## Non Uniform Multiresolution Method for Optical Flow and Phase Portrait Models: Environmental Applications (1999)

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Venue: | International Journal of Computer Vision |

Citations: | 13 - 2 self |

### BibTeX

@ARTICLE{Cohen99nonuniform,

author = {Isaac Cohen and Isabelle Herlin},

title = {Non Uniform Multiresolution Method for Optical Flow and Phase Portrait Models: Environmental Applications},

journal = {International Journal of Computer Vision},

year = {1999},

volume = {33},

pages = {29--49}

}

### OpenURL

### Abstract

. In this paper we dene a complete framework for processing large image sequences for a global monitoring of short range oceanographic and atmospheric processes. This framework is based on the use of a non quadratic regularization technique in optical ow computation that preserves ow discontinuities. We also show that using an appropriate tessellation of the image according to an estimate of the motion eld can improve optical ow accuracy and yields more reliable ows. This method denes a non uniform multiresolution approach for coarse to ne grid generation. It allows to locally increase the resolution of the grid according to the studied problem. Each added node renes the grid in a region of interest and increases the numerical accuracy of the solution in this region. We make use of such a method for solving the optical ow equation with a non quadratic regularization scheme allowing the computation of optical ow eld while preserving its discontinuities. The second part of th...

### Citations

1761 | Determining optical flow
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- 1981
(Show Context)
Citation Context .... This equation, based on the assumption that the pixel gray level remains constant, relates the temporal and spatial changes of the ! ! "# image gray level at point to the velocity at the same point =-=[12]-=-. Equation (1) ! is not sufficient for computing the image velocity at each point since the velocity components are constrained by only one equation; this is the aperture problem. Therefore, most of t... |

1491 |
The Finite Element Method for Elliptic Problems (North-Holland
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(Show Context)
Citation Context ...rary domain tessellation. The FEM method is generic and can be used for most minimization problems. However the domain tessellation must fulll the conform triangulation requirement of the FEM scheme [=-=5]-=-, i.e.: any face of any n-simplex T 1 in the triangulation is either a subset of the boundary of the domain, or a face of another n-simplex T 2 in the triangulation. This requirement restricts the typ... |

1401 |
Fatemi E.: Nonlinear total variation based noise removal algorithms
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(Show Context)
Citation Context ...(7) characterize, through a stationary solution, a solution of the initial nonlinear problem (Eq. 5). The evolution equations may be solved with asnite dierence or asnite element method. Rudin et al [=-=27-=-] proposed asnite dierence method for image deblurring, by using similar equations solved over a rectangular tessellation of the image. Such a method can be used for a uniform image tessellation but c... |

1179 |
The Design and Analysis of Spatial Data Structures
- Samet
- 1989
(Show Context)
Citation Context ...case of a quaternary decomposition, each triangle can be adjacent to a number of triangles on each of its sides and the resulting surface is not continuous unless all triangles are uniformly splitted =-=[28]-=-. Vasilescu and Terzopoulos [32] proposed an adaptive mesh scheme for subdividing and merging elements, based on regional properties of the intensity function. But this scheme is a quaternary scheme w... |

1057 | Performance of optical flow techniques
- Barron, Fleet, et al.
- 1994
(Show Context)
Citation Context ...nts are constrained by only one equation; this is the aperture problem. Therefore, most of the techniques use a regularity constraint that restrains the space of admissible solutions of equation (1) (=-=[1]-=- and references therein). This regularity constraint is generally quadratic and enforces the optical flow field to be continuous and smooth. But, true discontinuities can occur in the optical flow and... |

317 |
Image selective smoothing and edge detection by nonlin- ear diffusion
- Alvarez, Lions, et al.
- 1992
(Show Context)
Citation Context ...n quadratic regularizer: Z jr'j = jr'j L 1 (8) Non Uniform Multiresolution Method 5 where ' is the cartesian surface representation of the image (i.e '(x; y) = I(x; y)) in the case of image smoothing =-=[16, 18, 2-=-3] or the opticalsow components in our approach. The evolution equation of the mean curvature motion model is: @' @t +r r' jr'j = @' @t +D' = @' @t + jr'j = 0 (9) where is the gaussian curvature of '... |

