energy are indications of features in an image, [@Venkatesh1990]. The answer to Q1 can now be simplified as such that a potential criterion for parameter tuning of 2D-MSPC should be given in terms of $M$ and $m$.

From the potential relationships between momentums, singular values, local energies, 2D-MSPC and image features, it may be deduced that the image features detection can be enhanced by increasing the maximum and minimum moments of the PC. In other words, optimal values of 2D-MSPC parameters are those that maximize $M$ and $m$. Both $M$ and $m$ are matrices and their size is equal to the size of $\mbox{PC}_o$. Thus, the optimal parameters of 2D-MSPC should be obtained by solving a matrix optimization problem (an answer to Q2).

In order to formulate the problem, we define a cost function $\mathcal{M}$:$$\begin{aligned}
\label{Mm}
\mathcal{M}=\mu_1 \times  M+\mu_2 \times  m\end{aligned}$$ where, $\mu_i \in [0,1]$, $i=1,2$. The cost function enables us to not only maximize both $M$ and $m$, but also to manage their interactions.

As mentioned above, the question of parameter tuning for 2D-MSPC is a matrix optimization problem. Matrix maximization has widely been used in optimal (experimental) designs [@Fredov1972; @Goodwin1977; @Atkinson1982; @Pronzato1985; @Berger1994; @Pukelsheim1993], where the Fisher information 