Blind source separation with adaptive learning rates for image encryption

Abstract

In this study, we present a technique for image encryption problem based on the underdetermined blind source separation (BSS) principle with adaptive learning rates. The encryption process is transferred as the underdetermined BSS problem and is treated by means of the key images to achieve decryption. By properly generating the key images and constructing the underdetermined mixing matrix, the proposed BSS technique can achieve the security. The proposed BSS with adaptive learning rates approach is implemented by the interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC) and particle swarm optimization. The T2FCMAC system is a more generalized system with better learning ability to provide the adaptive learning rate of the BSS. Besides, the particle swarm optimization is utilized to enhance the performance of convergence. Computer simulation results are shown to illustrate the effectiveness of the proposed approach.

Keywords

Blind source separation image encryption fuzzy systems particle swarm optimization

1 Introduction

With the widespread use of the computer networks, the security of image transmission has become an import topic. The image encryptions are categorized as encryption without compression, encryption combining compression [4]. Recently, Lin et al. introduced the BSS for the image encryption [23 –26]. These articles utilized the intractability of the underdetermined BSS problem to present the BSS-based encryption by properly constructing the underdetermined mixing matrix for encryption [23 –26]. The basic idea is to mix multiple plaintexts with a number of secret key signals, in the hope that an attacker has to solve the underdetermined BSS problems. On the decryption side, if we have the information of key images, the original images can be separated well by the BSS. In addition, in order to reduce the complex calculation of the BSS for plenty image data, the pre-processing is considered to the mixed images in the BSS. In this paper, we propose a novel BSS with gradient algorithm which adopts the T2FCMAC-based adaptive learning rates technique for solving the image encryption.

In recent years, the BSS problem has received attentions from the signal processing community and the neural network community. It has become an active research area in both statistical signal processing and unsupervised neural learning [2 , 31–33]. The meaning of “blind” is that both the original sources and the way the sources were mixed are all unknown, and only mixed signals or mixtures can be measured and observed. The objective of the BSS is to recover the unknown source signals from observed mixed signals without knowing the mixing coefficients. The well-known independent component analysis (ICA) approach has been successfully applied to the BSS. The related BSS methods are also applied to many fields such as communications and biomedical engineering [6 , 33].

In literature [19, 27], a fuzzy-based learning rate determination was proposed for solving the BSS problems. The fuzzy-based method performs good signal separation but the membership functions and the fuzzy rule should be set up and need expert’s experience. In order to make the learning rate selection more flexible and perform better, we adopt a novel interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC) to adjust the learning rate for the BSS [29]. In this study, the T2FCMAC based-learning rate adjustment methods for the BSS are extended to image encryption problems. The T2FCMAC a generalization neural network and has better learning ability and lower computational complexity. According to the robust characteristics, the learning rate in the BSS can be selected by T2FCMAC system adaptively. Furthermore, to enhance the convergent performance of BSS problems, the particle swarm optimization (PSO) is adopted to adjust the parameters in the T2FCMAC system. To overcome the image encryption problem using BSS principle, the encryption process is formulated as the underdetermined BSS problem, and the key-images are introduced to achieve the decryption. Thus, the proposed approach has the ability of image encryption.

The rest of the paper is organized as follows. Section 2 gives the brief introduction of the underdetermined BSS principle. In Section 3, the image encryption system based on BSS is introduced. The proposed image encryption using T2FCMAC is introduced in Section 4. Simulation results are shown in Section 5. Finally, conclusion will be given.

2 Underdetermined BSS principle

Blind source separation (BSS) is a technique that recovers a set of unobserved signals from some observed mixtures [3]. Assume that there are n observed signals X (t) = [x₁ (t) , x₂ (t) , … , x_n (t)] ^T by a set of n sensors. The observed signals X (t) are mixed by full-rank n × m mixing matrix A with m unknown sources S (t) = [s₁ (t) , s₂ (t) , … , s_m (t)] ^T, thus the mixed model is defined as $X (t) = AS (t) .$ (1)

The goal of the BSS is to recover the unknown source signals S (t) from the observed mixed signals X (t) without knowing the mixing matrix A. There exists a separated model defined as $Y (t) = W (t) X (t)$ (2)where Y (t) = [y₁ (t) , y₂ (t) , … , y_m (t)] ^T is separated signals, W (t) is m × n separating matrix, and t is time instant. Signals Y (t) are separated from mixed signals X (t) by separating matrix W (t). According to Equations (1) and (2), we have $Y (t) = W (t) X (t) = W (t) AS (t) = B (t) S (t) .$ (3)

The ideal condition is W (t) = A^-1 if n = m then we have Y (t) = S (t). Consequently, the source signals can be separated from the mixed signals by finding the separating matrix W (t).