216 |
An investigation of smoothness constraints for the estimation of displacement vector fields from images sequences
- Nagel, Enkelmann
- 1986
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Citation Context ...uation; this is the aperture problem. Therefore, most of the techniques use a regularity constraint that restricts the space of admissible solutions of equation (1) ([1] and references therein) or in =-=[21]-=- for an investigation of the smoothness constraints in apparent motion estimation . This regularity constraint is generally quadratic and enforces the optical owseld to be continuous and smooth. But, ... |

202 |
Numerical Methods for Nonlinear Variational Problems
- Glowinski
- 1984
(Show Context)
Citation Context ...d in a particular way. An ecient method for solving this kind of nonlinear partial dierential equation is to consider the associated evolution equations, or equivalently, the gradient descent method [=-=11, 26]-=-. This time-dependent approach means that the evolution equations: 8 > > > > > > : @u @t +Du+ (uI 2 x + vI x I y + I x I t ) = 0 @v @t +Dv + (uI x I y + vI 2 y + I y I t ) = 0 +Boundary conditions:ru ... |

129 |
The gradient projection method for nonlinear programming, part II: Nonlinear constraints
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(Show Context)
Citation Context ...d in a particular way. An ecient method for solving this kind of nonlinear partial dierential equation is to consider the associated evolution equations, or equivalently, the gradient descent method [=-=11, 26]-=-. This time-dependent approach means that the evolution equations: 8 > > > > > > : @u @t +Du+ (uI 2 x + vI x I y + I x I t ) = 0 @v @t +Dv + (uI x I y + vI 2 y + I y I t ) = 0 +Boundary conditions:ru ... |

111 |
Detecting and tracking multiple moving objects using temporal integration
- Irani, Rousso, et al.
- 1992
(Show Context)
Citation Context ...to locate moving points. Indeed, w? is high near moving points and becomes small near stationary points. The denition of w? gives the theoretical proof of the motion measure D, dened by Irani et al [1=-=5]-=- and used by several authors [31, 20] : D = X (x;y)2W jI(x; y; t + 1) I(x; y; t)j jrI(x; y; t)j X (x;y)2W jrI(x; y; t)j 2 + C (22) where W is a small neighborhood and C a constant used to avoid numeri... |

110 | P.: Robust dynamic motion estimation over time
- Black, Anandan
- 1991
(Show Context)
Citation Context ...ient computation of the image tessellation can indeed increase the accuracy of the opticalsow near the discontinuities (see section 3) but does not allow the solution to be discontinuous. Black et al =-=[3, 4-=-] proposed several non-quadratic schemes for discontinuous motion estimation. These methods are based on the Lorentzian estimator which allows to reduce the eects of outliers (points which violates th... |

102 | Motion field and optical flow: Qualitative properties
- Verri, Poggio
- 1989
(Show Context)
Citation Context ...ronmental Applications + + + + is: Let represent the image brightness and + its gradient vector field. The optical flow equation where "# ! is the optical flow. This equation can then be rewritten as =-=[28]-=-: * * + * * where + is the norm of the component of the motion field along the direction of + If the flow constraint equation is satisfied (i.e. ) and + & , we obtain: . (19) representing the componen... |

64 |
Qualitative Theory of Differential Equations
- Niemytzki, Stepanov
- 1960
(Show Context)
Citation Context ... trajectories governed by Eq. (24), from the solution of the systemlinear characterizing the stationary points of the system while the description of the flow is completely given by the ofeigenvalues =-=[18]-=-. This description is useful, since in the general case (i.e. arbitrary polynomials) a classification of the stationary points can be obtained from the linearization of the polynomial model. 6.1 The L... |

54 | Motion estimation with quadtree splines
- Szeliski, Shum
- 1996
(Show Context)
Citation Context ...sion scheme that can be used for an automatic non-uniform cell subdivision. Indeed, dierent mesh renement techniques were proposed in computer vision but most of them do not fulll this requirement [32=-=, 31-=-]. Hierarchical triangular decomposition methods are dierentiated on the basis of whether the decomposition is into three (ternary) or four (quaternary) parts (see Figure 1). Ternary decompositions ar... |

