Image denoising via K-SVD with primal-dual active set algorithm

Quan Xiao, Canhong Wen, Zirui Yan

Abstract: K-SVD algorithm has been successfully applied to image denoising tasks dozens of years but the big bottleneck in speed and accuracy still needs attention to break. For the sparse coding stage in K-SVD, which involves ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> constraint, prevailing methods usually seek approximate solutions greedily but are less effective once the noise level is high. The alternative ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> optimization is proved to be powerful than ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , however, the time consumption prevents it from the implementation. In this paper, we propose a new K-SVD framework called K-SVD <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> by applying the Primal-dual active set (PDAS) algorithm to it. Different from the greedy algorithms based K-SVD, the K-SVD <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> algorithm develops a selection strategy motivated by KKT (Karush-Kuhn-Tucker) condition and yields to an efficient update in the sparse coding stage. Since the K-SVD <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> algorithm seeks for an equivalent solution to the dual problem iteratively with simple explicit expression in this denoising problem, speed and quality of denoising can be reached simultaneously. Experiments are carried out and demonstrate the comparable denoising performance of our K-SVD <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</sub> with state-of-the-art methods.