values of the cutoff point $c_o$, while other parameters are fixed.[]{data-label="fig:lena"}](lena-pc-co-p10 "fig:"){width="\linewidth"}

In the next few sections, these questions are addressed.

Parameter Optimization {#sec:opt}
======================

In this paper, several optimal frameworks are proposed for parameter tuning for the 2D-MSPC, which are based on the maximization of the PC momentums. The maximum and minimum moments of the 2D-MSPC, $M$ and $m$ respectively, are given by [@Kovesi2003]: $$\begin{aligned}
\label{maxmo}
& M=\frac{1}{2}\left(\alpha+\gamma+\sqrt{\beta^2+(\alpha-\gamma)^2}\right)\\
\label{minmo}
& m=\frac{1}{2}\left(\alpha+\gamma-\sqrt{\beta^2+(\alpha-\gamma)^2}\right)\end{aligned}$$ with $$\begin{aligned}
\label{alpha}
& \alpha=\textstyle \sum_{o}\left(\mbox{PC}_o \cos(\theta_o)\right)^2\\
\label{gamma}
& \gamma=\textstyle \sum_{o}\left(\mbox{PC}_o \sin(\theta_o)\right)^2\\
\label{beta}
& \beta=2 \textstyle \sum_{o}\left(\mbox{PC}_o \cos(\theta_o)\right)\left(\mbox{PC}_o \sin(\theta_o)\right).\end{aligned}$$ where, $\mbox{PC}_o$ and $\theta_o$ represent the PC and axis angle at orientation $o$.

The calculation of momentums corresponds to performing a singular value decomposition to the PC covariance matrix, and thus momentums correspond to the singular values, [@Kovesi2003]. On the other hand, singular values indicate the level of increase in energy that can occur between the input and output of a given system. The larger the maximum singular value of a system, the greater the increase in energy, (see Definition 40.2 and descriptions in §40.2.1, p. 652,[@Levine1995]). Therefore, it may be deduced that momentums represent the maximum of local energies. It was also shown that the maxima of local 