# Image Hiding Scheme Based on the Atrial Fibrillation Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model of the System

**Figure 1.**The diagram of interaction between cells within the network. Resting cells (black) and excited cells (white) interact in two directions with different probabilities. The resting cell becomes excited if it interacts with another excited cell. An excited cell (white) enters a refractory state (gray) for a time period τ.

**Figure 2.**Initial plane vertical wave is induced at left side by pacemaker cells (

**a**). However, one cell is dysfunctional and blocks the propagation of the wave front (

**b**). Due to the vertical connections between longitudinal cables, the resulting process forms a pattern (

**c**).

**Figure 3.**Initial cell state conditions (

**a**) and the resulting pattern after 20 iterations (

**b**). Parameter ν is set to $0.5$; $\tau =20$.

## 3. Self-Organizing Patterns

#### 3.1. Initial Conditions

**Figure 4.**Pseudorandom initial conditions generated sequentially by the Logistic map; the initial value of the Logistic map ${a}_{0}$ is set to 0.02.

#### 3.2. Parameters of the AF Model

- The cell excitation probability δ (in random initial conditions);
- The connection map with corresponding probability of transversal connections ν;
- Refractory period τ;
- The number of time-forward iterations n.

**Figure 5.**Comparison of self-organizing patterns (SOP) for different values of parameters. (

**a**) $\nu =0.1$, $\tau =20$, $n=20$, $\delta =0.001$ (

**b**) $\nu =0.9$, $\tau =20$, $n=20$, $\delta =0.001$ (

**c**) $\nu =0.5$, $\tau =20$, $n=10$, $\delta =0.001$ (

**d**) $\nu =0.5$, $\tau =20$, $n=20$, $\delta =0.001$ (

**e**) $\nu =0.5$, $\tau =20$, $n=30$, $\delta =0.001$.

**Figure 6.**Iterative processes at $\nu =0.5$, $\tau =10$, $\delta =0.1$. (

**a**) $n=1$ (

**b**) $n=2$ (

**c**) $n=3$ (

**d**) $n=4$ (

**e**) $n=5$ (

**f**) $n=6,7,8,9,10$ (

**g**) $n=11$.

## 4. A Communication Scheme Based on Self-Organizing Patterns

- Sender generates pseudo-random matrices of the initial cell excitation states and the connection map by using the Logistic map with the initial value ${a}_{0}$; the size of the matrix is set to ${L}_{x}\times {L}_{y}$; the parameters of the AF model are ν and δ. Initial random matrix is dichotomous (cells contain binary values 0 or 1); connection between cells are random.
- Sender modifies the pseudo-random matrix of initial cell excitation states by inverting pixels corresponding to the dot-skeleton representation of the secret image.
- Sender executes n time-forward iterations starting from the modified matrix of initial conditions and sends the image of the self-organized pattern to the Receiver.
- Receiver generates the identical copy of pseudo-random matrices of initial cell excitation states and the connection map by using the chaotic Logistic map with the initial value ${a}_{0}$ and parameters of the AF model (ν and δ); the size of the matrix is ${L}_{x}\times {L}_{y}$ (this is an identical copy of the matrix generated by the Sender in Step 1).
- Receiver executes n time-forward iterations starting from the non-modified initial conditions.
- Finally, the difference between the digital images of the patterns produced by the non-modified and modified initial conditions reveals the secret.

**Figure 7.**Big clusters of initially excited cells result in interpretable self-organized patterns ($\nu =0.5$, $n=\tau =20$, $\delta =0.01$). (

**a**) The secret image (

**b**) Initially excited cells (shown in white) (

**c**) The self-organized pattern (

**d**) The difference image (

**e**) The binarised initially excited cells.

**Figure 8.**The initial dot-skeleton image (

**a**) and SOP at different δ: the first row represents initial conditions; the second row—patterns; the third row—difference images. (

**a**) $\delta =0.001$ (

**b**) $\delta =0.01$ (

**c**) $\delta =0.1$ (

**d**) $\delta =0.2$.

**Figure 9.**The flow chart diagram of the communication algorithm. Original image (

**a**); dot-skeleton representation (

**b**); initial conditions (

**c**); perturbed initial conditions (

**d**); cover image (

**e**); perturbed self-organizing pattern (

**f**); perturbed cover image (

**g**); self-organizing pattern (

**h**); difference image (

**i**).

## 5. The Sensitivity of the Communication Scheme to Perturbations of System Parameters

**Figure 10.**The sensitivity of the communication scheme to the perturbation of system’s parameters. The difference images are shown when: (

