norm is denoted by a subscript, $\|.\|_c$.

([@Lyche2012], [@Schatten1960]) \[lemakf\] The Frobenius and all Schatten $p-$norms are sub-multiplicative, i.e., $$\begin{aligned}
\label{ableqab}
\| AB \|_F &\leq \|A\|_F ~\|B\|_F\\
\| AB \|_p & \leq \|A\|_p ~\|B\|_p\end{aligned}$$

\[lempc\] Let $\mbox{PC}_o$ denotes the PC at orientation $o$ and the 2D-MSPC is computed over $O$ orientations. Then, the following relationships hold between the maximum and minimum moments and $\mbox{PC}_o$. $$\begin{aligned}
M&=\frac{1}{2}\Big(\textstyle \sum_{o} \mbox{PC}_o^2 +\sqrt{\textstyle \sum_{o} \mbox{PC}_o^4 }\Big)\\
m&=\frac{1}{2}\Big(\textstyle \sum_{o} \mbox{PC}_o^2-\sqrt{\textstyle \sum_{o} \mbox{PC}_o^4 }\Big)\end{aligned}$$ [*Proof*]{}: Appendix \[p-lempc\].

\[thpcbnd\] Let $\mbox{PC}_o$ denotes the PC at orientation $o$. The 2D-MSPC is computed over $O$ orientations. If a consistent matrix norm is used, then $$\begin{aligned}
\label{Mbnd}
 &\left\| M \right\|_c  \leq  \textstyle \sum_{o} \left\| \mbox{PC}_o^2 \right\|_c\\
 \label{mbnd}
 &\left\| m \right\|_c  \leq  \textstyle \sum_{o} \left\| \mbox{PC}_o^2 \right\|_c\end{aligned}$$ where, $c$ can take $F$ or $p$. If the subscript $c$ is replaced with $F$, the $F-$norm is applied. If the subscript $c$ is replaced with $p$, the Schatten $p-$norm is applied. : Appendix \[p-pcbnd\].

\[thsubopt\] A sub-optimal solution to the optimization problem with the consistent norm is obtained by solving the following optimization problem. $$\begin{aligned}
\begin{aligned}\label{norm-subopt}
& \underset{\upsilon_o,~o=1,\ldots,O}{\text{minimize}}
& & -(|\mu_1|+|\mu_2|)\left\| \mbox{PC}_o^2(\upsilon_o) \right\|_c \\
& \text{subject to}
& & \underline{\upsilon}_o \leq \upsilon_o \leq 