Elsevier

Chaos, Solitons & Fractals

Volume 124, July 2019, Pages 78-85
Chaos, Solitons & Fractals

Frontiers
An image hiding scheme in a 2-dimensional coupled map lattice of matrices

https://doi.org/10.1016/j.chaos.2019.04.038Get rights and content

Highlights

  • An image hiding scheme is based on a 2-dimensional coupled map lattice of matrices.

  • Scalar nodal variables at each node of the lattice are replaced by nilpotent matrices.

  • The digital image hiding scheme does not require the difference image.

Abstract

An image hiding scheme in a 2-dimensional coupled map lattice of matrices is presented in this paper. Scalar variables at each node of the lattice are replaced by nilpotent matrices. The spatiotemporal divergence process is employed to hide the secret digital image in the state map of the nodal variables. The presented image hiding scheme does not require the computation of the difference image between two patterns produced by the perturbed and the unperturbed initial conditions. Computational experiments are used to demonstrate the efficacy of the presented technique.

Introduction

Coupled map lattices (CML) play an important role in the study of the chaotic dynamics of spatially extended systems. A CML generally incorporates a finite number of coupled nodes. The major difference of a CML from a cellular automata network is that each node of the CML is dependent upon its neighbors relative to the coupling term in the recurrence equation [1]. CMLs have been used to generate, illustrate and describe such complex phenomena as spatial bifurcations, frozen chaos, spatio-temporal intermittency, global travelling waves [2], [3], [4].

Scalar variables at each node of the CML can be replaced by matrix variables [5]. It is shown in [5] that such models of CMLs of matrices (CMLM) can diverge if initial nodal matrices are nilpotent matrices. Moreover, such CMLMs can generate fractal patterns representing spatiotemporal divergence that can be controlled by the coupling parameter between the nodes [5].

Self-organization is a process where some form of order arises from interactions between parts of initially disordered systems. Self-organization occurs in a variety of physical (granular material, liquid crystals), biological (growth of colonies, animal markings), chemical systems (Turing patterns, reaction-diffusion systems) [6], [7], [8]. Self-organizing patterns are widely exploited in computer science and informatics – particularly for hiding and communicating secret visual images. A fingerprint is used as the initial condition for the evolution of a self-organizing pattern in a network of cellular automata with elements representing the reaction-diffusion processes [9]. Beddington–DeAngelis type predator-prey model with self- and cross-diffusion is exploited in a steganographic digital image communication system developed in [10], [11]. Spatial 2 × 2 games, atrial fibrillation models, non-diffusively coupled nonlinear maps, breaking spiral waves are used to generate self-organizing patterns and to hide and to communicate a secret digital image [12], [13], [14], [15].

An alternative digital image hiding technique based on Abelian sandpiles is proposed in [16]. Perturbations of the digital image are self-erased in this scheme. That allows to heal the corrupted image (if only the cover image is encoded into the sandpile attractor). On the contrary, image hiding schemes based on self-organizing patterns do not self-erase perturbations. Perturbations are amplified and do propagate throughout the domain due to the nonlinear evolution of the pattern [12], [13], [14], [15].

The features, operation principles, the information capacity of communication schemes based on self-organizing patterns are all different. The “No-free-lunch theorem” [17] implies that every particular communication scheme does possess some or another sort of deficiencies. For example, the scheme based on Beddington–DeAngelis model is computationally ineffective and requires at least 10 000 time-forward integration steps of a system of nonlinear partial differential equations [10]. The scheme based on spatial 2 × 2 games is not sensitive to local perturbations of an individual pixel [13]. The information capacity of a scheme based on non-diffusively coupled nonlinear maps is comparatively low [14]. The scheme based on breaking spiral waves requires an iterative correction of individual perturbations of pixels [15].

However, a general feature of all discussed communication schemes (based on self-organizing patterns) is based on the computational processing of the difference image which is computed as the XOR difference between a pattern produced by the perturbed and non-perturbed initial conditions. The objective of this paper is to exploit the effect of the spatiotemporal divergence produced by a CMLM in such a way that the interpretation of the secret image would not require the difference image.

Section snippets

The simplified nilpotent model of the logistic coupled map lattice of matrices

The classical Kaneko model of Coupled Map Lattices (CML) with periodic boundary conditions [18] is a paradigmatic model representing the complex dynamics of spatiotemporal chaos:x(t+1)(i)=(1ε)f(x(t)(i))+ε2(f(x(t)(i+1))+f(x(t)(i1)))where t is a discrete time step; i is the nodal point of the lattice (i=1,2,,N, N is the size of the CML); ε is the coupling parameter, f(x) is a scalar mapping function (usually set as the Logistic mapping function: f(x)=ax(1x); 0 ≤ a ≤ 4 [19]). Properties of

The perturbation of a single pixel

The initial Eigenvalues λ0(0)(i,j) in Eq. (3) can be randomly distributed in interval [0, 1] – but all nilpotent parameters µ(0)(i, j) must be set to 1 (this is required by the structure of the simplified nilpotent model of a CMLM [5]). The perturbation of a single pixel at coordinates (k, l) at t=0 will be considered in this section. Two options are available – the perturbation of λ0(0)(k,l) or a(k, l).

An image hiding algorithm based on the 2D CMLM

The schematic diagram of the image communication algorithm based on the 2D CMLM is depicted in Fig. 6. Initially, the original image is transformed into its dot-skeleton representation [10] (the density of dots is predefined by the information capacity of the algorithm). Then, parameters a(i, j) are set to a value corresponding to the onset of chaos of the Logistic map – and are perturbed at pixels corresponding to the dot-skeleton representation of the original image. The magnitude of

Concluding remarks

This paper presents an image hiding scheme based on the simplified nilpotent model of the 2D CMLM. The essential feature of this scheme is the ability to reveal the secret in a single image produced during the evolution of the 2D lattice. In other words, the scheme does not require computing a difference image between two patterns (one developed from the perturbed initial conditions, another – from the non-perturbed initial conditions).

It is important to observe that the proposed image hiding

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research was supported by the Business Support Fund of Kaunas University of Technology (AlgebMIS).

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