## 3 Segmentation Results of Brain MRIs and Cell Culturing Images
To retain the characteristics of the proposed framework, only simple linear downsampling was used for the image compression and then all segmentation results were reconstructed using the bicubic interpolation. Then, the performances were checked using the common Jaccard index (JI) [48]:  
\[\mathrm{JI}=\frac{(\mathrm{result})\cap(\mathrm{ground\ truth})}{(\mathrm{ result})\cup(\mathrm{ground\ truth})}. \tag{7}\]Statistically, the numerator and the denominator of JI, respectively, represent the areas wherein pixels from segmentation result and ground truth are overlapped and union. Due to the lack of ground truth from the employed MR images, the segmentation result of each original MR image was assigned to be the ground truth. The comparison of various compression ratios is illustrated in Figure 2.  
Figure 1: Two original MR images, as shown in (**a1,b1**), sourced from Ref. [62] were employed to study performances of those utilized potential kernels. Regions of interest in each MR image were defined to avoid undesired segmentation results from skulls by red rectangular windows. According to the estimated morphologies of energy density functionals, shown in the late two columns, the segmentation results were obtained using the proposed framework and are as illustrated in second and third columns, as respectively shown from (**a2**) to (**a5**) and from (**b2**) to (**b5**). It is noted that the potential kernel used in (**a7**) and (**b7**) was the simple potential kernel, while that used in (**a8**) and (**b8**) was the square-root potential kernel. For estimating scores of the Jaccard index, the segmentation result illustrated in Figure 1(a3) was employed to be as a pseudo-ground truth. Then, four values of compression ratios, 125%, 167%, 250%, and 500%, were used to test the performance of the proposed method for tumor segmentation of tremendously small size medical images. The segmentation results with various compression ratios are sequentially shown in the first row of Figure 2. It should be emphasized that each segmentation image was reconstructed using a bicubic interpolation after corresponding image compression. The results, shown in Figure 2a, reveal the ability of the proposed method for automatically catching featured components from extreme small size medical images. However, the simplified reconstruction also caused some undesired consequences. As indicated by the red arrows shown in the last two columns of Figure 2a, some aliasing edges appeared. Those undesired aliasing edges significantly affected the scores of the Jaccard index. To estimate the scores of the Jaccard index and visually exhibit the consequences, the profiles of union and intersection areas between each reconstructed image and the pseudo-ground truth are, respectively, illustrated in Figure 2b,c. The curves of computational time and the Jaccard index are illustrated in Figure 2d. The illustration shows that, even when the compression ratio was set to be 500%, the score of the Jaccard index could have a nice value, about 0.7, while the computational time was only 0.3 s.  
Figure 2: The segmentation results with the compression rate of 125%, 167%, 250%, and 500% are respectively shown in **(a)** row. Then, all of them were reconstructed using the bicubic interpolation. It is obvious that aliasing edges also appeared, as indicated by the red arrows in **(a)** row, after the strong image compression. The **(b,c)** rows, respectively, illustrated the union and the intersection areas between the testing images and the ground truth. **(d)** The curves of computational times and scores of the Jaccard index with various compression ratios. Furthermore, to test the performance of the proposed method in the image segmentation of tremendously large size medical images, a set of cell culturing images was utilized. The pixel sizes of the original cell culturing images are 1200 pixels by 1600 pixels.