Divergence-free Wavelets and High Order Regularization

Souleymane Kadri Harouna (1) , Pierre Dérian (2) , Patrick Héas (3) , Etienne Mémin (3)

1 MIA - Mathématiques, Image et Applications - EA 3165
2 Atmospheric Lidar Research Group
3 FLUMINANCE - Fluid Flow Analysis, Description and Control from Image Sequences

Expanding on a wavelet basis the solution of an inverse problem provides several advantages. Wavelet bases yield a natural and efficient multiresolution analysis. The continuous representation of the solution with wavelets enables analytical calculation of regularization integrals over the spatial domain. By choosing differentiable wavelets, high-order derivative regularizers can be designed, either taking advantage of the wavelet differentiation properties or via the basis's mass and stiffness matrices. Moreover, differential constraints on vector solutions, such as the divergence-free constraint in physics, can be handled with biorthogonal wavelet bases. This paper illustrates these advantages in the particular case of fluid flows motion estimation. Numerical results on synthetic and real images of incompressible turbulence show that divergence-free wavelets and high-order regularizers are particularly relevant in this context.

Mots clés: Divergence-free wavelets High order derivatives regularization optic-flow estimation

Domaines: Sciences de l'environnement; Optimisation et contrôle [math.OC]; Océan, Atmosphère

Souleymane Kadri Harouna, Pierre Dérian, Patrick Héas, Etienne Mémin. Divergence-free Wavelets and High Order Regularization. International Journal of Computer Vision, 2013, 103 (1), pp.80-99. ⟨10.1007/s11263-012-0595-7⟩. ⟨hal-00646104v2⟩