retrospective**,

*AI(59)*, No. 1-2, January 1993, pp. 81-87.

Elsevier DOI Original paper important because it started the variational approach to optical
flow and other vision problems.
BibRef
**9301**

*Willick, D.[Darryl]*,
*Yang, Y.H.[Yee-Hong]*,

**Experimental Evaluation of Motion Constraint Equations**,

*CVGIP(54)*, No. 2, September 1991, pp. 206-214.

Elsevier DOI Evaluate

See also Determining Optical Flow.

See also Motion Constraint Equation for Optical Flow, The. and

See also On a Constraint Equation for the Estimation of Displacement Rates in Image Sequences. constraint equations and conclude that
the original was the best for optical flow.
BibRef
**9109**

*Longuet-Higgins, H.C.*, and
*Prazdny, K.*,

**The Interpretation of a Moving Retinal Image**,

*RoyalP(B-208)*, 1980, pp. 385-397.
*Optical Flow*. An early formulation of the flow pattern with rotation and translation.

See also Multiple Interpretations of a Pair of Images of a Surface.
BibRef
**8000**

*Ullman, S.*,

**The Interpretation of Visual Motion**,

Cambridge:
*MIT Press*1979.
BibRef
**7900**
*Ph.D.*Thesis (EE). His thesis as a
BibRef
*Book*
*Relaxation*.
A network of points are generated for each image with a relaxation
based matching scheme applied to find the 1-1
mapping between the views.

See also Interpretation of Structure from Motion, The.
BibRef

*Ullman, S.*,

**The Optical Flow of Planar Surfaces**,

*SV(1)*, 1986, pp. 263-276.
BibRef
**8600**

And:
*MIT AI Memo*-870, December 1985.
BibRef

*Ullman, S.*,

**Against Direct Perception**,

*MIT AI Memo*-574, March 1980.
BibRef
**8003**

*Hildreth, E.C.[Ellen C.]*,

**Computations Underlying the Measurement of Visual Motion**,

*AI(23)*, No. 3, August 1984, pp. 309-354.

Elsevier DOI
BibRef
**8408**

And:
*IU87*99-146).
BibRef

And:
*MIT AI Memo*-761, March 1984.
BibRef

Earlier:

**The Measurement of Visual Motion**,

Cambridge:
*MIT Press*1983.
BibRef
*Book*
BibRef

And:
Add A2:
*Ullman, S.[Shimon]*,
*MIT AI Memo*-699, December 1982.
BibRef

*Hildreth, E.C.*,

**The Computation of the Velocity Field**,

*RoyalP(B-221)*, 1984, pp. 189-220.
BibRef
**8400**

And:
*MIT AI Memo*-734, September 1983.

See also Computing the Velocity Field along Contours.
BibRef

*Hildreth, E.C.[Ellen C.]*,
*Koch, C.[Christof]*,

**The Analysis of Visual Motion:
From Computational Theory to Neuronal Mechanisms**,

*MIT AI Memo*-919, December 1986.

WWW Link.
BibRef
**8612**

*Mitiche, A.*, and
*Aggarwal, J.K.*,

**A Computational Analysis of Time-Varying Images**,

*HPRIP86*(311-332).
*Survey, Motion*.
*Motion, Survey*.
BibRef
**8600**

*Jacobson, L.[Lowell]*,
*Wechsler, H.[Harry]*,

**Derivation of Optical Flow Using a Spatiotemporal-Frequency Approach**,

*CVGIP(38)*, No. 1, April 1987, pp. 29-65.

Elsevier DOI
*Survey, Motion*.
*Motion, Survey*. The approach includes Hildreth and Schunck. The paper has a nice
survey of techniques and a lot of equations.
There may be something here if you want optical flow.
BibRef
**8704**

*Jacobson, L.[Lowell]*,
*Wechsler, H.[Harry]*,

**A Theory for Invariant Object Recognition in the Frontoparallel Plane**,

*PAMI(6)*, No. 3, May 1984, pp. 325-331.
BibRef
**8405**

And:

**A Paradigm for Invariant Object Recognition of Brightness, Optical Flow and
Binocular Disparity Images**,

*PRL(1)*, No. 1, October 1982, pp. 61-68.
BibRef

*Nagel, H.H.[Hans-Hellmut]*,

**On the Estimation of Optical Flow:
Relations between Different Approaches and Some New Results**,

*AI(33)*, No. 3, November 1987, pp. 299-324.

