A kinetic approach to consensus-based segmentation of biomedical images

TL;DR

This work introduces a kinetic, consensus-based framework for biomedical image segmentation, where pixels are modeled as interacting particles with time-dependent positions $\mathbf{x}_i$ and static grayscale features $c_i$. By deriving a Boltzmann-type equation and its quasi-invariant scaling to a surrogate Fokker-Planck form, the authors enable efficient direct simulation Monte Carlo (DSMC) parameter identification and segmentation via cluster means of gray levels. The approach is validated on HL60 nuclei, brain tumor, and thigh muscle MRI datasets, with high Dice scores achieved in several tasks and a patch-based extension improving challenging cases. The study also demonstrates the influence of diffusion functions $D(c)$ on segmentation performance and provides a data-driven workflow for optimizing segmentation parameters without heavy supervised learning assumptions.

Abstract

In this work, we apply a kinetic version of a bounded confidence consensus model to biomedical segmentation problems. In the presented approach, time-dependent information on the microscopic state of each particle/pixel includes its space position and a feature representing a static characteristic of the system, i.e. the gray level of each pixel. From the introduced microscopic model we derive a kinetic formulation of the model. The large time behavior of the system is then computed with the aid of a surrogate Fokker-Planck approach that can be obtained in the quasi-invariant scaling. We exploit the computational efficiency of direct simulation Monte Carlo methods for the obtained Boltzmann-type description of the problem for parameter identification tasks. Based on a suitable loss function measuring the distance between the ground truth segmentation mask and the evaluated mask, we minimize the introduced segmentation metric for a relevant set of 2D gray-scale images. Applications to biomedical segmentation concentrate on different imaging research contexts.

Figures (13)

• Figure 1: Results of the Hegselmann-Krause bounded confidence model for three different values of the $\Delta$ threshold. At the initial time we selected $N=100$ particles equally spaced in $[-1,1]$.
• Figure 2: Transient solutions of the Fokker-Planck equation \ref{['eq:FP']} approximated through a semi-implicit SP scheme over $[0,T]$, $T=50$, with time step $\Delta t = 3 \cdot 10^{-1}$. In the left panel we consider $\Delta_1=2$, $\Delta_2=1$ and $\sigma^2= 5 \cdot 10^{-2}$ while in the right panel $\Delta_1=0.5$, $\Delta_2=1$ and $\sigma^2= 10^{-2}$. The initial distribution is \ref{['eq:f0']}.
• Figure 3: Asymptotic solutions of the Fokker-Plank equation numerically computed with the SP scheme at the final time $T=50$. The left column represents the $xy$-projections, the middle column the $xc$-projections and the right column the $yc$-projections. In the top row we use $\Delta_1=2$, $\Delta_2=1$ and $\sigma^2= 5 \cdot 10^{-2}$ while in the bottom row $\Delta_1=0.5$, $\Delta_2=1$ and $\sigma^2= 10^{-2}$.
• Figure 4: Comparison between the numeric solution of the Fokker-Planck equation computed by the SP scheme (in blue) with final distribution provided by the MC algorithm for the Boltzmann-type equation with two different values of the parameter $\epsilon$ (in red and green). Both the panels represent the $x$-projections of the asymptotic distributions computed for $T=50$. In the left panel we use $\Delta_1=2$, $\Delta_2=1$ and $\sigma^2= 5 \cdot 10^{-2}$ while in the right panel $\Delta_1=0.5$, $\Delta_2=1$ and $\sigma^2= 10^{-2}$. The green distribution is computed with $\epsilon=10^{-1}$ and the red one with $\epsilon=10^{-2}$.
• Figure 5: Summary of the segmentation process. The first panel shows the input image; the second panel displays the multi-level segmentation mask produced by DSMC Algorithm \ref{['alg:Boltz']}; the third panel represents the mask after the binarization process; and the fourth panel shows the binary mask after the two morphological refinement steps.