# SOLVING ILL-POSED PROBLEMS USING DATA ASSIMILATION - Application to Optical Flow Estimation

### Dominique Béréziat, Isabelle Herlin

#### 2009

#### Abstract

Data Assimilation is a mathematical framework used in environmental sciences to improve forecasts performed by meteorological, oceanographic or air quality simulation models. Data Assimilation techniques require the resolution of a system with three components: one describing the temporal evolution of a state vector, one coupling the observations and the state vector, and one defining the initial condition. In this article, we use this framework to study a class of ill-posed Image Processing problems, usually solved by spatial and temporal regularization techniques. A generic approach is defined to convert an ill-posed Image Processing problem in terms of a Data Assimilation system. This method is illustrated on the determination of optical flow from a sequence of images. The resulting software has two advantages: a quality criterion on input data is used for weighting their contribution in the computation of the solution and a dynamic model is proposed to ensure a significant temporal regularity on the solution.

#### References

- Béréziat, D. and Herlin, I. (2008). Solving ill-posed image processing problems using data assimilation. Application to optical flow. Research Report 6477, INRIA.
- Hadamard, J. (1923). Lecture on Cauchy's Problem in Linear Partial Differential Equations. Yale University Press, New Haven.
- Horn, B. and Schunk, B. (1981). Determining optical flow. Artificial Intelligence, 17:185-203.
- Huot, E., Herlin, I., and Korotaev, G. (2008). Assimilation of sst satellite images for estimation of ocean circulation velocity. In Proceedings of IEEE International Geoscience and Remote Sensing Symposium (IGARSS), Boston, Massachusetts, U.S.A.
- Oliver, D. (1998). Calculation of the inverse of the covariance. Mathematical Geology, 30(7):911-933.
- Papadakis, N., Corpetti, T., and Mémin, E. (2007a). Dynamically consistent optical flow estimation. In Proceedings of International Conference on Computer Vision, Rio de Janeiro, Brazil.
- Papadakis, N., Héas, P., and Mémin, E. (2007b). Image assimilation for motion estimation of atmospheric layers with shallow-water model. In Proceedings of Asian Conference on Computer Vision, pages 864- 874, Tokyo, Japan.
- Papadakis, N. and Mémin, E. (2007). Variational optimal control technique for the tracking of deformable objects. In Proceedings of International Conference on Computer Vision, Rio de Janeiro, Brazil.
- Sethian, J. (1996). Level Set Methods. Cambridge University Press.
- Tarantola, A. (2005). Inverse Problem Theory and Methods for Model Parameter Estimation. Society for Industrial and Applied Mathematics.
- Tikhonov, A. N. (1963). Regularization of incorrectly posed problems. Sov. Math. Dokl., 4:1624-1627.
- Valur Hólm, E. (2003). Lectures notes on assimilation algorithms. Technical report, European Centre for Medium-Range Weather Forecasts Reading, U.K.
- Verwer, J. and Sportisse, B. (1998). A note on operator splitting in a stiff linear case. Technical Report MASR9830, Center voor Wiskunde en Informatica.
- Weickert, J. and Schnörr, C. (2001). Variational optic flow computation with a spatio-temporal smoothness constraint. Journal of Mathematical Imaging and Vision, 14:245-255.

#### Paper Citation

#### in Harvard Style

Béréziat D. and Herlin I. (2009). **SOLVING ILL-POSED PROBLEMS USING DATA ASSIMILATION - Application to Optical Flow Estimation** . In *Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)* ISBN 978-989-8111-69-2, pages 595-602. DOI: 10.5220/0001792205950602

#### in Bibtex Style

@conference{visapp09,

author={Dominique Béréziat and Isabelle Herlin},

title={SOLVING ILL-POSED PROBLEMS USING DATA ASSIMILATION - Application to Optical Flow Estimation},

booktitle={Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)},

year={2009},

pages={595-602},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001792205950602},

isbn={978-989-8111-69-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the Fourth International Conference on Computer Vision Theory and Applications - Volume 1: VISAPP, (VISIGRAPP 2009)

TI - SOLVING ILL-POSED PROBLEMS USING DATA ASSIMILATION - Application to Optical Flow Estimation

SN - 978-989-8111-69-2

AU - Béréziat D.

AU - Herlin I.

PY - 2009

SP - 595

EP - 602

DO - 10.5220/0001792205950602