## A low dimensional fluid motion estimator

Venue: | Int. J. Comp. Vision |

Citations: | 16 - 8 self |

### BibTeX

@ARTICLE{Cuzol_alow,

author = {Anne Cuzol and Pierre Hellier and Etienne Mémin},

title = {A low dimensional fluid motion estimator},

journal = {Int. J. Comp. Vision},

year = {},

volume = {75},

pages = {2007}

}

### OpenURL

### Abstract

In this paper we propose a new motion estimator for image sequences depicting fluid flows. The proposed estimator is based on the Helmholtz decomposition of vector fields. This decomposition consists in representing the velocity field as a sum of a divergence free component and a vorticity free component. The objective is to provide a low-dimensional parametric representation of optical flows by depicting them as deformations generated by a reduced number of vortex and source particles. Both components are approximated using a discretization of the vorticity and divergence maps through regularized Dirac measures. The resulting so called irrotational and solenoidal fields consist of linear combinations of basis functions obtained through a convolution product of the Green kernel gradient and the vorticity map or the divergence map respectively. The coefficient values and the basis function parameters are obtained by minimization of a functional relying on an integrated version of mass conservation principle of fluid mechanics. Results are provided on synthet-ic examples and real world sequences. 1

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Citation Context ...smoothness of the solution is generally imposed by considering an appropriate polynomial parameterization of the velocity field, or introducing various additional smoothing terms in the cost function =-=[13, 25, 42]-=-. For image sequences showing evolving fluid phenomena, the usual brightness consistency � ¦©¨� ¨¡s� assumption does not allow to model temporal distortions of luminance patterns caused by 3D matter t... |

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Citation Context ... shift each center. 5.4.5 Shifting the particles toward the pdf modes Mean shift procedure The mean shift procedure was first presented by [22], extended by [8] and popularized in image processing by =-=[11]-=-. It describes how to shift iteratively a point toward the nearest mode of a probability distribution function. This procedure relies on a non parametric estimate of the gradient of the underlying den... |

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Citation Context ...3), where each possible § value is weighted ¤ £ § ¦�¨ by � �¡� �s¥ ¦¨§�©�� ¡ . This remark is important since it will allow us to apply a modified expression of � £���� § ¦ � the mean shift procedure =-=[12]-=-, as described in the next paragraph. Application of the mean shift procedure These weights ¤ £ § ¦ can be included in the expression of the mean shift vector (44) as follows: � � © � £¥¤§¦ ¨ ¤s£¡ ¤s£... |

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Citation Context ...g � � ��£¥¤§¦ ��� � � ¤ � (21)sThe penalty functionsinvolved here is usually thes¡ norm but any robust function attenuating the effect of data that deviate significantly from the OFC-based data-model =-=[5, 33]-=- may be used instead. Without further specification on the motion field, the problem is ill-posed as we have to recover a vector function from a scalar constraint. In order to solve the problem a smoo... |

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Citation Context ...s using a Gaussian regularization [40] (demon’s), a first-order or second-order regularization [24] (penalization of the deformation discontinuities) or an intrinsically regularized deformation model =-=[37]-=- (B-splines deformation fields for instance). All these methods tend more or less to penalize the vorticity and the divergence of the motion field. This appears to be problematic when matter apparitio... |

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Citation Context ...in the next section the methodology used to shift each center. 5.4.5 Shifting the particles toward the pdf modes Mean shift procedure The mean shift procedure was first presented by [22], extended by =-=[8]-=- and popularized in image processing by [11]. It describes how to shift iteratively a point toward the nearest mode of a probability distribution function. This procedure relies on a non parametric es... |

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Citation Context ...of the standard motion estimators based on such a non linear formulation, we will consider an incremental minimization framework to remove the non linearity of the displaced image brightness function =-=[1, 5, 6, 35]-=-. This scheme consists in applying successive linearizations around previous estimates. This technique, similar to the Gauss-Newton non linear least squares, is in most cases embedded within a multire... |

