## The Computation of Optical Flow (1995)

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### BibTeX

@MISC{Beauchemin95thecomputation,

author = {S.S. Beauchemin and J.L. Barron},

title = {The Computation of Optical Flow},

year = {1995}

}

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### Abstract

Two-dimensional image motion is the projection of the three-dimensional motion of objects, relative to a visual sensor, onto its image plane. Sequences of time-ordered images allow the estimation of projected two-dimensional image motion as either instantaneous image velocities or discrete image displacements. These are usually called the optical flow field or the image velocity field. Provided that optical flow is a reliable approximation to two-dimensional image motion, it may then be used to recover the three-dimensional motion of the visual sensor (to within a scale factor) and the three-dimensional surface structure (shape or relative depth) through assumptions concerning the structure of the optical flow field, the three-dimensional environment and the motion of the sensor. Optical flow may also be used to perform motion detection, object segmentation, time-to-collision and focus of expansion calculations, motion compensated encoding and stereo disparity measurement. We investiga...

### Citations

3717 |
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
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Citation Context ...otion discontinuities. 2.4.1 Line Processes Other functionals, that attempt to estimate discontinuous motion, have beendeveloped. One strategy to handle occlusion involves using binary line processes =-=[51]-=- that explicitly model intensity discontinuities. Koch et al. [83] relax the imposition of a smoothness constraint at those pixels having a large spatial gradient. This prevents smoothing over discont... |

1734 | Determining optical flow - Horn, Schunck - 1981 |

1047 | Performance of optical flow techniques - Barron, Fleet, et al. - 1994 |

1015 | Laplacian Pyramid as a Compact Image Code
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Citation Context .... Their algorithm has four basic components: pyramid construction, motion estimation, image warping and coarse-to- ne re nement. A Laplacian pyramid is used to hierarchically represent the image data =-=[31]-=- and motion estimation is performed by SSD minimization with respect to a particular model of motion. Image warping uses the current parameter values to compute an optical ow eld at time t and then re... |

847 | E.Hildreth, “Theory of edge detection
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Citation Context ...es. Their approach, guided by psychophysical evidence of spatiotemporal ltering in human early visual processes, is a direct extension of the Marr and Hildreth centre-surround edge detection operator =-=[98]-=-. They compute spatiotemporal zero-crossings by convolving S(x�t)=; r 2 + 1 u 2 @ 2 @t 2 ! ! G(x�t) (2.19) with an image sequence, where the rst term is the d'Alembert operator and the second term is ... |

630 |
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Citation Context ...ned. The importance of motion in visual processing cannot be understated: approximations to image motion may be used to estimate 3-d scene properties and motion parameters from a moving visual sensor =-=[64, 91, 93, 121, 150, 149, 3, 16, 113, 66, 163, 165, 99, 52, 46, 73]-=-, to perform motion segmentation [24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45], to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated i... |

595 | Spatiotemporal Energy Models for the Perception o f M o t i o n . J o u r n a l o f t h e O p t i c a l S o c i e t y o f America, A , v o l . 2 , n o . 2 , p p
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Citation Context ...oral derivatives or local correlation surfaces and measurement integration either by regularization, correlation, or a least-squares computation. These approaches are thought to be broadly equivalent =-=[1, 2]-=- although di erences in implementation can lead to signi cant di erences in performance. Given this particular classi cation, this survey covers the optical ow techniques that do not require solving t... |

578 | Heirarchical model-based motion estimation
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Citation Context ...echnique to such a framework, Anandan [8] used a hierarchical area-based correlation method, Heeger [65] proposed a hierarchical energy-based ltering technique in a Gaussian pyramid and Bergen et al. =-=[20, 21]-=- proposed hierarchical parametric models for optical ow. 1.4 Problems and Issues Much progress has been made in optical ow computation and yet, its accurate estimation remains di cult because of numer... |

466 |
The interpetation of visual motion
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- 1979
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Citation Context ...ill-posed 3 . That is to say, only v ?, the motion component in the direction of the local gradient of the image intensity function, may be estimated. This phenomenon is known as the aperture problem =-=[151]-=- and only at image locations where there is su cient intensity structure (or Gaussian curvature) can the motion be fully estimated with the use of the optical ow constraint equation (see Figure 1.3). ... |

