Stein block thresholding for wavelet-based image deconvolution

Christophe Chesneau (1) , Jalal M. Fadili (2) , Jean-Luc Starck (3)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme
2 Equipe Image - Laboratoire GREYC - UMR6072
3 DAPNIA - Département d'Astrophysique, de physique des Particules, de physique Nucléaire et de l'Instrumentation Associée

In this paper, we propose a fast image deconvolution algorithm that combines adaptive block thresholding and Vaguelet-Wavelet Decomposition. The approach consists in first denoising the observed image using a wavelet-domain Stein block thresholding, and then inverting the convolution operator in the Fourier domain. Our main theoretical result investigates the minimax rates over Besov smoothness spaces, and shows that our block estimator can achieve the optimal minimax rate, or is at least nearlyminimax in the least favorable situation. The resulting algorithm is simple to implement and fast. Its computational complexity is dominated by that of the FFT in the Fourier-domain inversion step. We report a simulation study to support our theoretical findings. The practical performance of our block vaguelet-wavelet deconvolution compares very favorably to existing competitors on a large set of test images.

Mots clés: Wavelets; Minimax; Image deconvolution; Block thresholding

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Christophe Chesneau, Jalal M. Fadili, Jean-Luc Starck. Stein block thresholding for wavelet-based image deconvolution. Electronic Journal of Statistics , 2010, 4, pp.415-435. ⟨10.1214/09-EJS550⟩. ⟨hal-00436661v2⟩