52 |
Optical flow estimation: advances and comparisons
- Otte, Nagel
- 1994
(Show Context)
Citation Context ...s algorithmic complexity. This is due to the presence of objects having different velocities in the image sequence. This no longer true for image sequences with uniform motion (like “Otte’s Sequence” =-=[20]-=-). For this kind of sequence the quadtree spline approach [26] seems to give a better accuracy/complexity ratio since the image is subdivided in regions having the same velocity. RR n ˚ 281920 Isaac ... |

51 |
Computerized flow field analysis: Oriented texture fields
- Rao, Jain
- 1992
(Show Context)
Citation Context ...to characterize the flow field through the particles trajectories and their stationary points. RR n ˚ 2819+ - 26 Isaac COHEN, Isabelle HERLIN Different works were led on linear phase portrait models =-=[8, 21, 25]-=- and their use for characterizing oriented texture fields. Recently, Zhong et al [30] proposed an application of linear phase portrait for structure’s analysis in a fluid flow. The main drawback is th... |

47 | Recursive non-linear estimation of discontinuous flow fields
- Black
- 1994
(Show Context)
Citation Context ...ient computation of the image tessellation can indeed increase the accuracy of the opticalsow near the discontinuities (see section 3) but does not allow the solution to be discontinuous. Black et al =-=[3, 4-=-] proposed several non-quadratic schemes for discontinuous motion estimation. These methods are based on the Lorentzian estimator which allows to reduce the eects of outliers (points which violates th... |

41 | Adaptive meshes and shells: Irregular triangulation, discontinuities and hierarchical subdivision
- Vasilescu, Terzopoulos
- 1992
(Show Context)
Citation Context ...sion scheme that can be used for an automatic non-uniform cell subdivision. Indeed, dierent mesh renement techniques were proposed in computer vision but most of them do not fulll this requirement [32=-=, 31-=-]. Hierarchical triangular decomposition methods are dierentiated on the basis of whether the decomposition is into three (ternary) or four (quaternary) parts (see Figure 1). Ternary decompositions ar... |

41 | Geometric heat equation and nonlinear diffusion of shapes and images
- Kimia, Siddiqi
- 1994
(Show Context)
Citation Context ...otion scheme. Indeed, both approaches use the non quadratic regularizer: * (7) ' ) *,+ * */+ "# ! ! "# where is the cartesian surface representation of the image (i.e ) in the case of image smoothing =-=[14, 16, 19]-=- or the optical flow components in our approach. The evolution equation of the mean curvature motion model is: (8) * + * where is the gaussian curvature of . This equation is closely related to the ev... |

39 |
Determining Optical
- Horn, Schunck
- 1981
(Show Context)
Citation Context ...ased on the assumption that the pixel gray level remains constant, relates the temporal and spatial changes of the image gray level I(x; y; t) at point (x; y) to the velocity (u; v) at the same point =-=[14]-=-. Equation (1) is not sucient for computing the image velocity (u; v) at each point since the velocity components are constrained by only one equation; this is the aperture problem. Therefore, most of... |

38 | A multigrid approach for hierarchical motion estimation
- Mémin, Pérez
- 1998
(Show Context)
Citation Context ...nd does not fulll the previous conform triangulation requirement since cracks orsrst-order discontinuities in the interpolated function will arise unless a crack-lling strategy is used. Memin et al [1=-=9-=-] proposed a method relying on adaptive multigrid and representing a tradeo between the spline approach and the global parametrization approaches. After reviewing the methods used for hierarchical dec... |

33 |
Differential Equations: Geometric Theory
- Lefschetz
- 1977
(Show Context)
Citation Context ...2 ; (39) where O 2 is a polynomial of the second order. This linearization of the initial system allows to study the behavior of the particles trajectories only in the neighborhood of critical points =-=[17]-=-. The behavior is the same as for thesrst order approximation (i.e. linear case) except when the characteristic roots are pure complex. In this case, we have a center or a focus [17]. In section 5 we ... |