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Citation Context ...minance can usually be classified according to the image similarity and the deformation field regularization. Most often, methods tend to regularize deformation fields using a Gaussian regularization =-=[40]-=- (demon’s), a first-order or second-order regularization [24] (penalization of the deformation discontinuities) or an intrinsically regularized deformation model [37] (B-splines deformation fields for... |

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Citation Context ...of the standard motion estimators based on such a non linear formulation, we will consider an incremental minimization framework to remove the non linearity of the displaced image brightness function =-=[1, 5, 6, 35]-=-. This scheme consists in applying successive linearizations around previous estimates. This technique, similar to the Gauss-Newton non linear least squares, is in most cases embedded within a multire... |

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Citation Context ...equently the whole vector field) may be recovered through a convolution product knowing the vorticity and the divergence of the velocity field. 3 Vortex particles The idea of vortex particles methods =-=[9, 29]-=- consists in representing the vorticity distribution of a field by a set of discrete amounts of vorticity (vortices). Using these vortices, the vorticity distribution is mathematically approximated by... |

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Citation Context ...of the standard motion estimators based on such a non linear formulation, we will consider an incremental minimization framework to remove the non linearity of the displaced image brightness function =-=[1, 5, 6, 35]-=-. This scheme consists in applying successive linearizations around previous estimates. This technique, similar to the Gauss-Newton non linear least squares, is in most cases embedded within a multire... |

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Citation Context ...g � � ��£¥¤§¦ ��� � � ¤ � (21)sThe penalty functionsinvolved here is usually thes¡ norm but any robust function attenuating the effect of data that deviate significantly from the OFC-based data-model =-=[5, 33]-=- may be used instead. Without further specification on the motion field, the problem is ill-posed as we have to recover a vector function from a scalar constraint. In order to solve the problem a smoo... |

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Citation Context ...alization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have been proposed for fluid flow velocity estimation =-=[13, 26, 44]-=-. Unlike most of the motion estimators based on the brightness consistency assumption and a first order smoothness function, these techniques rely on a data-model derived from the continuity equation ... |

86 |
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Citation Context ...to a non-rigid registration issue of medical images. This is a very active field of research and numerous methods have been proposed so far. We refer the reader for comprehensive surveys on this area =-=[30, 32, 41, 46]-=-. This process aims at estimating a non-linear geometric transformation that puts two images into correspondence. Beyond rigid transformations, non-rigid registration is needed when deformable phenome... |

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Citation Context ...e support, whose mass is essentially concentrated on a disc of radius � . Then, for � some proper choices � of , the property of radial symmetry leads to an explicit forms� for . In fact we can write =-=[15]-=-:s� £¥¤§¦ ¨ ¤ ¡ ¤�� � � (10) ¨ �¡¨� � ��� £ � ¦�� � � (11) © and consequently, if an analytic expression of the integral is available, the entire expression fors� is known. Rectangle basis function an... |

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Citation Context ... �¡ � � � � ¡ ¡�§�¡�� (35) where ¡ is a scalar, corresponding to the Fletcher-Reeves variant of the non linear conjugate � gradient algorithm. Various forms of this coefficient exist (see for example =-=[23]-=-) leading to different versions of the algorithm. Different implementation schemes may be chosen, based on exact or inexact line searches, or on the possibility to restart the iteration (35) every � s... |

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Citation Context ...smoothness of the solution is generally imposed by considering an appropriate polynomial parameterization of the velocity field, or introducing various additional smoothing terms in the cost function =-=[13, 25, 42]-=-. For image sequences showing evolving fluid phenomena, the usual brightness consistency � ¦©¨� ¨¡s� assumption does not allow to model temporal distortions of luminance patterns caused by 3D matter t... |

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Citation Context ...equently the whole vector field) may be recovered through a convolution product knowing the vorticity and the divergence of the velocity field. 3 Vortex particles The idea of vortex particles methods =-=[9, 29]-=- consists in representing the vorticity distribution of a field by a set of discrete amounts of vorticity (vortices). Using these vortices, the vorticity distribution is mathematically approximated by... |

