#### DMCA

## Discontinuity-preserving computation of variational optic flow in real-time (2005)

### Cached

### Download Links

- [www.cvgpr.uni-mannheim.de]
- [www.mia.uni-saarland.de]
- DBLP

### Other Repositories/Bibliography

Venue: | Scale-Space and PDE Methods in Computer Vision, volume 3459 of Lecture Notes in Computer Science |

Citations: | 9 - 2 self |

### Citations

2902 |
Robust Statistics,
- Huber
- 1981
(Show Context)
Citation Context ...viations from the smoothness assumption with the L1 norm of the flow gradient magnitude. This corresponds to total variation regularisation [23] and can be related to statistically robust error norms =-=[17]-=-. Thereby large deviations are penalised less severely than in the frequently used quadratic setting (L2 norm). As a consequence, large gradient features such as edges are better preserved. The energy... |

2367 | Determining optical flow,
- Horn, Schunck
- 1981
(Show Context)
Citation Context ...of strategies to regularise the non-unique solution of this data term: Homogeneous regularisation that assumes overall smoothness and does not adapt to semantically important image or flow structures =-=[16]-=-, image-driven regularisation that assumes piecewise smoothness and respects discontinuities in the image data [1, 21] and flow-driven regularisation that assumes piecewise smoothness and respects dis... |

2232 | Nonlinear total variation based noise removal algorithms
- Rudin, Osher, et al.
- 1992
(Show Context)
Citation Context ...tic flow techniques is an isotropic method that penalises deviations from the smoothness assumption with the L1 norm of the flow gradient magnitude. This corresponds to total variation regularisation =-=[23]-=- and can be related to statistically robust error norms [17]. Thereby large deviations are penalised less severely than in the frequently used quadratic setting (L2 norm). As a consequence, large grad... |

1315 | Performance of optical flow techniques.
- Barron, Fleet, et al.
- 1994
(Show Context)
Citation Context ...ong the discontinuity but not across it. Although recent developments [7, 20] have shown that variational methods are among the best techniques for computing the optic flow in terms of error measures =-=[3]-=-, they are often considered to be too slow for real-time applications. In particular the computational costs for solving the resulting linear and nonlinear system of equations by standard iterative so... |

1002 |
Multi-grid methods and applications
- Hackbusch
- 1985
(Show Context)
Citation Context ...tions by using standard iterative solvers are regarded as too high. In [8] we have already demonstrated for variational methods with homogeneous regularisation that bidirectional multigrid strategies =-=[5, 6, 19, 31, 35]-=- do allow for real-time performance. These techniques that create a sophisticated hierarchy of equation systems with excellent error reduction properties belong to the fastest numerical schemes for so... |

752 |
Iterative Solution of Nonlinear Equations
- Ortega, Rheinboldt
- 1970
(Show Context)
Citation Context ...igrid 4.1 Basic Concept In general, the obtained linear and nonlinear systems of equations are solved by using non-hierarchical iterative schemes; e.g. variants of the Jacobi or the GaußSeidel method =-=[22, 30]-=-. However, such techniques are not suitable for equation systems that are only coupled via a small local neighbourhood: It may take thousands of iterations to transport local information between unkno... |

677 |
A multigrid tutorial
- Briggs, Henson, et al.
- 2000
(Show Context)
Citation Context ...f equations by standard iterative solvers are regarded as too high. In [8] we have already demonstrated for variational methods with homogeneous regularisation that bidirectional multigrid strategies =-=[5, 6, 29]-=- do allow for real-time performance. These techniques, that create a sophisticated hierarchy of equation systems, belong to the fastest numerical schemes for solving linear or nonlinear systems of equ... |

525 |
A computational framework and an algorithm for the measurement of visual motion.
- Anandan
- 1989
(Show Context)
Citation Context ... summary in Section 6 concludes this paper.sRelated Work. In the literature on variational optic flow methods, coarseto-fine strategies are quite common to speed up the computation. (see e.g. Anandan =-=[2]-=-, Luettgen et al. [19]). They are based on a successive refinement of the problem whereby coarse grid solutions serve as initial guesses on finer grids. However, from a numerical viewpoint such unidir... |

521 |
Multi-level adaptive solutions to boundary-value problems
- Brandt
- 1977
(Show Context)
Citation Context ...f equations by standard iterative solvers are regarded as too high. In [8] we have already demonstrated for variational methods with homogeneous regularisation that bidirectional multigrid strategies =-=[5, 6, 29]-=- do allow for real-time performance. These techniques, that create a sophisticated hierarchy of equation systems, belong to the fastest numerical schemes for solving linear or nonlinear systems of equ... |