455 | Representing Moving Images with Layers
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- 1994
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Citation Context ...eatedly merge neighbouring regions based on similarity of their a ne parameters. Then, given two su ciently large planar regions, a motion and structure calculation can be performed. Wang and Adelson =-=[153]-=- use a clustering algorithm to group velocities into layers, each consistent with an a ne motion. Bober and Kittler [28] use a block-based Hough transform in a robust estimation framework (redescendin... |

451 |
A computational framework and an algorithm for the measurement of visual motion
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- 1989
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Citation Context ... : : : : : : : : : : : : : : : : : : : : : : 8 1.4 The hierarchical computational model. Coarse motion estimates are projected to ner levels where they serve as initial estimates subject to re nement =-=[8]-=- .: : : : 10 2.5 If at least two normal ow estimates along a contour are di erent, then the full velocity of the contour can be uniquely determined. : : : : : : : : : : : : : : : : : 20 2.6 Hierarchic... |

313 |
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Citation Context ...ned. The importance of motion in visual processing cannot be understated: approximations to image motion may be used to estimate 3-d scene properties and motion parameters from a moving visual sensor =-=[64, 91, 93, 121, 150, 149, 3, 16, 113, 66, 163, 165, 99, 52, 46, 73]-=-, to perform motion segmentation [24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45], to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated i... |

298 |
Computation of Component Image Velocity from Local Phase Information. Int
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- 1990
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Citation Context ...orm of the brightness consistency assumption. Highlights, shadows, variable illumination and surface translucency are phenomena violating the assumption and have only been studied to a limited extent =-=[102, 21, 48, 78]-=-. Occluding surfaces and independently moving objects The problem posed by occluding surfaces is currently being addressed by the research community. Occlusion is di cult to analyze, despite the fact ... |

294 |
The interpretation of a moving retinal image
- Longuet-Higgins, Prazdny
- 1980
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Citation Context ...ned. The importance of motion in visual processing cannot be understated: approximations to image motion may be used to estimate 3-d scene properties and motion parameters from a moving visual sensor =-=[64, 91, 93, 121, 150, 149, 3, 16, 113, 66, 163, 165, 99, 52, 46, 73]-=-, to perform motion segmentation [24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45], to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated i... |

238 |
Determining three-dimensional motion and structure from optical flow generated by several moving objects
- Adiv
- 1985
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Citation Context |

219 | Computing occluding and transparent motions
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Citation Context ...-layer, cooperative robust estimation framework. Jespon and Blackuse an expectation-maximization (EM) algorithm to group a wide variety of component velocities intoa xednumber of layers. Irani et al. =-=[72]-=- determine a dominant motion in an image using a least-squares approach and then group and segment the outlying motions. Their approach assumes that there is only one dominant motion and many outlying... |

216 |
Enkelmann, An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences
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- 1986
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Citation Context ...uities [120]. Black [23] also shows how line processes could be used in a robust (Kalman lter-like) framework. The non-binary inhibition of smoothness across intensity contours, was proposed by Nagel =-=[106, 107, 108, 111]-=-. This approach is based on the minimization of the functional where Z D (rI v + It) 2 + 2 tr((rv) T W (rv)))dx (2.42) 1 W = krIk 2 2 +2 0 @ Iy 2 + ;IxIy ;IxIy Ix 2 + in which the quantity (uxIy ; uyI... |

177 | Probability distributions of optical flow - Simoncelli, Adelson, et al. - 1991 |

168 | Passive navigation
- Bruss, Horn
- 1983
(Show Context)
Citation Context ...sional translation vectors, rotation matrices and surface attributes [91, 150]. Intensity Image intensities and their derivatives are sometimes used directly to obtain motion and structure parameters =-=[6, 7, 71, 112, 67, 166]-=-, thus avoiding an explicit intermediate representation of image motion such as optical ow or disparity elds. Usually, relating image motion estimates or intensity derivatives to three-dimensional mot... |