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Citation Context ...atterns caused by 3D matter transportation. For these sequences, it has been shown that a data model based on mass conservation (also known as the continuity equation) constitutes a better modeling ¦ =-=[2, 4, 13, 38, 39, 43]-=-. The continuity equation links the variation of mass density, ¡ , within an infinitesimal enclosing volume to the flux of matter through the volume boundary surface: � � ¡ � � Let us not that the vec... |

45 |
The pyramid as a structure for efficient computations
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Citation Context ... the correction field ¡ is parameterized on the basis of a set of vortex and source particles. In practice, this kind of scheme is embedded into a pyramidal multiresolution data representation scheme =-=[7]-=-. Such a representation is obtained through low-pass filtering and subsampling. At a given level, the known motion ¢sestimate is fixed to be the projected estimate obtained at the previous level. At t... |

44 |
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Citation Context ...larity and the deformation field regularization. Most often, methods tend to regularize deformation fields using a Gaussian regularization [40] (demon’s), a first-order or second-order regularization =-=[24]-=- (penalization of the deformation discontinuities) or an intrinsically regularized deformation model [37] (B-splines deformation fields for instance). All these methods tend more or less to penalize t... |

38 | Discrete orthogonal decomposition and variational fluid flow estimation
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(Show Context)
Citation Context ...alization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have been proposed for fluid flow velocity estimation =-=[13, 26, 44]-=-. Unlike most of the motion estimators based on the brightness consistency assumption and a first order smoothness function, these techniques rely on a data-model derived from the continuity equation ... |

37 |
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Citation Context ...ssumed to hold as well for the bidimensional motion field captured by the image sequence. This latter assumption has been theoretically established in the case of transmittance imaging by Fitzpatrick =-=[20]-=-. These two hypotheses cannot be easily justified for all types of images. Nevertheless, this continuity equation constitutes an interesting alternative to the brightness constancy equation. As a matt... |

35 |
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Citation Context ...rom multiple sclerosis (MS). The characterization of the lesion growth in MR images is important to assess the evolution of the disease. Two ¡¡sT2 were acquired within ¢ months and rigidly registered =-=[31]-=-. The MR volumes have been registered with a ¡¡srigid transformation and an axial � � © � +¢ © slice has been extracted. The lesion appears as a white stain in the left part of the brain at . Its grow... |

34 | The role of image registration in brain mapping
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Citation Context |

32 | Direct estimation of non-rigid registration
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(Show Context)
Citation Context ...ence maps. This discretization enables to define implicitly adapted regularizers for fluid motion estimation problems. Such a technique is related to motion estimators based on radial basis functions =-=[3]-=-. Nevertheless, contrary to techniques that operate directly a regularized discretization of the motion field, the proposed method relies on the dis2scretization of the curl and the divergence of the ... |

27 | A generalized optical flow constraint and its physical interpretation
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(Show Context)
Citation Context ...atterns caused by 3D matter transportation. For these sequences, it has been shown that a data model based on mass conservation (also known as the continuity equation) constitutes a better modeling ¦ =-=[2, 4, 13, 38, 39, 43]-=-. The continuity equation links the variation of mass density, ¡ , within an infinitesimal enclosing volume to the flux of matter through the volume boundary surface: � � ¡ � � Let us not that the vec... |

25 | Fluid structure and motion analysis from multi-spectrum 2D cloud images sequences
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(Show Context)
Citation Context ...rements are difficult. The analysis of dynamic structures and the estimation of velocities for fluid image sequences have received great attention from the computer vision community for several years =-=[19, 21, 28, 34, 43, 45]-=-. Application domains range from experimental visualization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have... |

24 | Extraction of singular points from dense motion fields: an analytic approach - Corpetti, Mémin, et al. |

21 |
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(Show Context)
Citation Context ...rements are difficult. The analysis of dynamic structures and the estimation of velocities for fluid image sequences have received great attention from the computer vision community for several years =-=[19, 21, 28, 34, 43, 45]-=-. Application domains range from experimental visualization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have... |