505 | High accuracy optical flow estimation based on a theory for warping, in:
- Brox, Bruhn, et al.
- 2004
(Show Context)
Citation Context ...oothness terms. While isotropic regularisers do not impose any smoothness at discontinuities, anisotropic ones permit smoothing along the discontinuity but not across it. Although recent developments =-=[7, 20]-=- have shown that variational methods are among the best techniques for computing the optic flow in terms of error measures [3], they are often considered to be too slow for real-time applications. In ... |

470 |
An introduction to multigrid methods
- Wesseling
- 1992
(Show Context)
Citation Context ...f equations by standard iterative solvers are regarded as too high. In [8] we have already demonstrated for variational methods with homogeneous regularisation that bidirectional multigrid strategies =-=[5, 6, 29]-=- do allow for real-time performance. These techniques, that create a sophisticated hierarchy of equation systems, belong to the fastest numerical schemes for solving linear or nonlinear systems of equ... |

444 |
Iterative Solutions of Large Linear Systems
- Young
- 1971
(Show Context)
Citation Context ...igrid 4.1 Basic Concept In general, the obtained linear and nonlinear systems of equations are solved by using non-hierarchical iterative schemes; e.g. variants of the Jacobi or the GaußSeidel method =-=[22, 30]-=-. However, such techniques are not suitable for equation systems that are only coupled via a small local neighbourhood: It may take thousands of iterations to transport local information between unkno... |

274 |
An investigation of smoothness constraints for the estimation of displacement vector fields from image sequences
- Nagel, Enkelmann
- 1986
(Show Context)
Citation Context ...all smoothness and does not adapt to semantically important image or flow structures [16], image-driven regularisation that assumes piecewise smoothness and respects discontinuities in the image data =-=[1, 21]-=- and flow-driven regularisation that assumes piecewise smoothness and respects discontinuities in the flow field; see e.g. [9, 24, 27]. Moreover, when considering image and flow-driven regularisation,... |

224 | Lucas/Kanade Meets Horn/Schunck: Combining Local and Global Optic Flow Methods
- Bruhn, Weickert, et al.
(Show Context)
Citation Context ...oothness terms. While isotropic regularisers do not impose any smoothness at discontinuities, anisotropic ones permit smoothing along the discontinuity but not across it. Although recent developments =-=[7, 9, 25]-=- have shown that variational methods are among the best techniques for computing the optic flow in terms of error measures [3], they are often considered to be too slow for real-time applications. In ... |

106 |
Investigations of multigrid algorithms for the estimation of optical flow fields
- Enkelmann
- 1988
(Show Context)
Citation Context ...le there is at least some literature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [15], Terzopoulos [25], Enkelmann =-=[12]-=-, Ghosal and Vaněk [14]), only the work of Borzi et al. [4] is known to the authors where nonlinear bidirectional multigrid schemes (FAS) were used as solver for an optic flow problem that was motivat... |

105 | Efficient multiscale regularization with applications to the computation of optical flow
- Luettgen, Karl, et al.
- 1994
(Show Context)
Citation Context ... concludes this paper.sRelated Work. In the literature on variational optic flow methods, coarseto-fine strategies are quite common to speed up the computation. (see e.g. Anandan [2], Luettgen et al. =-=[19]-=-). They are based on a successive refinement of the problem whereby coarse grid solutions serve as initial guesses on finer grids. However, from a numerical viewpoint such unidirectional schemes are n... |

102 |
Variational optic flow computation with a spatio-temporal smoothness constraint
- Weickert, Schnörr
- 2001
(Show Context)
Citation Context ... by � E(u, v) = Ω � (fxu + fyv + ft) 2 + α � |∇u| 2 + |∇v| 2 + ɛ 2 � dxdy, (7) where ɛ serves as small regularisation parameter. Related functionals that approximate TV regularisation are proposed in =-=[28]-=-, while variational approaches for rotational invariant TV regularisation have been researched in [9, 10, 18]. At first glance, the corresponding Euler-Lagrange equations that are given by 0 = f 2 x u... |

99 | A theoretical framework for convex regularizers in PDE-based computation of image motion
- Weickert, Schnörr
(Show Context)
Citation Context ...be expressed in terms of the optic flow constraint fxu + fyv + ft = 0, (1) where (u, v) ⊤ (x, y, t) is the displacement field we search for and subscripts denote partial derivatives. As classified in =-=[27]-=-, there are basically three different types of strategies to regularise the non-unique solution of this data term: Homogeneous regularisation that assumes overall smoothness and does not adapt to sema... |

