## 3 Segmentation Results of Brain MRIs and Cell Culturing Images
The pixel intensity in each case of the cell culturing image was first normalized to \([0,1]\), its global mean was subtracted, and the values lower than zero were set to be zero for preliminary denoising. Since the cells are specifically the only substance of foreground image, thus differing from the mentioned approach, the Yukawa potential kernel was assigned for measuring the specific local property. Figure 3 illustrates the segmentation results with and without dealing with the compression processes. The profiles of kinetic and potential energy density functionals of the original image, as shown in Figure 3a, are depicted in Figure 3b,c, respectively. As expected, the profile of the potential energy density functional was similar to that of the kinetic one. The reason is that the Yukawa potential is an extreme short potential; thus, it can significantly exhibit local properties of a physical particle of interest. The difference between the Yukawa potential energy density functional and the kinetic one is that the kinetic one has an extreme abrupt edge while the Yukawa one has a relatively alleviating edge due to the slight interaction adjacent to the boundaries. This means that the image boundaries can also be defined using the proposed Lagrangian density functional. In other words, the employed Yukawa potential kernel is suitable for measuring specific substance with strong localization. The segmented background and foreground are, respectively, shown in Figure 3d,e, while the corresponding compressed images with a 10-time compression ratio are, respectively, shown in Figure 3f,g. Then, the profiles of corresponding pixel areas of union and intersection are sequentially shown in Figure 3h,i. Figure 3j exhibits the curves of scores of the Jaccard index and log-computational time. As expected, the compressed results from the large size image had high scores of the Jaccard index as they still had enough featured components even after a 10-time image compression. To test the performance of anti-noising of the proposed method, a cell-culturing image that has severe light and noise pollution was employed as an input image in the study. As illustrated in Figure 4a, the original size of the employed image is also 1200 pixels by 1600 pixels. Thus, computational times of this set should be similar to that of the image shown in Figure 3a. The noise was randomly distributed in the image, and the red arrows depicted the regions that have severe light pollution. The segmented background and foreground images of the original are illustrated in Figure 4b,c, respectively. As expected, the regions having severe pollution, such as indicated by the red arrow in Figure 4b, were also mapped back to the original image. The Yukawa potential kernel also enhanced the noise and the light spots. Then, several speckles caused by the noise also appeared in the foreground and background images. To inhibit the undesired featured components caused by the noise and the light pollution, the original image was compressed with a 10-time compression ratio and the segmentation results are illustrated in Figure 4d,e. Obviously, the noise and light pollution were sufficiently inhibited, thus the undesired featured components from those issues were filtered out by the global density functional estimations. The score of the Jaccard index of this reconstructed background image was 0.9178 and the computational time was about 5.7 s. To exhibit the comparison between the contemporary method and the proposed algorithm, a segmentation result by employing the [PERSON]'s method [63] is illustrated in Figure 4f. The [PERSON]'s method exhibited superior performance in time saving. It only costed 1.0 s to extract all of the borders of the cells. On other hand, it was also obvious that the method filtered out all information about intercellular substances inside each cell. It might cause an inconvenience on studying cell morphologies.