Percentile half thresholding

Recent research suggests that it's possible to reconstruct the seismic data using non-convex $L_p$-norm minimization, $0< p <1$. Particularly, iterative half thresholding has been developed both in signal processing and exploration geophysics communities (Yang et al., 2013a; Xu et al., 2012). The difference between half thresholding and the conventional soft thresholding is just the thresholding operator. When $p=1/2$, $\gamma(\tau,1/2)=\frac{3}{2}\tau^{2/3}$, $\mathbf{R}'(\cdot)=\tau \Arrowvert \cdot \Arrowvert_{1/2}^{1/2}$, $\mathbf{T}_{\gamma(\tau,1/2)}$ becomes a half-thresholding operator:

  $\displaystyle \mathbf{T}_{\gamma(\tau,1/2)}[v(\mathbf{x})] =$ (10)
  $\displaystyle \left\{ \begin{array}{l}
... \vert v(\mathbf{x})\vert \le \gamma(\tau,1/2) \\
%\quad &
\end{array}\right..$ (11)

The threshold $\gamma(\tau,p)$ can be constant, linear-decreasing (Abma and Kabir, 2006) and exponential-decreasing (Gao et al., 2010). However, all of these defining criterion are based on a prior knowledge about the data and is often not easy to choose. Instead, a processing-convenient percentile-thresholding strategy can be selected to overcome this inconvenience (Wang et al., 2008). $\gamma(\tau,p)$ is selected as the $k$th largest absolute value among all $N$ values in the transformed domain, where the predefined percentile threshold $pc=k/N$ such that $\gamma(\tau,p)=prctile(\vert v(\mathbf{x})\vert,pc)$. Here, $prctile$ returns percentile $pc$ of the values in $\vert v(\mathbf{x})\vert$.