Elsevier DOI
*Optical Flow*.
A unifying approach to optical flow.
Nagel (

See also Displacement Vectors Derived from Second-Order Intensity Variations in Image Sequences. ),
Haralick-Lee (

See also Facet Approach to Optic Flow, The. ),
Tretiak-Pastor (

See also Velocity Estimation from Image Sequences with Second Order Differential Operators. ),
Hildreth (

See also Computations Underlying the Measurement of Visual Motion. ).
BibRef
**8711**

*Werkhoveh, P.*,
*Toet, A.*, and
*Koenderink, J.J.*,

**Displacement Estimates Through Adaptive Affinities**,

*PAMI(12)*, No. 7, July 1990, pp. 658-663.

IEEE DOI Replace iterative approach of

See also On the Estimation of Optical Flow: Relations between Different Approaches and Some New Results. with a noniterative scheme.
BibRef
**9007**

*Horn, B.K.P.*,

**Motion Fields Are Hardly Ever Ambiguous**,

*IJCV(1)*, No. 3, October 1987, pp. 239-258.

Springer DOI The cases where a flow field can be ambiguous are difficult to
construct and thus are not a major concern for the solution.
BibRef
**8710**

*Negahdaripour, S.*,

**Critical Surface Pairs and Triplets**,

*IJCV(3)*, No. 4, November 1989, pp. 293-312.

Springer DOI Where can the field have multiple interpretations. At most, for a
curved surface, it is three interpretations.
BibRef
**8911**

*Negahdaripour, S.[Shahriar]*,

**Multiple Interpretations of the Shape and
Motion of Objects from Two Perspective Images**,

*PAMI(12)*, No. 11, November 1990, pp. 1025-1039.

IEEE DOI
BibRef
**9011**

Earlier:

**Ambiguities of a Motion Field**,

*ICCV87*(607-611).
BibRef

And:
*MIT AI Memo*-940, January 1987.
Cases with ambiguous perspective motion fields are
limited with know flow for all points on the surface.
BibRef

*Singh, A.*, and
*Allen, P.K.*,

**Image-Flow Computation: An Estimation-Theoretic Framework
and a Unified Perspective**,

*CVGIP(56)*, No. 2, September 1992, pp. 152-177.

Elsevier DOI Two categories: conservation information and neighborhood information.
BibRef
**9209**

*Singh, A.*,

**Incremental Estimation of Image-Flow Using a Kalman Filter**,

*JVCIR(3)*, 1992, pp. 39-57.
BibRef
**9200**

Earlier:
*Motion91*(36-43).
BibRef

*Singh, A.*,

**An Estimation-Theoretic Framework for Image-Flow Computation**,

*ICCV90*(168-177).

IEEE DOI
BibRef
**9000**

And:
*DARPA90*(314-328).
*Kalman Filter*. Generate the depths from a spatio-temporal sequence.
BibRef

*Singh, A.*,

**Image-Flow Computation:
An Estimation-Theoretic Framework, Unification and Integration**,

*MVA(4)*, 1991, pp. 55.
BibRef
**9100**

*Singh, A.*,

**Robust Computation of Image-Motion and Scene Depth**,

*CRA91*(2730-2737).
BibRef
**9100**

*Singh, A.*,

**Optic Flow Computation: A Unified Perspective**,

*IEEE_Press*1990.
BibRef
**9000**

And:
From the thesis:

**Image-Flow Computation: Estimation-Theoretic Framework,
Unification and Integration**,

*Ph.D.*Thesis (CS), Columbia, Univ., May 1990.
BibRef

*Ioka, M.*,
*Kurokawa, M.*,

**Estimation Of Motion Vectors And Their Application To Scene Retrieval**,

*MVA(7)*, No. 3, 1994, pp. 199-208.
BibRef
**9400**

*Fitzpatrick, J.M.[J. Michael]*,

**The Existence of Geometrical Density-Image Transformations
Corresponding to Object Motion**,

*CVGIP(44)*, No. 2, November 1988, pp. 155-174.

Elsevier DOI Geometrical image
transformation is identical to change in density image produced by motion of
the object. (Primarily medical imagery.)
BibRef
**8811**

*Fitzpatrick, J.M.[J. Michael]*,

**A Method for Calculating Velocity in Time Dependent Images
Based on the Continuity Equation**,

*CVPR85*(78-81). (Vanderbilt Univ.)
CT or X-ray data is preferred, more equations than results.
BibRef
**8500**

*Sozou, P.D.*, and
*Loizou, G.*,

**New Perspectives on Optical-Flow**,

*PR(26)*, No. 8, August 1993, pp. 1125-1136.

Elsevier DOI Non-uniform medium. Refractive index changes the computation.
BibRef
**9308**

*Arnspang, J.[Jens]*,

**Motion Constraint Equations Based on Constant Image Irradiance**,

*IVC(11)*, No. 9, November 1993, pp. 577-587.