40 | A multigrid approach for hierarchical motion estimation
- Mémin, Pérez
- 1998
(Show Context)
Citation Context ...oothness terms. While isotropic regularisers do not impose any smoothness at discontinuities, anisotropic ones permit smoothing along the discontinuity but not across it. Although recent developments =-=[7, 20]-=- have shown that variational methods are among the best techniques for computing the optic flow in terms of error measures [3], they are often considered to be too slow for real-time applications. In ... |

37 |
Calculus of variations
- Elsgolc
- 1962
(Show Context)
Citation Context ...2 fy + ɛ2 � � � −fxfy a b =: . (3) + ɛ2 b c −fxfy f 2 x In this context ɛ serves as regularisation parameter that prevents the matrix D(∇f) from getting singular. Following the calculus of variations =-=[11]-=-, the minimisation of this convex functional comes down to solving its Euler–Lagrange equations that are given by 0 = f 2 x u + fxfy v + fxft − α LNEu, (4) 0 = fxfy u + f 2 y v + fyft − α LNEv (5)swit... |

35 |
Segmentation of visual motion by minimizing convex non-quadratic functionals, in:
- Schnörr
- 1994
(Show Context)
Citation Context ...s piecewise smoothness and respects discontinuities in the image data [1, 21] and flow-driven regularisation that assumes piecewise smoothness and respects discontinuities in the flow field; see e.g. =-=[9, 24, 27]-=-. Moreover, when considering image and flow-driven regularisation, one can distinguish between isotropic and anisotropic smoothness terms. While isotropic regularisers do not impose any smoothness at ... |

34 |
Scalable algorithm for discontinuous optical flow estimation
- Ghosal, Vaněk
- 1996
(Show Context)
Citation Context ...me literature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [15], Terzopoulos [25], Enkelmann [12], Ghosal and Vaněk =-=[14]-=-), only the work of Borzi et al. [4] is known to the authors where nonlinear bidirectional multigrid schemes (FAS) were used as solver for an optic flow problem that was motivated from control theory.... |

33 | Nonlinear variational method for optical flow computation
- Cohen
- 1993
(Show Context)
Citation Context ...s piecewise smoothness and respects discontinuities in the image data [1, 21] and flow-driven regularisation that assumes piecewise smoothness and respects discontinuities in the flow field; see e.g. =-=[9, 24, 27]-=-. Moreover, when considering image and flow-driven regularisation, one can distinguish between isotropic and anisotropic smoothness terms. While isotropic regularisers do not impose any smoothness at ... |

33 | Optical–flow estimation while preserving its discontinuities: A variational approach
- Deriche, Kornobst, et al.
- 1995
(Show Context)
Citation Context ... regularisation parameter. Related functionals that approximate TV regularisation are proposed in [28], while variational approaches for rotational invariant TV regularisation have been researched in =-=[9, 10, 18]-=-. At first glance, the corresponding Euler-Lagrange equations that are given by 0 = f 2 x u + fxfy v + fxft − α 2 LTV(u, v), (8) 0 = fxfy u + f 2 y v + fyft − α 2 LTV(v, u) (9) have a very similar str... |

29 | An algebraic multigrid approach for image analysis
- Kimmel, Yavneh
(Show Context)
Citation Context ...cheme (FAS). Also for other tasks in image processing and computer vision multigrid methods have been used successfully. In the context of photometric stereo and image biniarisation Kimmel and Yavneh =-=[22]-=- developed an algebraic multigrid method, while Chan et al. [10] researched geometric multigrid schemes for variational deconvolution with TV regularisation. For TV denoising Vogel [32] proposed the u... |

28 |
Multilevel relaxation in low-level computer vision, in Multi-resolution image processing and Analysis
- Glazer
- 1984
(Show Context)
Citation Context ...btain useful correction steps. While there is at least some literature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer =-=[15]-=-, Terzopoulos [25], Enkelmann [12], Ghosal and Vaněk [14]), only the work of Borzi et al. [4] is known to the authors where nonlinear bidirectional multigrid schemes (FAS) were used as solver for an o... |

27 |
A PDE model for computing the optical flow, in:
- Alvarez, Esclarin, et al.
- 1999
(Show Context)
Citation Context ...all smoothness and does not adapt to semantically important image or flow structures [16], image-driven regularisation that assumes piecewise smoothness and respects discontinuities in the image data =-=[1, 21]-=- and flow-driven regularisation that assumes piecewise smoothness and respects discontinuities in the flow field; see e.g. [9, 24, 27]. Moreover, when considering image and flow-driven regularisation,... |

