Effective Wavelet-Based Regularization of Divergence-free Fractional Brownian Motion

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

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

This paper presents a method for regularization of inverse problems. The vectorial bi-dimensional unknown is assumed to be the realization of an isotropic divergence-free fractional Brownian Motion (fBm). The method is based on fractional Laplacian and divergence-free wavelet bases. The main advantage of these bases is to enable an easy formalization in a Bayesian framework of fBm priors, by simply sampling wavelet coe cients according to Gaussian white noise. Fractional Laplacians and the divergence-free projector can naturally be implemented in the Fourier domain. An interesting alternative is to remain in the spatial domain. This is achieved by the analytical computation of the connection coefficients of divergence-free fractional Laplacian wavelets, which enables to easily rotate this simple prior in any sufficiently "regular" wavelet basis. Taking advantage of the tensorial structure of a separable fractional wavelet basis approximation, isotropic regularization is then computed in the spatial domain by low-dimensional matrix products. The method is successfully applied to fractal image restoration and turbulent optic-flow estimation.

Mots clés: image restoration; Bayesian modeling; fractional Brownian motion; divergence-free wavelets; fractional Laplacian; wavelet connection coefficients; fast recursive ltering; optic-flow; image restoration.

Domaines: Analyse numérique [math.NA]; Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]; Océan, Atmosphère

Patrick Héas, Pierre Dérian, Souleymane Kadri Harouna. Effective Wavelet-Based Regularization of Divergence-free Fractional Brownian Motion. 2011. ⟨hal-00649989v2⟩