Accelerating ℓ1−ℓ2 deblurring using wavelet expansions of operators

Image deblurring is a fundamental problem in imaging, usually solved with computationally intensive optimization procedures. The goal of this paper is to provide new efficient strategies to reduce computing times for simple deblurring models regularized using orthogonal wavelet transforms. We show that the minimization can be significantly accelerated by leveraging the fact that images and blur operators are compressible in the same orthogonal wavelet basis. The proposed methodology consists of three ingredients: (i) a sparse approximation of the blur operator in wavelet bases, (ii) a diagonal preconditioner and (iii) an implementation on massively parallel architectures. Combining the three ingredients leads to acceleration factors ranging from 4 to 250 on a typical workstation. For instance, a 1024 × 1024 image can be deblurred in 0.15 s.