Image segmentation based on self-organization

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

The popular reaction–diffusion model for the negative bias temperature instability is discussed from the viewpoint of stochastic chemical kinetics. We present a microscopic formulation of the reaction–diffusion model based on the reaction–diffusion master equation and solve it using the stochastic simulation algorithm. The calculations are compared to the macroscopic version as well as established experimental data. The degradation predicted by the microscopic reaction–diffusion model strongly deviates from the macroscopic version and the experimentally observed behavior. Those deviations are explained as necessary consequences of the physical processes involved. The presented results show the impact of the unphysical assumptions in the reaction–diffusion model. Further, we generally question the suitability of the mathematical framework of reaction rate equations for a reactive-diffusive system at the given particle densities.

In the area of musculoskeletal MR images analysis, the image denoising plays an important role in enhancing the spatial image area for further processing. Recent studies have shown that non-local means (NLM) methods appear to be more effective and robust when compared with conventional local statistical filters, including median or average filters, when Rician noise is presented. A significant limitation of NLM is the fact that thy have the tendency to suppress tiny objects, which may represent clinically important information. For this reason, we provide an extensive quantitative and objective analysis of a novel NLM algorithm, taking advantage of pixel and patch similarity information with the optimization procedure for optimal filter parameters selection to demonstrate a higher robustness and effectivity, when comparing with NLM and conventional local means methods, including average and median filters. We provide extensive testing on variable noise generators with dynamical noise intensity to objectively demonstrate the robustness of the method in a noisy environment, which simulates relevant, variable and real conditions. This work also objectively evaluates the potential and benefits of the application of NLM filters in contrast to conventional local-mean filters. The final part of the analysis is focused on the segmentation performance when an NLM filter is applied. This analysis demonstrates a better performance of tissue identification with the application of smoothing procedure under worsening image conditions.

Asymptotic stability results of generalized discrete time reaction diffusion system applied to Lengyel-Epstein and Dagn Harrison models

Particle-based Brownian dynamics simulations offer the opportunity to not only simulate diffusion of particles but also the reactions between them. They therefore provide an opportunity to integrate varied biological data into spatially explicit models of biological processes, such as signal transduction or mitosis. However, particle based reaction-diffusion methods often are hampered by the relatively small time step needed for accurate description of the reaction-diffusion framework. Such small time steps often prevent simulation times that are relevant for biological processes. It is therefore of great importance to develop reaction-diffusion methods that tolerate larger time steps while maintaining relatively high accuracy. Here, we provide an algorithm, which detects potential particle collisions prior to a BD-based particle displacement and at the same time rigorously obeys the detailed balance rule of equilibrium reactions. We can show that for reaction-diffusion processes of particles mimicking proteins, the method can increase the typical BD time step by an order of magnitude while maintaining similar accuracy in the reaction diffusion modelling.

Evaluating Denoising Performances of Fundamental Filters for T2-Weighted MRI Images

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Numerical bifurcation analysis and pattern formation in a minimal reaction–diffusion model for vegetation

We generalize our recent discrete cellular automata (CA) model of excitable media [Y. B. Chernyak, A. B. Feldman, and R. J. Cohen, Phys. Rev. E 55, 3215 (1997)] to incorporate the effects of inhibitory processes on the propagation of the excitation wave front. In the common two variable reaction-diffusion (RD) models of excitable media, the inhibitory process is described by the $v$ ``controller'' variable responsible for the restoration of the equilibrium state following excitation. In myocardial tissue, the inhibitory effects are mainly due to the inactivation of the fast sodium current. We represent inhibition using a physical model in which the ``source'' contribution of excited elements to the excitation of their neighbors decreases with time as a simple function with a single adjustable parameter (a rate constant). We sought specific solutions of the CA state transition equations and obtained (both analytically and numerically) the dependence of the wave-front speed $c$ on the four model parameters and the wave-front curvature $\ensuremath{\kappa}$. By requiring that the major characteristics of $c(\ensuremath{\kappa})$ in our CA model coincide with those obtained from solutions of a specific RD model, we find a unique set of CA parameter values for a given excitable medium. The basic structure of our CA solutions is remarkably similar to that found in typical RD systems (similar behavior is observed when the analogous model parameters are varied). Most notably, the ``turn-on'' of the inhibitory process is accompanied by the appearance of a solution branch of slow speed, unstable waves. Additionally, when $\ensuremath{\kappa}$ is small, we obtain a family of ``eikonal'' relations $c(\ensuremath{\kappa})$ that are suitable for the kinematic analysis of traveling waves in the CA medium. We compared the solutions of the CA equations to CA simulations for the case of plane waves and circular (target) waves and found excellent agreement. We then studied a spiral wave using the CA model adjusted to a specific RD system and found good correspondence between the shapes of the RD and CA spiral arms in the region away from the tip where kinematic theory applies. Our analysis suggests that only four physical parameters control the behavior of wave fronts in excitable media.

