Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain

Paul Escande (1) , Pierre Weiss (2) , François Malgouyres (3)

1 ITAV - Institut des Technologies Avancées en sciences du Vivant
2 PRIMO (ITAV)
3 IMT - Institut de Mathématiques de Toulouse UMR5219

Restoration of images degraded by spatially varying blurs is an issue of increasing importance in the context of photography, satellite or microscopy imaging. One of the main difficulty to solve this problem comes from the huge dimensions of the blur matrix. It prevents the use of naive approaches for performing matrix-vector multiplications. In this paper, we propose to approximate the blur operator by a matrix sparse in the wavelet domain. We justify this approach from a mathematical point of view and investigate the approximation quality numerically. We finish by showing that the sparsity pattern of the matrix can be pre-defined, which is central in tasks such as blind deconvolution.

Mots clés: Deblurring; Integral operator; Spatially varying blur; Numerical analysis; Wavelet compression; Image restoration; Matrix approximation; Sparse approximation

Domaines: Traitement des images [eess.IV]; Imagerie médicale; Modélisation et simulation; Optimisation et contrôle [math.OC]; Analyse numérique [math.NA]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Paul Escande, Pierre Weiss, François Malgouyres. Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain. Journal of Physics: Conference Series, 2013, vol. 464, ⟨10.1088/1742-6596/464/1/012004⟩. ⟨hal-00794274v2⟩