## 3 Segmentation Results of Brain MRIs and Cell Culturing Images  
An open source dataset of brain MRIs with distinct tumor profiles [62] was applied in the potential kernel study. For denoising in the physical space, a global average of Lagrangian density functional \(\langle\mathcal{L}[n]\rangle\) was used as a threshold value for the component segmentation in each corresponding MRI dataset. Values of Lagrangian density functional lower than \(\langle\mathcal{L}[n]\rangle\) were directly filtered out Meanwhile, effects of sharpened kinetic energy density functionals were also introduced into the datasets. The pixel intensity was first normalized to \([0,1]\), and then to take its squared value for component sharpening. Additionally, parallel computing was used for reduction of computational time in the MATLAB programming environment. Since there is no ground truth for each employed image, each segmentation result without compressing was utilized as the ground truth and then a common measurement of similarity, Jaccard index [48], was used for similarity comparisons. Meanwhile, the computational time of aacg employed image with a specific compression ratio was recorded for exhibiting the relationship between the Jaccard index and the compression ratio. Figure 1 illustrates the results of image segmentation utilizing various potential kernels and sharpened kinetic energy density functionals. The original MR images are illustrated in Figure 1(a1,b1), respectively, and regions of interest (ROI) are also indicated in both original images using red rectangular windows. The size of each window was set to be 80% length and width of each image to avoid catching undesired featured components from skulls. The segmentation results, shown in Figure 1(a2,a4,b2,b4), were obtained using the simple potential kernel in Equation (2), while those shown in Figure 1(a3,a5,b3,b5) were obtained using the square-root potential kernel in Equation (6). Morphologies of kinetic energy density functional of employed original MR images are, respectively, illustrated in Figure 1(a6,b6). Then, morphologies of potential energy density functionals with the simple potential kernel are, respectively, shown in Figure 1(a7,b7), while those with the square-root potential kernel are, respectively, shown in Figure 1(a8,b8). From the profiles of the potential energy density functionals, it is obvious that the smooth performance of using the square-root potential kernel was better than that of the simple potential kernel. For instance, the spot indicated by the red arrow inserted in Figure 1(a7) became smooth when replacing the simple kernel using the square-root potential kernel, and the result is shown in Figure 1(a8). Thus, the segmentation result was also smoothened, as shown in Figure 1(a3). By comparing Figure 1(a2,a3), the fragmentary parts of the segmentation result using the simple kernel was mended by means of the square-root kernel. Similarly, the high energy part indicated by the red arrow shown in Figure 1(b7) was smoothened by taking the square-root kernel, and the smoothing segmentation result is shown in Figure 1(b3). It is noted that, in the tentative experiments, the segmentation consequences using the Yukawa potential kernel did not exhibit obvious benefits compared to the simple one in the employed MR images. Additionally, the unique property of the global estimation of the proposed framework might have a benefit on the issues of image compression. Differing from the wavelet-transform-based techniques to partition imageries into hierarchical structures or the neural-network-based methods to map imageries into features space, the proposed method directly reconstruct the structural features of imageries in specific physical spaces. Thus, the proposed framework has no additional prior assumptions onto the imageries and need not to extract image features in advance.