## 2 Theoretical Framework of Data Density Functionals  
Each pixel of a medical image is referred to as a physical particle under the framework of the data density functionals, and the structural features are measured by mapping all of the pixels into specific physical spaces. Local pixel intensities of a specific region in a pixel space signify corresponding local intrinsic inertia in a physical space. A region retaining high intrinsic inertia will similarly have high pixel intensity, and thus these physical particles in this region should have large data weightings. In other words, the local inertia emphasizes the local significance in a perspective of machine learning. Physically, the intrinsic inertia of particles can be measured by the kinetic energy. Thus, under the proposed framework, a kinetic energy density functional was employed to measure the image significance. On the other hand, from a physical perspective, potential energy represents particle-particle interaction by measuring local physical intensities as well as particle-particle distances. High intensities or short particle-particle distances will both cause high potential energy. In the pixel space, the fact of short pair-pixel distances represents that these pixels have similar image features. Pixels that have similar image features in a specific region simply indicate that they might belong to an image cluster. Thus, a potential energy density functional is employed to measure the image similarity. Theoretically, the mentioned image significance and the image similarity between image pixels can be, respectively, expressed using two-dimensional kinetic and potential energy density functionals [11; 12; 32]:  
\[t[n]=2\pi^{2}\cdot n(r), \tag{1}\]  
and  
\[u[n]\cong\frac{1}{2}\sum_{i=1}^{N}\frac{n\left(r_{i}^{\prime}\right)}{\|r-r_{ i}^{\prime}\|_{r\
eq r_{i}^{\prime}}}, \tag{2}\]  
where \(n(r)\) and \(r_{i}^{\prime}\) are a pixel intensity located at \(r\) and a location of a source point, respectively. The symbol \(r\) represents an observed point in the physical space. The factor, \(N\), is the data length and \(i\) is the location index of \(i\)th pixel point. Generally, the data length \(N=H\times W\), where \(H\) and \(W\) are the height and the width of an image, respectively. For image segmentation, a nonlinear combination of kinetic and potential energy density functionals is utilized to catch boundaries of particle-clusters, and then the result is mapped back into the original pixel space. Thus, a studied image can be divided into several featured components depending on the segmented results in the physical space. Physically, the theoretical form of a combination of kinetic energy and potential energy is called a Lagrangian, and it physically represents the actions of a physical system. Subtracting the potential energy from the kinetic energy formulates the mathematical form of the so-called action. Under the proposed framework, similarly, a Lagrangian density functional can be estimated by taking the kinetic energy density functional to subtract the potential one. A set of zero-points estimated form the Lagrangian density functional indicates a balance between the local significance and the local similarity. According to the potential energy density functional shown in Equation (2), \(u[n](r)\) is a global sum of the multiplication of the pixel intensity \(n\left(r_{i}^{\prime}\right)\) and the kernel \(1/\|r-r_{i}^{\prime}\|\). It means that the value of potential energy density functional at \(r\) is an average of \(n\left(r_{i}^{\prime}\right)\) from \(r_{i}^{\prime}\)s with respect to \(\|r-r_{i}^{\prime}\|\). Thus, the values of the potential energy density functionals will be smoothed by the averages estimated by the surrounding pixel intensities with respect to their corresponding distances. In other words, the profile of similarity of an image is smoothing. Thus, the balance between the local significance and the local similarity is equal to a zero-point estimated by taking a local weighting to subtract a local mean.