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In this study, we propose a novel discrete-time coupled model to generate oscillatory responses via periodic points with a high periodic order. Our coupled system comprises one-dimensional oscillators based on the Rulkov map and a single globally coupled oscillator. Because the waveform of a one-dimensional oscillator has sharply defined peaks, the coupled system can be applied to dynamic image segmentation. Our proposed system iteratively transforms the coupling of each oscillator based on an input value that corresponds to the pixel value of an input image. This approach enables our system to segment image regions in which pixel values gradually change with respect to a connected region. We conducted a bifurcation analysis of a single oscillator and a three-coupled model. Through simulations, we demonstrated that our system works well for gray-level images with three isolated image regions.

Image segmentation is one of the most important techniques used in image processing. Many studies have addressed methods of improving the accuracy and effectiveness of image segmentation using various approaches [

To address these problems, discrete-time dynamical systems are used as an alternative approach for simulating coupled oscillators. Zhao et al. proposed a model that used a network of coupled logistic maps to achieve multi scale image segmentation [

In contrast with these methods, we previously proposed a discrete-time coupled model that can generate oscillatory responses via periodic points with a high periodic order [

Our system has a network structure in which two-dimensional (2D) oscillators, based on chaotic neurons [

In this study, we investigated a novel discrete-time coupled model comprising one-dimensional oscillators based on the Rulkov map [

In this section, we present the architecture of our proposed discrete-time coupled model with adaptive coupling.

The coupled model comprises a global oscillator and N one-dimensional oscillators, where N denotes the number of pixels in an input image. With the exception of the global oscillator, the one-dimensional oscillators are arranged on the grid so that each corresponds to a pixel. They are connected to the eight neighboring oscillators with similar pixel values. The global oscillator connects all the other oscillators and acts as a relay between them. Oscillators with similar pixel values in the eight-neighborhood connection are coupled together.

Here,

and

where

Function f is based on the Rulkov map, where h denotes the effect of the global oscillator on each of the other oscillators, and k and d in (2) are tunable system parameters.

where

where

The dynamics of an N-coupled system are described by the P-dimensional discrete-time dynamical system (

or, equivalently, by an iterated map defined by

where

where

The coupling of oscillators defined by (5) is uniformly based on the pixel value of the input image. We replaced this fixed coupling with an adaptive coupling based on the clustering method proposed in [

where

In this section, we describe our analysis, in which we used qualitative bifurcation theory and the order parameter. Note that these analyses must be used to determine the optimum system parameters for dynamic image segmentation, but do not need to be applied every time an image is input.

In our bifurcation analysis, the point

becomes a fixed point of

where

Next, we considered the topological classification of a hyperbolic fixed point. Let

Bifurcation sets of a fixed point were computed by solving the simultaneous Equations (15) and (16). For the numerical determination [

To investigate the bifurcation phenomena in (2), we used finite-time Lyapunov exponents in which local expansion rates are defined by

Here,

To investigate the relationship between the coupling coefficients and the phase difference of oscillators (in- phase or out-of-phase), we used the order parameter defined by

where

which represents the phase difference between

We first investigated the bifurcation of the fixed point observed in a single oscillator defined by (2) with no connections. Next, we analyzed the coupled model corresponding to the input image shown in

with

regions could be controlled by adjusting parameter a.

Simulation were used to demonstrate that dynamic image segmentation could be achieved using our adaptive coupled model with appropriate parameter values. The parameter values were set as follows:

In this study, we proposed a novel discrete-time coupled model for use in dynamic image segmentation. The mechanisms underlying the generation of oscillatory responses in a single oscillator were revealed by a bifurcation analysis. We also investigated the bifurcation sets for the fixed point observed in a three-coupled model. Using order parameters to show the phase differences between the oscillators, we elucidated the relationship between the oscillatory responses and the coupling coefficients of oscillators in the three-coupled model. We used this bifurcation analysis to set appropriate parameter values and applied our model to dynamic image segmentation. Data from simulations demonstrate that our proposed model is capable of segmenting regions of a gray-level image in which the pixel values change gradually. In future work, we will analyze our proposed model in greater detail, for example, by applying it to input images with more isolated image regions.

Mio Kobayashi,Tetsuya Yoshinaga, (2016) Discrete-Time Dynamic Image Segmentation Using Oscillators with Adaptive Coupling. International Journal of Modern Nonlinear Theory and Application,05,93-103. doi: 10.4236/ijmnta.2016.52010