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 $\ell_{0}$ constraint, prevailing methods usually seek approximate solutions greedily but are less effective once the noise level is high. The alternative $\ell_{1}$ optimization is proved to be powerful than $\ell_{0}$, however, the time consumption prevents it from the implementation. In this paper, we propose a new K-SVD framework called K-SVD$_P$ by applying the Primal-dual active set (PDAS) algorithm to it. Different from the greedy algorithms based K-SVD, the K-SVD$_P$ 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$_P$ 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$_P$ with state-of-the-art methods.

中文翻译：

---

使用原始对偶活动集算法通过 K-SVD 进行图像去噪

K-SVD算法在图像去噪任务中成功应用了几十年，但速度和精度的大瓶颈仍有待突破。对于 K-SVD 中的稀疏编码阶段，涉及 $\ell_{0}$ 约束，主流方法通常贪婪地寻求近似解，但一旦噪声水平高则效果不佳。替代的 $\ell_{1}$ 优化被证明比 $\ell_{0}$ 强大，但是，时间消耗阻止了它的实现。在本文中，我们通过将原始对偶活动集（PDAS）算法应用于它，提出了一种称为 K-SVD$_P$ 的新 K-SVD 框架。与基于 K-SVD 的贪婪算法不同，K-SVD$_P$ 算法开发了一种由 KKT（Karush-Kuhn-Tucker）条件驱动的选择策略，并在稀疏编码阶段产生了有效的更新。由于K-SVD$_P$算法在这个去噪问题中用简单的显式表达式迭代地寻求对偶问题的等效解，因此可以同时达到去噪的速度和质量。进行了实验并证明了我们的 K-SVD$_P$ 与最先进的方法具有可比性的去噪性能。