By introducing the methods of machine learning into the density functional theory, we made a detour for the construction of the most probable density function, which can be estimated by learning relevant features from the system of interest. Using the properties of universal functional, the vital core of density functional theory, the most probable cluster numbers and the corresponding cluster boundaries in a studying system can be simultaneously and automatically determined and the plausibility is erected on the Hohenberg-Kohn theorems. For the method validation and pragmatic applications, interdisciplinary problems from physical to biological systems were enumerated. The amalgamation of uncharged atomic clusters validated the unsupervised searching process of the cluster numbers and the corresponding cluster boundaries were exhibited likewise. High accurate clustering results of the Fisher’s iris dataset showed the feasibility and the flexibility of the proposed scheme. Brain tumor detections from low-dimensional magnetic resonance imaging datasets and segmentations of high-dimensional neural network imageries in the Brainbow system were also used to inspect the method practicality. The experimental results exhibit the successful connection between the physical theory and the machine learning methods and will benefit the clinical diagnoses.

Techniques of automatic medical image segmentation are the most important methods for clinical investigation, anatomic research, and modern medicine. Various image structures constructed from imaging apparatus achieve a diversity of medical applications. However, the diversified structures are also a burden of contemporary techniques. Performing an image segmentation with a tremendously small size (<25 pixels by 25 pixels) or tremendously large size (>1024 pixels by 1024 pixels) becomes a challenge in perspectives of both technical feasibility and theoretical development. Noise and pixel pollution caused by the imaging apparatus even aggravate the difficulty of image segmentation. To simultaneously overcome the mentioned predicaments, we propose a new method of medical image segmentation with adjustable computational complexity by introducing data density functionals. Under this theoretical framework, several kernels can be assigned to conquer specific predicaments. A square-root potential kernel is used to smoothen the featured components of employed images, while a Yukawa potential kernel is applied to enhance local featured properties. Besides, the characteristic of global density functional estimation also allows image compression without losing the main image feature structures. Experiments on image segmentation showed successful results with various compression ratios. The computational complexity was significantly improved, and the score of accuracy estimated by the Jaccard index had a great outcome. Moreover, noise and regions of light pollution were mostly filtered out in the procedure of image compression.