the Rayleigh that describes the noise energy response; $k_o$ is a scaling factor used to estimate the maximum degree of the noise response.

When the spread of frequencies is narrow, $E_o$ and $\sum_n A_{no}$ become very small, making the computations to become ill-conditioned. The parameter $\varepsilon$ is to address this issues.

Table \[table:par\] summarizes the parameters discussed above which are fundamental and need to be fine-tuned in the 2D-MSPC calculations, including some hints from [@Kovesi1996; @Kovesi1999; @Kovesi2003; @Kovesi_url], mainly for manual tuning.

Hereafter, we denote all parameters in a vector format, $$\begin{aligned}
\label{parvec}
\upsilon_o= \left[ \begin{array}{ccccccccc}
    c_o &  g_o & \lambda_{\min o} & \sigma_{no} & \eta_o & k_o & \varepsilon & N_o & O \\ 
  \end{array}\right]\end{aligned}$$ where, $\upsilon_o$ denote the parameter vector at orientation $o$.

As usual in the numerical optimization, we assume that upper and lower limits of the parameter vector are known [*a priori*]{}. The upper and lower limits of $\upsilon$ in the vector format are denoted by $\overline{\upsilon}_o$ and $\underline{\upsilon}_o$ respectively, such that $$\begin{aligned}
\underline{\upsilon}_o \leq \upsilon_o \leq \overline{\upsilon}_o\end{aligned}$$ In this paper, $\upsilon_o^*$ denotes the optimal parameter vector and $\mbox{PC}_o^*$ denotes the PC matrix computed with the optimal parameter vector $\upsilon_o^*$, both at orientation $o$.

Description of 