22 |
Performance of optical techniques
- Barron, Fleet, et al.
- 1994
(Show Context)
Citation Context ...nts are constrained by only one equation; this is the aperture problem. Therefore, most of the techniques use a regularity constraint that restricts the space of admissible solutions of equation (1) (=-=[1]-=- and references therein) or in [21] for an investigation of the smoothness constraints in apparent motion estimation . This regularity constraint is generally quadratic and enforces the optical owseld... |

21 |
Vector field analysis for oriented patterns
- Shu, Jain
- 1994
(Show Context)
Citation Context ...to characterize the flow field through the particles trajectories and their stationary points. RR n ˚ 2819+ - 26 Isaac COHEN, Isabelle HERLIN Different works were led on linear phase portrait models =-=[8, 21, 25]-=- and their use for characterizing oriented texture fields. Recently, Zhong et al [30] proposed an application of linear phase portrait for structure’s analysis in a fluid flow. The main drawback is th... |

18 | Optical flow and phase portrait methods for environmental satellite image sequences
- Cohen, Herlin
- 1996
(Show Context)
Citation Context ...of the infrared image sequence. described in section 6.3 gives an accurate localization of the vortex. This processing was also applied to a SST image sequence in order to characterize ocean vortices =-=[6]-=-. The location of the detected vortex may further be used for a complete modeling of the vortex. Indeed, Herlin et al [11] use a geometric modeling of the vortex structure based on the location of vor... |

16 |
Image models for 2-d flow visualization and compression
- Ford, Strickland, et al.
- 1994
(Show Context)
Citation Context ...to characterize the flow field through the particles trajectories and their stationary points. RR n ˚ 2819+ - 26 Isaac COHEN, Isabelle HERLIN Different works were led on linear phase portrait models =-=[8, 21, 25]-=- and their use for characterizing oriented texture fields. Recently, Zhong et al [30] proposed an application of linear phase portrait for structure’s analysis in a fluid flow. The main drawback is th... |

13 | Curves Matching Using Geodesic Paths
- Cohen, Herlin
- 1998
(Show Context)
Citation Context ...rroir et al [2] may be viewed as a simpli cation of the true physical model of motion. A more elaborate model that can deal with changes of topology and large deformations was proposed by Cohen et al =-=[8, 7]-=-. This model is based on the computation of geodesic paths matching the two occurrences of the tracked object. 7. Conclusion In this paper we have addressed some problems encountered in processing ver... |

9 |
Qualitative Theory of Di erential Equations
- Nemytskii, Stepanov
- 1989
(Show Context)
Citation Context ...governed by Eq. (26), from the solution of the linear system AX + b = 0. The characterization of the stationary points ofsow and its description is completely given by the eigenvalues of the matrixsA =-=[22-=-]. This description is useful, since in the general case (i.e. arbitrary polynomials) a classication of the stationary points can be obtained from the linearization of the polynomial model. 18 Cohen a... |

8 | Tracking Meteorological Structures Through Curves Matching Using Geodesic Paths
- Cohen, Herlin
- 1998
(Show Context)
Citation Context ...rroir et al [2] may be viewed as a simpli cation of the true physical model of motion. A more elaborate model that can deal with changes of topology and large deformations was proposed by Cohen et al =-=[8, 7]-=-. This model is based on the computation of geodesic paths matching the two occurrences of the tracked object. 7. Conclusion In this paper we have addressed some problems encountered in processing ver... |

8 |
Salient structure analysis of fluid flow
- Zhong, Huang, et al.
- 1994
(Show Context)
Citation Context ...s. RR n ˚ 2819+ - 26 Isaac COHEN, Isabelle HERLIN Different works were led on linear phase portrait models [8, 21, 25] and their use for characterizing oriented texture fields. Recently, Zhong et al =-=[30]-=- proposed an application of linear phase portrait for structure’s analysis in a fluid flow. The main drawback is that it can handle only one critical point. This limitation led to local [21] and globa... |