164 | Multidimensional orientation estimation with applications to texture analysis and optical flow - Bigun, Granlund, et al. - 1991 |

160 | A framework for the robust estimation of optical flow - Black, Anandan - 1993 |

154 | Layered representation for motion analysis - Wang, Adelson - 1993 |

147 | On the computation of motion from sequences of images - A review - Aggarwal, Nandhakumar - 1988 |

146 | Mixture models for optical flow computation - Jepson, Black - 1993 |

145 | Optical flow using spatiotemporal filters - Heeger - 1988 |

143 | Velocity determination in scenes containing several moving objects - Fennema, Thompson - 1979 |

139 |
Scene segmentation from visual motion using global optimization
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Citation Context ...on may be used to estimate 3-d scene properties and motion parameters from a moving visual sensor [64, 91, 93, 121, 150, 149, 3, 16, 113, 66, 163, 165, 99, 52, 46, 73], to perform motion segmentation =-=[24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45]-=-, to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated image encoding [35, 41, 101, 105, 115, 164], to compute stereo disparity [12, 39, ... |

135 | On the estimation of optical flow: Relations between different approaches and some new results - Nagel - 1987 |

129 | Disparity Analysis of Images
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Citation Context ... 126, 45], to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated image encoding [35, 41, 101, 105, 115, 164], to compute stereo disparity =-=[12, 39, 77, 85]-=-, to measure blood ow and heart-wall motion in medical imagery [123] and, recently, to measure minute amounts of growth in corn seedlings [17, 87]. 1.1 Motion and Structure Paradigms Traditionally, ap... |

124 |
Motion segmentation and qualitative dynamic scene analysis from an image sequence
- Bouthemy, Francois
- 1993
(Show Context)
Citation Context ...on may be used to estimate 3-d scene properties and motion parameters from a moving visual sensor [64, 91, 93, 121, 150, 149, 3, 16, 113, 66, 163, 165, 99, 52, 46, 73], to perform motion segmentation =-=[24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45]-=-, to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated image encoding [35, 41, 101, 105, 115, 164], to compute stereo disparity [12, 39, ... |

122 |
A three-frame algorithm for estimating twocomponent image motion
- Bergen, Burt, et al.
- 1992
(Show Context)
Citation Context ...s of this model: Hanna [61, 62] shows that the rigidity assumption can be used to overcome the aperture problem in most cases. These parametric models are presented in a uni ed hierarchical framework =-=[21]-=-. The hierarchy yields increased computational e ciency and also allows for increased accuracy and robustness via coarse-to- ne re nements and image warping. Black and Jepson [27] determine coarse opt... |

111 | Displacement vectors derived from second-order intensity variations in image sequences - Nagel - 1983 |

107 | Robust dynamic motion estimation over time
- Black, Anandan
- 1991
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Citation Context ...s within an image sequence being acquired: given two or more images, optical ow is computed only for one of the images. Recently, there has been some interest in incremental computation of optical ow =-=[23, 25, 135, 50, 36]-=-. The advantages include instantaneous access to optimal velocity estimates, accuracy improvement as the integration of optical ow over time is performed, computational e ciency gained by updating the... |

97 |
Measurement of Image Velocity
- Fleet
- 1992
(Show Context)
Citation Context ...ples have been applied to transparent motions [21, 78, 132]. Practical issues in computing optical ow were addressed in a recent study [14, 15] which analyzed nine techniques dating from 1981 to 1990 =-=[70, 106, 108, 152, 95, 48, 47, 65, 8, 134, 156]-=- for accuracy, density and reliability of measurements. To test the implementations of these algorithms, both synthetic and real data were used. The observed performance of these algorithms lead to th... |

97 | Estimating Optical Flow in Segmentated Images Using Variable-Order Parametric Models With Local Deformations - Black, Jepson - 1996 |

97 | Efficient multiscale regularization with applications to the computation of optical flow - Luettgen, Karl, et al. - 1994 |

95 | Robust computation of optical flow in a multi-scale differential framework - Weber, Malik - 1995 |