For the BSS, there are some restrictions about the source signals. One of the restrictions is that the number of observed signals (sensor number) n should not be smaller than the number of source signals m (n ≥ m). If the sensor number is smaller than source number (n < m), this is the so-called underdetermined problem and it is difficult for BSS [6, 24]. However, the underdetermined problem is treated as an advantage to establish the security of many cryptosystems [9 , 23–26]. The secret key signals introduced by Lin et al. make the determination of signals become an underdetermined problem that the key signals is unknown in the BSS [23 –26]. In other words, there is an underdetermined problem, less information of the mixed signals (n < m), in the BSS for the cryptosystem, but the problem can be solved if we have the key signals to make the mixed signals to be complete (n ≥ m).

Many independent component analysis (ICA) algorithms have been proposed to solve the BSS problem [7 , 29–33]. The ICA approaches measure the non-Gaussian signals to find the independent component. That is, the mixed signals can be separated well by ICA algorithm if the source signals are independent to each other. The gradient algorithm is one of popular methods for ICA and it is usually used to update the separating matrix W with $W (t + 1) = W (t) + η (t) \frac{\partial J (W)}{\partial W} | W = W (t)$ (4)where η (t) is the learning rate, and J (·) is the objective function of ICA. In this paper, the mutual information is chosen as the objective function which will be introduced in Section 3). Herein, the equivariant adaptive separation via independence (EASI) algorithm is used [25] $\begin{matrix} W (t + 1) = W (t) + η (t) [I - f (Y (t)) Y^{T} (t) \\ + Y (t) f^{T} (Y (t)) - Y (t) Y^{T} (t)] W (t) \end{matrix}$ (5)where I is identity matrix, f (Y (t)) = Y³ (t) is a nonlinear transformation function in array power. In order to avoid the amplitude of the separated signals Y (t) over magnifying, we normalize each vector w_i of the separating matrix W (t) as ${\hat{w}}_{i} = \frac{w_{i}}{∥ w_{i} ∥}$ (6)where w_i = [w_i1, w_i2, …, w_in], ∥ .∥ denotes Euclidean norm operation. The separating matrix W will be replaced by $W = [w_{1}^{T}, w_{2}^{T}, \dots, w_{n}^{T}]^{T}$ .

In this paper, we develop a novel BSS with gradient algorithm which adopts the adaptive learning rates technique for solving the image encryption, i.e., η (t) is time-varying and provides by the T2FCMAC and optimized by PSO algorithm.

3 BSS-based image encryption system

In this section, we introduce the BSS-based image encryption system. In literature [23 –26], Lin et al. construct the intractable underdetermined BSS problem in encryption that can only be solved with the key signals in decryption. According to the BSS system, there are source images S (t), mixed images X (t), and separated images Y (t) for the encryption system. In addition, the key images K (t) are used to protect the security in the image system. The data of key images can be generated by some pseudorandom method. To simplify, we generate the key images data randomly. The proposed image encryption system based on the BSS is shown in Fig. 1 and the models of encryption, pre-process, and decryption in image are introduced as follows.

3.1 Encryption

Assume that there are m original images S (t) = [s₁ (t) , s₂ (t) , …, s_m (t)], t = 1, 2, … , T where T is the size (data length) of each image. In order to protect the source images, the key images number is the same as m, i.e., K (t) = [k₁ (t) , k₂ (t) , …, k_m (t)]. Therefore, the encrypted model $X (t) = A_{e} S^{2 m} (t)$ (7)where X (t) = [x₁ (t) , x₂ (t) , …, x_m (t)] are the encrypted images, S^2m (t) = [s₁ (t) , s₂ (t) , …, s_m (t) , k₁ (t) , k₂ (t) , …, k_m (t)] are the source images, and $A_{e} \in R^{m \times 2 m}$ is the underdetermined mixing matrix defined as $A_{e} = [\begin{matrix} A & β A \end{matrix}]$ (8)where A is the mixing matrix, β is a scalar to adjust the degree of encryption. Obviously, the encrypted model (7) presents the underdetermined problem of the BSS since source images number (2m) is great than the number of mixed images (m). In addition, the underdetermined mixing matrix A_e not only masks the original images with key images in the encrypted images but also makes the original images immune from the BSS without the key images. Therefore, the underdetermined BSS problem ensures the security of the encrypted images. Note that β ≥ 10 to ensure the key images recovering over the mixed images [24].

