is a matrix with elements between 0 and 1, the determinant is expected to be small. Note that, the larger the image, the smaller the determinant is most likely achieved. Thus, from the numerical point of view, solving such an optimization problem should be more sophisticated than solving the norm-based optimization problem.

     $c_o^*$   $g_o^*$   $\lambda_{\min o}^*$   $\sigma_{no}^*$   $\eta_o^*$   $N_o^*$   $O^*$   Cost function opt. value
  -- --------- --------- ---------------------- ----------------- ------------ --------- ------- ---------------------------------------------
     0.1       49.9994   3.055                  0.4               3.8315       4         1       $\|\mathcal{M}\|_F^*=442.0584$
     0.1       50.00     2.6894                 0.4               4.0          4         1       $\|\mbox{PC}_o^2\|_F^*=6.2033\times 10^{4}$
                                                                                                 $\|\mathcal{M}\|_F^*=1.2407\times 10^{5}$

\[table:resultex2\]

[0.25]{} ![The results of the optimization of 7 parameters by solving Frobenius norm-based methods in Example 2. Figures \[fig:pc-Fnorm-mpm-full\] and \[fig:pc-Fnorm-pc-full\] shows the images of $\mathcal{M}^*$ with optimal parameters obtained through and , respectively. Figure \[fig:diff\] shows the difference by subtracting the images. []{data-label="fig:lena-7par"}](pc_lena_Fnorm_Mpm_full "fig:"){width="\linewidth"}

[0.25]{} ![The results of the optimization of 7 parameters by solving Frobenius norm-based methods in Example 2. Figures \[fig:pc-Fnorm-mpm-full\] and \[fig:pc-Fnorm-pc-full\] shows the images of $\mathcal{M}^*$ with optimal parameters obtained through and , respectively. Figure \[fig:diff\] shows the difference by subtracting the images. []{data-label="fig:lena-7par"}](pc_lena_Fnorm_PC_full "fig:"){width="\linewidth"}

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[0.25]{} ![The results of the optimization of 7 parameters by solving Frobenius norm-based methods in Example 