## 2 Theoretical Framework of Data Density Functionals
An enclosed curve constitutedby a set of zero-points constructs a boundary of a specific featured pixel component. The value of Lagrangian density functional adjacent to the boundary should transfer its plus/minus sign. Physically, the boundary represents a limitation of the action, and particles enclosed by the boundary should not interact with other particles outside. Under the same concept, the featured pixel components adjacent to the boundary should have a significant discrepancy. Thus, under the proposed method, a discrepancy of a studied medical image can be expressed as [32]:  
\[\mathcal{L}[n]=\gamma^{2}t[n]-\gamma u[n]. \tag{3}\]  
The adaptive scaling factor \(\gamma\) in Equation (3) is used to resolve the issue of dimensional mismatches occurred in mapping the pixel into a specific physical space. The theoretical form is expressed as:  
\[\gamma=\frac{1}{2}\frac{\langle u[n]\rangle}{\langle t[n]\rangle}. \tag{4}\]  
It should be emphasized that the adaptive scaling factor is automatically defined by the system of interest. \(\langle u[n]\rangle\) and \(\langle t[n]\rangle\), respectively, represent the global averages of the utilized tow-dimensional kinetic and potential energy density functionals. According to the mathematical postulation of density functional theory [32], the dimensional mismatch between kinetic and potential energy density functionals can be balanced using the adaptive scaling factor \(\gamma\) in Equation (4). Thus, the postulation provides a convenience to directly employ any useful form of energy density functional. As mentioned, the kernel \(1/\|\mathbf{r}-\mathbf{r}_{i}^{\prime}\|\) of the simple potential energy density functional in Equation (2) smoothens the profile of the similarity. Thus, various mathematical forms of kernels can be used to construct other structural features within a studied image. For the purpose, the Yukawa-like and square-root potential forms are, respectively, further utilized to construct the structural featured properties of local edge sharpening and global fuzzification. The mathematical form of the Yukawa potential provides an elegant property to explain physical phenomena occurred in an extreme short distance [61]; thus, the potential form is employed to construct sharpened profiles of boundary edges within a medical image. For the simplification and the convenience of usage, the form of Yukawa potential energy density functional is expressed as:  
\[u_{Yukawa}[n]\cong\frac{1}{2}\sum_{i=1}^{N}\frac{n\left(\mathbf{r}_{i}^{\prime} \right)e^{-\mathbf{r}_{i}^{\prime}}}{\|\mathbf{r}-\mathbf{r}_{i}^{\prime}\|_{\mathbf{r}\mathbf{r} _{i}^{\prime}}}. \tag{5}\]  
On the contrary, to study the blur characteristics, a square-root potential kernel is used, and the mathematical form of the square-root potential energy density functional is expressed simply as:  
\[u_{SR}[n]\cong\frac{1}{2}\sum_{i=1}^{N}\frac{n\left(\mathbf{r}_{i}^{\prime}\right) }{\sqrt{\|\mathbf{r}-\mathbf{r}_{i}^{\prime}\|_{\mathbf{r}\mathbf{r}_{i}^{\prime}}}}. \tag{6}\]  
Two type medical imageries, a set of MRIs having brain tumors and cell culturing images, were utilized for the potential study and the method validation. Experimentally, each potential kernel revealed their superior properties in each kind of medical image dataset. Meanwhile, sharpened kinetic energy density functionals were also used for the discrepancy study, and specific intensity normalization procedure was then proposed for the tasks of image segmentation.