94 |
Robust estimation of a multi-layered motion representation
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- 1991
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Citation Context ...ity sensitive lters may contribute to the solving of this particular problem, as they provide multiple measurements for each location [48]. Alternatively, superposition principles and layered motions =-=[40, 132, 153]-=- are promising frameworks. However, segmenting multiple motion distributions remains di cult if no a priori assumption is made on the number of distributions present within a support region. Lastly, w... |

90 |
A computational approach to motion perception
- Uras, Girosi, et al.
- 1988
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Citation Context ...ples have been applied to transparent motions [21, 78, 132]. Practical issues in computing optical ow were addressed in a recent study [14, 15] which analyzed nine techniques dating from 1981 to 1990 =-=[70, 106, 108, 152, 95, 48, 47, 65, 8, 134, 156]-=- for accuracy, density and reliability of measurements. To test the implementations of these algorithms, both synthetic and real data were used. The observed performance of these algorithms lead to th... |

89 |
Investigations of multigrid algorithms for the estimation of optical flow fields in image sequences
- Enkelmann
- 1988
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Citation Context ...tical ow techniques in a hierarchical, coarseto- ne framework. Hierarchical frameworks allow the images to be decomposed in di erent scales of resolution in the form of Gaussian or Laplacian pyramids =-=[8, 18, 44, 53]-=-. Because ofalow frequency representation at coarser resolutions, the optical ow constraint equation becomes applicable in the case of large image motions at the nest resolution [80]. In addition to h... |

79 | Robust incremental optical flow - Black - 1992 |

78 | Optical flow estimation: an error analysis of gradient-based methods with local optimization - Kearney, Thompson, et al. |

76 |
Determining optical ow
- Horn, Schunk
- 1981
(Show Context)
Citation Context ...thesis in measuring image motion is that the intensity structures of local timevarying image regions are approximately constant under motion for at least a short durationsBeauchemin and Barron 1995 6 =-=[70]-=-. Formally, ifI(x�t) is the image intensity function, then I(x�t) I(x + x�t+ t)� (1.1) where x is the displacement of the local image region at (x�t) after time t. Expanding the left-hand side of this... |

71 | Stability of phase information - Fleet, Jepson - 1993 |

67 | Optic flow computation: a unified perspective - Singh - 1991 |

66 |
A locally adaptive window for signal matching
- Okutomi, Kanade
- 1992
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Citation Context ...owmust remain negligible as local matches operate under the hypothesis of a constant velocity model. The optimal window size then depends on the structure of the underlying signal. Okutomi and Kanade =-=[117]-=- propose a statistical model of disparity within correlation windows which assumes that disparity values are constant but exhibit increasing uncertainty as they are farther from the central point of t... |

63 | Recovery of ego-motion using image stabilization
- Irani, Rousso, et al.
- 1997
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Citation Context |

61 |
Detection of binocular disparities
- Prazdny
- 1985
(Show Context)
Citation Context ...or disappear in time, possibly leading to mismatches. Furthermore, local optical ow constraints such as (1.3) and local correlation methods are often coupled with global requirements which 12 Prazdny =-=[122]-=-was one of the rst to explicitly allow for disparity discontinuities locally in two stereo images.sBeauchemin and Barron 1995 34 impose a spatial continuity on optical ow. It is obvious that such isot... |

58 |
The sampling and reconstruction of time-varying imagery with application in video systems
- Dubois
- 1985
(Show Context)
Citation Context ...otion segmentation [24, 116, 125, 104, 138, 42, 76, 29, 9, 126, 45], to compute the focus of expansion and time-to-collision [124, 118, 141, 75, 143, 30], to perform motion-compensated image encoding =-=[35, 41, 101, 105, 115, 164]-=-, to compute stereo disparity [12, 39, 77, 85], to measure blood ow and heart-wall motion in medical imagery [123] and, recently, to measure minute amounts of growth in corn seedlings [17, 87]. 1.1 Mo... |

58 | Image flow segmentation and estimation by constraint line clustering - Schunck - 1989 |