25 | Optical flow: a curve evolution approach
- Kumar, Tannenbaum, et al.
- 1996
(Show Context)
Citation Context ... regularisation parameter. Related functionals that approximate TV regularisation are proposed in [28], while variational approaches for rotational invariant TV regularisation have been researched in =-=[9, 10, 18]-=-. At first glance, the corresponding Euler-Lagrange equations that are given by 0 = f 2 x u + fxfy v + fxft − α 2 LTV(u, v), (8) 0 = fxfy u + f 2 y v + fyft − α 2 LTV(v, u) (9) have a very similar str... |

22 | Multigrid for differential-convolution problems arising from image processing
- Chan, Chan, et al.
- 1997
(Show Context)
Citation Context ...er vision multigrid methods have been used successfully. In the context of photometric stereo and image biniarisation Kimmel and Yavneh [22] developed an algebraic multigrid method, while Chan et al. =-=[10]-=- researched geometric multigrid schemes for variational deconvolution with TV regularisation. For TV denoising Vogel [32] proposed the use of a linear multigrid method within a nonlinear fixed-point i... |

21 | Variational optical flow computation in realtime
- Bruhn, Weickert, et al.
- 2005
(Show Context)
Citation Context ...oo slow for real-time applications. In particular the computational costs for solving the resulting linear and nonlinear system of equations by standard iterative solvers are regarded as too high. In =-=[8]-=- we have already demonstrated for variational methods with homogeneous regularisation that bidirectional multigrid strategies [5, 6, 29] do allow for real-time performance. These techniques, that crea... |

19 |
Optimal control formulation for determining optical flow
- Borzì, Ito, et al.
(Show Context)
Citation Context ...nt schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [15], Terzopoulos [25], Enkelmann [12], Ghosal and Vaněk [14]), only the work of Borzi et al. =-=[4]-=- is known to the authors where nonlinear bidirectional multigrid schemes (FAS) were used as solver for an optic flow problem that was motivated from control theory. In the related field of image proce... |

17 |
Nonlinear multigrid methods for total variation image denoising
- Frohn-Schauf, Henn, et al.
(Show Context)
Citation Context ...e context of nonlinear problems: Vogel [26] proposed the use of a linear multigrid method within a nonlinear fixed-point iteration for the purpose of TV denoising, while, very recently, Frohn-Schnauf =-=[13]-=- et al. investigated a nonlinear multigrid scheme (FAS) for the same task. 2 Prototypes for Variational Methods 2.1 The Method of Nagel and Enkelmann As prototype for the class of optic flow methods w... |

16 | A multigrid method for total variation-based image denoising
- Vogel
- 1995
(Show Context)
Citation Context ...r for an optic flow problem that was motivated from control theory. In the related field of image processing, there are two works that are worth mentioning in the context of nonlinear problems: Vogel =-=[26]-=- proposed the use of a linear multigrid method within a nonlinear fixed-point iteration for the purpose of TV denoising, while, very recently, Frohn-Schnauf [13] et al. investigated a nonlinear multig... |

8 |
Application of continuum theory and multi-grid methods to motion evaluation from 3D echocardiography
- Zini, Sarti, et al.
- 1997
(Show Context)
Citation Context ...e there is at least some literature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [18], Terzopoulos [30], Zini et a. =-=[37]-=-, El Kalmoun and Rüde [13], Enkelmann [15], Ghosal and Vaněk [17]), only the work of Borzi et al. [4] is known to the authors where an optic flow problem was solved by means of a nonlinear bidirection... |

7 |
Image analysis using multigrid relaxation
- Terzopoulos
- 1986
(Show Context)
Citation Context ...ction steps. While there is at least some literature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [15], Terzopoulos =-=[25]-=-, Enkelmann [12], Ghosal and Vaněk [14]), only the work of Borzi et al. [4] is known to the authors where nonlinear bidirectional multigrid schemes (FAS) were used as solver for an optic flow problem ... |

7 | A variational multigrid for computing the optical flow, In: Vision, Modeling and Visualization
- Kalmoun, Rüde
- 2003
(Show Context)
Citation Context ...iterature on these highly efficient schemes for variational optic flow techniques with homogeneous and image-driven regularisation (Glazer [18], Terzopoulos [30], Zini et a. [37], El Kalmoun and Rüde =-=[13]-=-, Enkelmann [15], Ghosal and Vaněk [17]), only the work of Borzi et al. [4] is known to the authors where an optic flow problem was solved by means of a nonlinear bidirectional multigrid scheme (FAS).... |