19 | Variational dense motion estimation using the helmholtz decomposition
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Citation Context ...alization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have been proposed for fluid flow velocity estimation =-=[13, 26, 44]-=-. Unlike most of the motion estimators based on the brightness consistency assumption and a first order smoothness function, these techniques rely on a data-model derived from the continuity equation ... |

19 | Fluid motion recovery by coupling dense and parametric motion fields
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(Show Context)
Citation Context ...rements are difficult. The analysis of dynamic structures and the estimation of velocities for fluid image sequences have received great attention from the computer vision community for several years =-=[19, 21, 28, 34, 43, 45]-=-. Application domains range from experimental visualization in fluid mechanics, environmental sciences (oceanography, meteorology, ...), to medical imagery. Recently, several dedicated approaches have... |

18 |
A method for calculating velocity in time dependent images based on the continuity equation
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Citation Context |

17 |
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Citation Context |

16 | Vortex and source particles for fluid motion estimation
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(Show Context)
Citation Context ...l examples. Results on fluid motion analysis are shown for applications in meteorology and fluid mechanics. Results on medical images are also presented. This paper is an extended version of [16] and =-=[18]-=-, respectively dedicated to the estimation of fluid motions and to a non-rigid registration problem in medical imaging. 2 Definitions and properties of vector fields In this section, known analytic re... |

15 | A stochastic filter for fluid motion tracking
- Cuzol, Mémin
- 2005
(Show Context)
Citation Context .... The tracking can then be done within a sequential Bayesian filtering framework, with a continuous evolution law adapted to fluid flows and discrete error measurements provided by the image sequence =-=[17]-=-. Acknowledgements This work was supported by the European community through the IST Fet open FLUID project (http://fluid.irisa.fr). The authors would like to thank the ONERA for having provided them ... |

15 |
Physically based fluid flow recovery from image sequences
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(Show Context)
Citation Context |

13 |
The motion constraint equation for optical flow
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(Show Context)
Citation Context ...atterns caused by 3D matter transportation. For these sequences, it has been shown that a data model based on mass conservation (also known as the continuity equation) constitutes a better modeling ¦ =-=[2, 4, 13, 38, 39, 43]-=-. The continuity equation links the variation of mass density, ¡ , within an infinitesimal enclosing volume to the flux of matter through the volume boundary surface: � � ¡ � � Let us not that the vec... |

12 |
A novel parametric method for non-rigid image registration
- Cuzol, Hellier, et al.
- 2005
(Show Context)
Citation Context ...c and real examples. Results on fluid motion analysis are shown for applications in meteorology and fluid mechanics. Results on medical images are also presented. This paper is an extended version of =-=[16]-=- and [18], respectively dedicated to the estimation of fluid motions and to a non-rigid registration problem in medical imaging. 2 Definitions and properties of vector fields In this section, known an... |

10 | A scalar function formulation for optical flow
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(Show Context)
Citation Context |

9 |
Field theory approach for determining optical flow
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(Show Context)
Citation Context ...¨ � � � ��¤�� This model which states that the velocity is linear inside a disc of radius � and decreases � ¡ ��¤�� in outside of the circular domain has been already used in image analysis studies � =-=[14, 34, 36]-=-. That kind of model has been used to segment fluid motion fields in oder to characterize vortex or sink/source areas. Such a simple radial basis function can be used when centered 6 (13)son critical ... |

6 |
Direct numerical simulation of interactions between a mixing layer and a wake around a cylinder
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(Show Context)
Citation Context ...eld describing the 2D evolution of a turbulent wake behind a circular cylinder. The field has been obtained by simulation of the Navier-Stokes equation with a DNS method (Direct Numerical Simulation) =-=[27]-=-. A synthetic pair is created from one initial image of particles (see figure 7(a)). The known displacement and the associated vorticity map are represented figure 7(b,c). To compare the accuracy of t... |