optimal values of the parameter vector and cost functions. Recall that the solution to and results in one optimal vector for all orientations, and the solution to or results in one optimal vector for each orientation. The results of Frobenius-norm optimization are given in Table \[table:resultex1\]. Recall that the superscript $^*$ represents the optimal value.

Figure \[fig:lena-opt\] shows the image of $\mathcal{M}^*$ for , , and . Features such as lines, edges, Mach bands, corners, and blobs are satisfactorily detected in all images. Visually, it is seen that all four optimization methods result in the same image.

Figure \[fig:diff-2par\] shows the comparison between determinant-based optimization methods and , and norm-based optimization methods and . Differences are demonstrated in the subtraction images between the two approaches. Notably, although the determinant and norm optimal frameworks result in a similar performance visually, their subtraction images are not empty, meaning not identical.

[0.25]{} ![Detection of features from the Lena’s image. This is done by optimizing the parameters of the weighting function for 2D-MSPC in Example 1. The image of $\mathcal{M}^*$ is obtained through methods , , and .[]{data-label="fig:lena-opt"}](pc_lena_det_Mpm_6or "fig:"){width="\linewidth"}

[0.25]{} ![Detection of features from the Lena’s image. This is done by optimizing the parameters 