Elsevier DOI
BibRef
**9311**

*Malladi, R.*,
*Sethian, J.A.*,

**Image-Processing: Flows under Min/Max Curvature and Mean-Curvature**,

*GMIP(58)*, No. 2, March 1996, pp. 127-141.
*Level Set Methods*.
BibRef
**9603**

*Malladi, R.*,
*Sethian, J.A.*,

**Flows under Min/Max Curvature Flow and Mean Curvature:
Applications in Image Processing**,

*ECCV96*(I:251-262).

Springer DOI Image enhancement, noise suppression.
BibRef
**9600**

*Heikkonen, J.*,

**A Computer Vision Approach to Air-Flow Analysis**,

*PRL(17)*, No. 4, April 4 1996, pp. 369-385.
**9605**

BibRef

*Ma, J.*,
*Lu, X.*,
*Wu, C.*,

**A Motion Constraint Equation under Space-Varying or
Time-Varying Illumination**,

*PRL(5)*, 1987, pp. 203-205.
BibRef
**8700**

*Zanker, J.M.*,

**Second-Order Motion Perception in the Peripheral Visual-Field**,

*JOSA-A(14)*, No. 7, July 1997, pp. 1385-1392.
**9708**

BibRef

*Brodský, T.*,
*Fermüller, C.*,
*Aloimonos, Y.*,

**Directions of Motion Fields Are Hardly Ever Ambiguous**,

*IJCV(26)*, No. 1, January 1998, pp. 5-24.

DOI Link
**9804**

BibRef

Earlier:
*ECCV96*(II:119-128).

Springer DOI
BibRef

And:
*UMD*TR3501, 1995.

WWW Link.
BibRef

*Gros, B.L.*,
*Blake, R.*,
*Hiris, E.*,

**Anisotropies in Visual Motion Perception: A Fresh Look**,

*JOSA-A(15)*, No. 8, August 1998, pp. 2003-2011.
**9808**

BibRef

*Åström, K.[Kalle]*,
*Heyden, A.[Anders]*,

**Continuous Time Matching Constraints for Image Streams**,

*IJCV(28)*, No. 1, June 1998, pp. 85-96.

DOI Link
**9807**

Multilinear constraints for optical flow.

See also Simplifications of Multilinear Forms for Sequences of Images.
BibRef

*Mitiche, A.*,
*Mansouri, A.R.*,

**On convergence of the Horn and Schunck optical-flow estimation method**,

*IP(13)*, No. 6, June 2004, pp. 848-852.

IEEE DOI
**0406**

See also Determining Optical Flow. Analyze the equations to prove convergence via both the
Jacobi and the Gauss-Seidel methods.
BibRef

*Bayerl, P.[Pierre]*,
*Neumann, H.[Heiko]*,

**Disambiguating Visual Motion by Form-Motion Interaction:
A Computational Model**,

*IJCV(72)*, No. 1, April 2007, pp. 27-45.

Springer DOI
**0001**

BibRef

Earlier:

**Neural Mechanisms of Visual Flow Integration and Segregation:
Insights from the Pinna-Brelstaff Illusion and Variations of It**,

*BMCV02*(301 ff.).

Springer DOI
**0303**

Computational model of neural mechanisms for visual flow.
BibRef

*Beck, C.[Cornelia]*,
*Gottbehuet, T.[Thomas]*,
*Neumann, H.[Heiko]*,

**Integration of Multiple Temporal and Spatial Scales for Robust Optic
Flow Estimation in a Biologically Inspired Algorithm**,

*CAIP07*(53-60).

Springer DOI
**0708**

BibRef

*Beck, C.[Cornelia]*,
*Bayerl, P.[Pierre]*,
*Neumann, H.[Heiko]*,

**Optic Flow Integration at Multiple Spatial Frequencies:
Neural Mechanism and Algorithm**,

*ISVC06*(I: 741-750).

Springer DOI
**0611**

BibRef

*Bayerl, P.[Pierre]*,
*Neumann, H.[Heiko]*,

**A Fast Biologically Inspired Algorithm for Recurrent Motion Estimation**,

*PAMI(29)*, No. 2, February 2007, pp. 246-260.

IEEE DOI
**0701**

Sparse coding framework to implement the method.
BibRef

*Meinhardt-Llopis, E.[Enric]*,
*Sánchez Pérez, J.[Javier]*,
*Kondermann, D.[Daniel]*,

**Horn-Schunck Optical Flow with a Multi-Scale Strategy**,

*IPOL(2012)*, No. 2012, pp. xx-yy.

DOI Link
**1309**

*Code, Optical Flow*.