7 |
Nonlinear phase portrait models for oriented textures
- Ford, Strickland
- 1993
(Show Context)
Citation Context ...he main drawback of the method is that it can handle only one critical point. This limitation led to local [25] and global [30] implementations of the linear phase portrait model. Ford and Strickland =-=[9]-=- proposed a nonlinear phase portrait model allowing multiple critical points, but this model is computationally expensive and cannot be generalized to arbitrary polynomials. In the following, we propo... |

4 |
Geometric heat equation and non-linear di usion of shapes and images
- Kimia, Siddiqi
- 1995
(Show Context)
Citation Context ...n quadratic regularizer: Z jr'j = jr'j L 1 (8) Non Uniform Multiresolution Method 5 where ' is the cartesian surface representation of the image (i.e '(x; y) = I(x; y)) in the case of image smoothing =-=[16, 18, 2-=-3] or the opticalsow components in our approach. The evolution equation of the mean curvature motion model is: @' @t +r r' jr'j = @' @t +D' = @' @t + jr'j = 0 (9) where is the gaussian curvature of '... |

4 |
Adaptive-complexity registration of images
- Muller, Anandan, et al.
- 1994
(Show Context)
Citation Context ...w? is high near moving points and becomes small near stationary points. The denition of w? gives the theoretical proof of the motion measure D, dened by Irani et al [15] and used by several authors [3=-=1, 20]-=- : D = X (x;y)2W jI(x; y; t + 1) I(x; y; t)j jrI(x; y; t)j X (x;y)2W jrI(x; y; t)j 2 + C (22) where W is a small neighborhood and C a constant used to avoid numerical instabilities. This motion measur... |

4 |
Computation of discontinuous optical by domain decomposition and shape optimization
- Schnorr
- 1992
(Show Context)
Citation Context ...ontinuous. This regularization technique (Eq. 2) is used in most cases due to its computational simplicity, since the minimization of (2) requires one to solve a linear set of equations [14]. Schnorr =-=[29]-=- have proposed an approach for discontinuous opticalsow computation by domain decomposition and shape optimization. The ecient computation of the image tessellation can indeed increase the accuracy of... |

3 |
Image processing for sequence of oceanographic images
- Herlin, Cohen, et al.
- 1996
(Show Context)
Citation Context ...d successfully to a SST image sequence in order to characterize ocean vortices [6]. The location of the detected vortex may further be used for a complete modeling of the vortex. Indeed, Herlin et al =-=[13]-=- use a geometric modeling of the vortex structure based on the location of vortex rolls. This approach is very stable and 20 Cohen and Herlin Fig. 12. A frame of the infrared image sequence. Fig. 13. ... |

3 |
Optical estimation: Advances and comparisons
- Otte, Nagel
- 1994
(Show Context)
Citation Context ...s algorithmic complexity. This is due to the presence of objects having different velocities in the image sequence. This no longer true for image sequences with uniform motion (like \Otte's Sequence&q=-=uot; [24]-=-). For this kind of sequence the quadtree spline approach [31] seems to give a better accuracy/complexity ratio since the image is subdivided in regions having the same velocity. 4.2. Environmental Im... |

2 |
Tracking highly deformable structures: a surface model applied to vortex evolution within satellite oceanographic images
- Berroir, Herlin, et al.
- 1995
(Show Context)
Citation Context ...eometrical evolution model that enables one to generate a surface interpolating the two successive contours of the object during its temporal evolution. This geometrical model dened by Berroir et al [=-=2]-=- may be viewed as a simpli cation of the true physical model of motion. A more elaborate model that can deal with changes of topology and large deformations was proposed by Cohen et al [8, 7]. This mo... |

2 |
Motion and optical Qualitative properties
- Verri, Poggio
- 1989
(Show Context)
Citation Context ... I represent the image brightness and ! rI its gradient vectorseld. The opticalsow equation is: dI dt = @I @t + ! rI : ! w where ! w = (u; v) is the opticalsow. This equation can then be rewritten as =-=[33-=-]: dI dt = @I @t + jj ! rI jjw ? where w? is the norm of the component ! w? of the motionseld ! w along the direction of ! rI . If thesow constraint equation is satised (i.e. dI dt = 0) and ! rIs6= 0,... |

