Accelerated dual gradient-based methods for total variation image denoising/deblurring problems

Donghwan Kim, Jeffrey A. Fessler

Abstract: We study accelerated dual gradient-based methods for image denoising/deblurring problems based on the total variation (TV) model. For the TV-based denoising problem, combining the dual approach and Nesterov's fast gradient projection (FGP) method has been found effective. The corresponding denoising method minimizes the dual function with FGP's optimal rate O(1/k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) where k denotes the number of iterations, and guarantees a rate O(1/k) for the primal function decrease. Considering that the dual projected gradient decrease is closely related to the primal function decrease, this paper proposes new accelerated gradient projection methods that decrease the projected gradient norm with a fast rate O(1/k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> ) and that are as efficient as FGP. The proposed approach also decreases the primal function with a faster rate O(1/k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.5</sup> ). We provide preliminary results on image denoising/deblurring problems with a TV regularizer, where the fast and efficient denoising solver is iteratively used for solving a deblurring problem as the inner proximal update of a fast iterative shrinkage/thresholding algorithm (FISTA).