See also Determining Optical Flow.
BibRef

*Le Tarnec, L.*,
*Destrempes, F.*,
*Cloutier, G.*,
*Garcia, D.*,

**A Proof of Convergence of the Horn-Schunck Optical Flow Algorithm in
Arbitrary Dimension**,

*SIIMS(7)*, No. 1, 2014, pp. 277-293.

DOI Link
**1404**

See also Determining Optical Flow.
BibRef

*Fortun, D.[Denis]*,
*Bouthemy, P.[Patrick]*,
*Kervrann, C.[Charles]*,

**Optical flow modeling and computation: A survey**,

*CVIU(134)*, No. 1, 2015, pp. 1-21.

Elsevier DOI
**1504**

*Survey, Optical Flow*. Optical flow
BibRef

*Fortun, D.[Denis]*,
*Bouthemy, P.[Patrick]*,
*Kervrann, C.[Charles]*,

**A Variational Aggregation Framework for Patch-Based Optical Flow
Estimation**,

*JMIV(56)*, No. 2, October 2016, pp. 280-299.

Springer DOI
**1609**

BibRef

Earlier:

**Sparse Aggregation Framework for Optical Flow Estimation**,

*SSVM15*(323-334).

Springer DOI
**1506**

BibRef

*Zhu, B.[Bin]*,
*Tian, L.F.[Lian-Fang]*,
*Du, Q.L.[Qi-Liang]*,
*Wu, Q.X.[Qiu-Xia]*,
*Sahl, F.Z.[Farisi Zeyad]*,
*Yeboah, Y.[Yao]*,

**Adaptive dual fractional-order variational optical flow model for
motion estimation**,

*IET-CV(13)*, No. 3, April 2019, pp. 277-284.

DOI Link
**1904**

BibRef

*Bao, W.*,
*Zhang, X.*,
*Chen, L.*,
*Gao, Z.*,

**KalmanFlow 2.0: Efficient Video Optical Flow Estimation via
Context-Aware Kalman Filtering**,

*IP(28)*, No. 9, Sep. 2019, pp. 4233-4246.

IEEE DOI
**1908**

BibRef

Earlier:

**KalmanFlow: Efficient Kalman Filtering for Video Optical Flow**,

*ICIP18*(3343-3347)

IEEE DOI
**1809**

image sequences, Kalman filters, motion estimation,
video signal processing, KalmanFlow 2.0,
convolutional neural networks.
Estimation, Coherence, Optical imaging,
Noise measurement, Adaptive optics, Optical filters,
time-variant system
BibRef

Springer DOI

BibRef

*Güney, F.[Fatma]*,
*Sevilla-Lara, L.[Laura]*,
*Sun, D.[Deqing]*,
*Wulff, J.[Jonas]*,

**'What Is Optical Flow For?': Workshop Results and Summary**,

*OpticalFlow18*(VI:731-739).

Springer DOI
**1905**

BibRef

*Zikic, D.[Darko]*,
*Kamen, A.[Ali]*,
*Navab, N.[Nassir]*,

**Revisiting Horn and Schunck: Interpretation as Gauss-newton
Optimisation**,

*BMVC10*(xx-yy).

HTML Version.
**1009**

See also Determining Optical Flow.
BibRef

*Govindu, V.M.[Venu Madhav]*,

**Revisiting the Brightness Constraint:
Probabilistic Formulation and Algorithms**,

*ECCV06*(III: 177-188).

Springer DOI
**0608**

BibRef

*Giaccone, P.R.*,
*Jones, G.A.*,

**Spatio-Temporal Approaches to Computation of Optical Flow**,

*BMVC97*(xx-yy).

HTML Version.
**0209**

BibRef

*Randriantsoa, A.*,
*Berthoumieu, Y.*,

**Optical Flow Estimation Using Forward-backward Constraint Equation**,

*ICIP00*(Vol II: 578-581).

IEEE DOI
**0008**

BibRef

*Iu, S.L.[Siu-Leong]*,
*Lin, Y.T.[Yun-Ting]*,

**Re-examining the Optical Flow Constraint:
A New Optical Flow Algorithm with Outlier rejection**,

*ICIP99*(III:727-731).

IEEE Abstract.
BibRef
**9900**

*Moons, T.*,
*Pauwels, E.J.*,
*Van Gool, L.J.*, and
*Oosterlinck, A.*,

**Towards a General Framework for Feature Extraction**,

*CVPR92*(865-868).

IEEE DOI Merge optical flow and recognition.
BibRef
**9200**

Chapter on Optical Flow Field Computations and Use continues in

Optical Flow Field Computation and Analysis .

Last update:Jan 17, 2021 at 16:22:28