1 |
Optical and phase portrait methods for environmental satellite image sequences
- Cohen, Herlin
- 1996
(Show Context)
Citation Context ...d in section 6.3 gives an accurate localization of the vortex (upper left of thesowseld). This processing was also applied successfully to a SST image sequence in order to characterize ocean vortices =-=[6]-=-. The location of the detected vortex may further be used for a complete modeling of the vortex. Indeed, Herlin et al [13] use a geometric modeling of the vortex structure based on the location of vor... |

1 |
Image models for 2-D visualization and compression
- Ford, Strickland, et al.
- 1994
(Show Context)
Citation Context ...n ofsow's orientation using a dynamic system characterizes thesowseld through the particles trajectories and their stationary points. Dierent work have been performed on linear phase portrait models [=-=10, 25, 30]-=- and their use for characterizing oriented textureselds. Recently, Zhong et al [35] proposed an application of linear phase portraits for structures analysis in asuidsow. The main drawback of the meth... |

1 |
Salden B.M. Romeny. Nonlinear diusion of scalar images using well-posed dierential operators
- Niessen, H
- 1994
(Show Context)
Citation Context ...n quadratic regularizer: Z jr'j = jr'j L 1 (8) Non Uniform Multiresolution Method 5 where ' is the cartesian surface representation of the image (i.e '(x; y) = I(x; y)) in the case of image smoothing =-=[16, 18, 2-=-3] or the opticalsow components in our approach. The evolution equation of the mean curvature motion model is: @' @t +r r' jr'j = @' @t +D' = @' @t + jr'j = 0 (9) where is the gaussian curvature of '... |

1 |
Computerized analysis: Oriented textures
- Rao, Jain
- 1992
(Show Context)
Citation Context ...n ofsow's orientation using a dynamic system characterizes thesowseld through the particles trajectories and their stationary points. Dierent work have been performed on linear phase portrait models [=-=10, 25, 30]-=- and their use for characterizing oriented textureselds. Recently, Zhong et al [35] proposed an application of linear phase portraits for structures analysis in asuidsow. The main drawback of the meth... |

1 |
Vector analysis for oriented patterns
- Shu, Jain
- 1992
(Show Context)
Citation Context ...n ofsow's orientation using a dynamic system characterizes thesowseld through the particles trajectories and their stationary points. Dierent work have been performed on linear phase portrait models [=-=10, 25, 30]-=- and their use for characterizing oriented textureselds. Recently, Zhong et al [35] proposed an application of linear phase portraits for structures analysis in asuidsow. The main drawback of the meth... |

1 | Temporal tracking with implicit templates
- Yahia, Herlin, et al.
- 1996
(Show Context)
Citation Context ...2 ) polynomial. is used for tracking vortex shape in an image se- quence. Another boundary tracking model based Non Uniform Multiresolution Method 21 on implicit functions was proposed by Yahia et al =-=[34]-=-. This model can handle topological changes of the structures being tracked. Finally, a pointwise tracking of the vortex shape can be achieved through a geometrical evolution model that enables one to... |

1 |
Salient structure analysis of
- Zhong, Huang, et al.
- 1994
(Show Context)
Citation Context ... trajectories and their stationary points. Dierent work have been performed on linear phase portrait models [10, 25, 30] and their use for characterizing oriented textureselds. Recently, Zhong et al [=-=35]-=- proposed an application of linear phase portraits for structures analysis in asuidsow. The main drawback of the method is that it can handle only one critical point. This limitation led to local [25]... |

1 |
Salden B.M. Romeny. Nonlinear diffusion of scalar images using well-posed differential operators
- Niessen, H
- 1994
(Show Context)
Citation Context ...otion scheme. Indeed, both approaches use the non quadratic regularizer: * (7) ' ) *,+ * */+ "# ! ! "# where is the cartesian surface representation of the image (i.e ) in the case of image smoothing =-=[14, 16, 19]-=- or the optical flow components in our approach. The evolution equation of the mean curvature motion model is: (8) * + * where is the gaussian curvature of . This equation is closely related to the ev... |