**a**) ${a}_{0}^{ex}=0.021$ is used instead of $0.02$; (

**b**) ${a}_{0}^{con}=0.021$ is used instead of $0.02$; (

**c**) $n=21$ is used instead of 20; (

**d**) $\tau =19$ is used instead of 20.

## 6. Concluding Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Suzuki, Y.; Takayama, T.; Motoike, I.N.; Asai, T. Striped and spotted pattern generation on reaction-diffusion cellular automata—Theory and LSI implementation. Int. J. Unconv. Comput.
**2007**, 3, 1713–1719. [Google Scholar] - Saunoriene, L.; Ragulskis, M. Secure steganographic communication algorithm based on self-organizing patterns. Phys. Rev. E
**2011**, 84, 056213. [Google Scholar] [CrossRef] [PubMed] - Ishimura, K.; Komuro, K.; Schmid, A.; Asai, T.; Motomura, M. Image steganography based on reaction diffusion models toward hardware implementation. NOLTA
**2014**, 5, 456–465. [Google Scholar] [CrossRef] - Ziaukas, P.; Ragulskis, T.; Ragulskis, M. Communication scheme based on evolutionary spatial games. Phys. A Stat. Mech. Appl.
**2014**, 403, 177–188. [Google Scholar] [CrossRef] - Vaidelys, M.; Ziaukas, P.; Ragulskis, M. Competitively coupled maps for hiding secret visual information. Phys. A Stat. Mech. Appl.
**2016**, 443, 91–97. [Google Scholar] [CrossRef] - Luke, R.A.; Saffitz, J.E. Remodeling of ventricular conduction pathways in healed canine infarct border zones. J. Clin. Investig.
**1991**, 87, 1594–1602. [Google Scholar] [CrossRef] [PubMed] - Verheule, S.; Wilson, E.; Everett, T.; Shanbhag, S.; Golden, C.; Olgin, J. Alterations in Atrial Electrophysiology and Tissue Structure in a Canine Model of Chronic Atrial Dilatation Due to Mitral Regurgitation. Circulation
**2003**, 107, 2615–2622. [Google Scholar] [CrossRef] [PubMed] - Nakamura, K.; Funabashi, N.; Uehara, M.; Ueda, M.; Murayama, T.; Takaoka, H.; Komuro, I. Left atrial wall thickness in paroxysmal atrial fibrillation by multislice-CT is initial marker of structural remodeling and predictor of transition from paroxysmal to chronic form. Int. J. Cardiol.
**2011**, 148, 139–147. [Google Scholar] [CrossRef] [PubMed] - Clayton, R.; Bernus, O.; Cherry, E.; Dierckx, H.; Fenton, F.; Mirabella, L.; Panfilov, A.; Sachse, F.; Seemann, G.; Zhang, H. Models of cardiac tissue electrophysiology: Progress, challenges and open questions. Prog. Biophys. Mol. Biol.
**2011**, 104, 22–48. [Google Scholar] [CrossRef] [PubMed] - Clayton, R.H. Computational models of normal and abnormal action potential propagation in cardiac tissue: Linking experimental and clinical cardiology. Physiol. Meas.
**2001**, 22, R15–R34. [Google Scholar] [CrossRef] [PubMed] - Christensen, K.; Manani, K.A.; Peters, N.S. Simple Model for Identifying Critical Regions in Atrial Fibrillation. Phys. Rev. Lett.
**2015**, 114, 028104. [Google Scholar] [CrossRef] [PubMed] - May, R.M. Simple mathematical models with very complicated dynamics. Nature
**1976**, 261, 459–467. [Google Scholar] [CrossRef] [PubMed] - Wang, X.; Luan, D. A novel image encryption algorithm using chaos and reversible cellular automata. Commun. Nonlinear Sci. Numer. Simul.
**2013**, 18, 3075–3085. [Google Scholar] [CrossRef] - Pareek, N.; Patidar, V.; Sud, K. Image encryption using chaotic logistic map. Image Vis. Comput.
**2006**, 24, 926–934. [Google Scholar] [CrossRef] - Kocarev, L.; Jakimoski, G. Logistic map as a block encryption algorithm. Phys. Lett. A
**2001**, 289, 199–206. [Google Scholar] [CrossRef] - Fridrich, J.; Goljan, M.; Du, R. Reliable detection of LSB steganography in color and grayscale images. In Proceedings of the 2001 Workshop on Multimedia and Security, New Challenges, Ottawa, ON, Canada, 5 October 2001; pp. 27–30.

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**MDPI and ACS Style**

Vaidelys, M.; Ragulskiene, J.; Ziaukas, P.; Ragulskis, M.
Image Hiding Scheme Based on the Atrial Fibrillation Model. *Appl. Sci.* **2015**, *5*, 1980-1991.
https://doi.org/10.3390/app5041980

**AMA Style**

Vaidelys M, Ragulskiene J, Ziaukas P, Ragulskis M.
Image Hiding Scheme Based on the Atrial Fibrillation Model. *Applied Sciences*. 2015; 5(4):1980-1991.
https://doi.org/10.3390/app5041980

**Chicago/Turabian Style**

Vaidelys, Martynas, Jurate Ragulskiene, Pranas Ziaukas, and Minvydas Ragulskis.
2015. "Image Hiding Scheme Based on the Atrial Fibrillation Model" *Applied Sciences* 5, no. 4: 1980-1991.
https://doi.org/10.3390/app5041980