3.2 Pre-process for encrypted images

In order to reduce the complex calculation of the BSS, especially for lots of image signals, the pre-process of the mixed signal matrix X is adopted [27]. The pre-process translates the correlative matrix X into uncorrelated matrix Z using centering and whitening. The first step-centering is $x (t) = x (t) - E {x (t)}$ (9)where t = 1, 2, …, T. After centering, the matrix X becomes to be zero mean signals. The second step-whitening is to have elements of matrix X are uncorrelated. The method utilizes the eigenvalue decomposition defined as $E {{XX}^{T}} = C_{x} = E Λ E^{T}$ (10)where C_x is the covariance matrix of X, E is the orthogonal matrix of eigenvectors of C_x, and Λ is the diagonal matrix of eigenvalues of C_x. The whitening process is expressed as $Z = PX$ (11)where Z is the corresponding uncorrelated matrix and P is the whitening matrix defined as $P = Λ^{- \frac{1}{2}} E^{T} .$ (12)

After whitening, the elements in the matrix Z are uncorrelated, that is,

$\begin{matrix} E {{ZZ}^{T}} & = P E {{XX}^{T}} P^{T} \\ = Λ^{- \frac{1}{2}} E^{T} E Λ E^{T} E Λ^{- \frac{1}{2}} = I . \end{matrix}$ (13)

Thus, the mixing matrix A is replaced by $\tilde{A}$ and $Z = PX = PAS = \tilde{AS} .$ (14)

Furthermore, the matrix B introduced in Section 2 is replaced by $B = WPA = W \tilde{A} .$ (15)

3.3 Decryption

To decrypt images from the encrypted images based on the BSS, the underdetermined problem should be solved. At first, Equation (7) is replaced by $X^{2 m} (t) = A_{d} S^{2 m} (t)$ (16)where $X^{2 m} (t) = [X (t) K (t)] \in R^{2 m \times 2 m}$ , the mixing matrix $A_{d} \in R^{2 m \times 2 m}$ is defined as $A_{d} = [\begin{matrix} A & β A \\ 0 & I \end{matrix}] .$ (17)

As above, the information of the key images makes the model determinedly. Thus, the decryption can be treated well by the BSS technique.

3.4 Objective function and performance index

In this paper, the mutual information is chosen as the objective function. The so-called Kullback–Leibler divergence for mutual information is defined as [31, 33] $I (W) = \int p (Y; W) log \frac{p (Y; W)}{\prod_{i = 1}^{n} p_{i} (y_{i}; W)} d Y$ (18)

In general, the mutual information is nonnegative when the components of separated signals Y are independent.

To evaluate the statistical performance of BSS, the cross-talking error is adopted to be the performance index (PI) [13 , 28]

$\begin{matrix} PI & = \sum_{i = 1}^{n} (\sum_{j = 1}^{n} \frac{| b_{ij} |}{{max}_{k} | b_{ik} |} - 1) \\ + \sum_{i = 1}^{n} (\sum_{j = 1}^{n} \frac{| b_{ij} |}{{max}_{k} | b_{kj} |} - 1) \end{matrix}$ (19)where B (t) = {b_ij} = W (t) A is an n × n matrix, $max_{k} | b_{ik} | = max {| b_{i 1} |, \dots, | b_{in} |},$ , and $max_{k} | b_{kj} | = max {| b_{1 j} |, \dots, | b_{nj} |} .$ When PI is close to zero, the BSS will have a good performance.

