Table 2 Regularization penalties and the corresponding proximity operator (\(\lambda >0\))
From: Image denoising based on global image similar patches searching and HOSVD to patches tensor
Penalty name | Penalty formulation | Proximity operator |
|---|---|---|
Soft-thresholding | \(P_\lambda (x)=\lambda |x|\) | \({\text{proxp}}_{\lambda }(t)={\text{sign}}(t)\max \{|t|-\lambda ,0\}\) |
Hard thresholding | \(P_{\lambda }(x)=\lambda [2-(|x|-\sqrt{2})^{2}I(|x|<\sqrt{2})]\) or \(P_{\lambda }(x)=\lambda |x|_{0}\) | \({\text{proxp}}_{\lambda }(t)={\left\{ \begin{array}{ll} 0 &{}\quad |t|<\sqrt{2\lambda } \\ \{0,t\} &{}\quad |t|=\sqrt{2\lambda }\\ t &{}\quad |t|>\sqrt{2\lambda } \end{array}\right. }\) |