Medical Image Segmentation with Adjustable Computational Complexity Using Data Density Functionals

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.

Potential kernel, hitting probabilities and distributional asymptotics

$\mathbb{Z}^{d}$-extensions of probability-preserving dynamical systems are themselves dynamical systems preserving an infinite measure, and generalize random walks. Using the method of moments, we prove a generalized central limit theorem for additive functionals of the extension of integral zero, under spectral assumptions. As a corollary, we get the fact that Green–Kubo’s formula is invariant under induction. This allows us to relate the hitting probability of sites with the symmetrized potential kernel, giving an alternative proof and generalizing a theorem of Spitzer. Finally, this relation is used to improve, in turn, the assumptions of the generalized central limit theorem. Applications to Lorentz gases in finite horizon and to the geodesic flow on Abelian covers of compact manifolds of negative curvature are discussed.

Sharp estimates of the potential kernel for the harmonic oscillator with applications

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that theLp–Lqestimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.

Sharp estimates of the potential kernel for the harmonic oscillator with applications

We prove qualitatively sharp estimates of the potential kernel for the harmonic oscillator. These bounds are then used to show that the Lp–Lq estimates of the associated potential operator obtained recently by Bongioanni and Torrea are in fact sharp.