This is a pedagogical account on reaction-diffusion systems and their relationship with integrable quantum spin chains. Reaction-diffusion systems are paradigmatic examples of non-equilibrium systems. Their long-time behaviour is strongly influenced through fluctuation effects in low dimensions which renders the habitual mean-field cinetic equations inapplicable. Starting from the master equation rewritten as a Schr\"odinger equation with imaginary time, the associated quantum hamiltonian of certain one-dimensional reaction-diffusion models is closely related to integrable magnetic chains. The relationship with the Hecke algebra and its quotients allows to identify integrable reaction-diffusion models and, through the Baxterization procedure, relate them to the solutions of Yang-Baxter equations which can be solved via the Bethe ansatz. Methods such as spectral and partial integrability, free fermions, similarity transformations or diffusion algebras are reviewed, with several concrete examples treated explicitly. An outlook on how the recently-introduced concept of local scale invariance might become useful in the description of non-equilibrium ageing phenomena is presented, with particular emphasis on the kinetic Ising model with Glauber dynamics.

We previously showed that the adult ocellated lizard skin colour pattern is effectively generated by a stochastic cellular automaton (CA) of skin scales. We additionally suggested that the canonical continuous 2D reaction-diffusion (RD) process of colour pattern development is transformed into this discrete CA by reduced diffusion coefficients at the borders of scales (justified by the corresponding thinning of the skin). Here, we use RD numerical simulations in 3D on realistic lizard skin geometries and demonstrate that skin thickness variation on its own is sufficient to cause scale-by-scale coloration and CA dynamics during RD patterning. In addition, we show that this phenomenon is robust to RD model variation. Finally, using dimensionality-reduction approaches on large networks of skin scales, we show that animal growth affects the scale-colour flipping dynamics by causing a substantial decrease of the relative length scale of the labyrinthine colour pattern of the lizard skin.

The stability and slow dynamics of spot patterns in the 2D Brusselator model: The effect of open systems and heterogeneities

Spatially localized 1D spike patterns occur for various two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. A competition instability of a steady-state spike pattern is a linear instability that locally preserves the sum of the heights of the spikes. This instability, which results from a zero-eigenvalue crossing of a nonlocal eigenvalue problem at a certain critical value of the inhibitor diffusivity, has been implicated from full PDE numerical simulations of various RD systems of triggering a nonlinear event leading to spike annihilation. As a result, this linear instability is believed to be a key mechanism for initiating a coarsening process of 1D spike patterns. As an extension of the linear theory, we develop and implement a weakly nonlinear theory to analyse competition instabilities associated with symmetric two-boundary spike equilibria on a finite 1D domain for the Gierer–Meinhardt and Schnakenberg RD models. Two symmetric boundary spikes interacting through a long-range bulk diffusion field is the simplest spatial configuration of interacting localized spikes that can undergo a competition instability. Within a neighborhood of the parameter value for the competition instability threshold, a multi-scale asymptotic expansion is used to derive an explicit amplitude equation for the heights of the boundary spikes. This amplitude equation confirms that the competition instability is subcritical and, moreover, it shows that the competition instability threshold corresponds to a symmetry-breaking bifurcation point where an unstable branch of asymmetric two-boundary spike equilibria emerges from the symmetric branch. Results from our weakly nonlinear analysis are confirmed from full numerical solutions of the steady-state problem using numerical bifurcation software.

Biologically inspired algorithms have attracted a large number of researchers and found to have application in many areas of computer science including image processing. This paper presents the application of a biologically inspired algorithm, bacterial foraging optimization algorithm (BFOA), to optimize the filtering of images for denoising and its performance comparison with existing noise reduction techniques like average filter. Usually, images suffer from noises which will corrupt image quality and appearance. Hence, denoising of images plays a great role in the image processing and the frequently used filters are average filter, median filter, etc., to remove noises. This research paper explores the suitability of applying BFOA on the filtered image produced by average filter to result a further denoised image. In this proposed method of bacterial foraging-based optimization, peak signal-to-noise ratio (PSNR) is used as fitness function to denoise the noisy images. The implemented code is tested for noisy images (Gaussian noise and salt–pepper noise) filtered with average filter, and results show the optimization capability of BFOA-based method and that it improves the denoised images produced by average filter.

In this paper, the spatial, temporal and spatiotemporal dynamics of a reaction–diffusion predator–prey system with mutual interference described by the Crowley–Martin-type functional response, under homogeneous Neumann boundary conditions, are studied. Preliminary analysis on the local asymptotic stability and Hopf bifurcation of the spatially homogeneous model based on ordinary differential equations is presented. For the reaction–diffusion model, firstly the invariance, uniform persistence and global asymptotic stability of the coexistence equilibrium are discussed. Then it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Next it is proved that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Furthermore, at the points where the Turing instability curve and Hopf bifurcation curve intersect, it is demonstrated that the model undergoes Turing–Hopf bifurcation and exhibits spatiotemporal patterns. Finally, the existence and non-existence of positive non-constant steady states of the reaction–diffusion model are established. Numerical simulations are presented to verify and illustrate the theoretical results.