4 Adaptive learning rate technique and optimization

This section introduces the interval type-2 fuzzy cerebellar model articulation controller (T2FCMAC)-based learning rate for the BSS problem. At first, we give a brief introduction of the learning rate adjustment method for the BSS. Subsequently, we introduce the T2FCMAC to adjust the learning rate for the BSS. Finally, the particle swarm optimization algorithm is presented to optimize the T2FCMAC for enhancing the performance of the BSS problem.

4.1 Adaptive learning rates technique

Several adaptive learning rate methods have been developed to speed up the convergence of BSS algorithms [5 , 26]. As above description, we adopt the EASI algorithm to solve the BSS problem and develop a fuzzy-based adaptive learning rate method for the EASI algorithm. In addition, techniques for selecting different step-sizes during the different stages of adaptation have been proposed. Herein, we would like to determine the learning rates with the separation states by using the T2FCMAC. The T2FCMAC implements the interval type-2 fuzzy system in CMAC structure. It can be simplified to an interval type-2 fuzzy neural network, a fuzzy neural network, and a fuzzy cerebellar model articulation controller (FCMAC) in some special cases [20]. Hence, this T2FCMAC is a more generalized network and has better learning ability to adjust the learning rate of the BSS. The proposed T2FCMAC-based learning rates adjustment for the BSS system is shown in Fig. 2. The learning rate η (t) = diag [η₁ (t) , η₂ (t) , …, η_n (t)] of the EASI algorithm is adjusted by each T2FCMAC with correlation measure D_i (t) and HD_i (t) inputs, i = 1, 2, …, n. The calculation of measure D and HD will be introduced in Appendix.

Note that different output components require different learning rates. Therefore, the separated signals with higher dependence between the components should be adjusted by larger learning rate. On the other hand, the signals with lower dependence between the components should be adjusted by smaller learning rate. In addition, the T2FCMAC is optimized by the particle swam optimization algorithm to enhance the performance of our approach.

4.2 T2FCMAC system for learning rate generation

Herein, we introduce the interval type-2 fuzzy cerebeller model articulation controller (T2FCMAC), which is fed by D and HD to generate the proper learning rate for EASI algorithm. The T2FCMAC has two inputs, (D_i (t) and HD_i (t)), and one output, (η_i (t)). The network structure of T2FCMAC is shown in Fig. 3(a). From Fig. 3(a), there are five spaces in the T2FCMAC: Input Space, Fuzzification Space, Receptive-field Space, Weight Space, and Output space. The T2FCMAC is a more generalized network with better learning ability and has lower computational complexity for practical implementation. The signal propagation and operation functions of each space are described briefly as follows. More detailed discussion and comparisons about the T2FCMAC system can be found in literature [20].

Input Space: The given control space is uniformly divided into n_E regions in this space. The T2FCMAC in two dimensions with n_E = 9 is shown in Fig. 3(b). The number of n_E is termed as the resolution.

Fuzzification Space: This space shows the fuzzification operation of interval type-2 fuzzy systems. According to the concept of CMAC, n_λ elements form a block and n_λ layer presented in CMAC. The illustration example in Fig. 3(b) shows four elements form a full block. Therefore, there are four layer (n_λ = 4) in the fuzzification space of T2FCMAC and three blocks (n_B = 3) in each layer. Herein, we use the interval type-2 triangular asymmetric fuzzy membership function in each block [1 , 21]

$\begin{matrix} φ_{r λ b} & = μ_{{\tilde{F}}_{1 λ b}} (q_{r}) = [{\underline{φ}}_{r λ b} {\bar{φ}}_{r λ b}] \\ = [{\underline{μ}}_{{\tilde{F}}_{r λ b}} (q_{r}) {\bar{μ}}_{{\tilde{F}}_{r λ b}} (q_{r})] \end{matrix}$ (20)where r = 1, 2, [q₁, q₂] = [D_i (t) , HD_i (t)], λ = 1, 2, …, n_λ, and b = 1, 2, …, n_B. The interval type-2 triangular asymmetric triangular membership function is defined as (shown in Fig. 4) ${\bar{μ}}_{{\tilde{F}}_{r λ b}} (q_{r}) = {\begin{matrix} \frac{q_{r} - {\bar{p}}_{r λ b}^{1}}{{\bar{p}}_{r λ b}^{2} - {\bar{p}}_{r λ b}^{1}}, {\bar{p}}_{r λ b}^{1} \leq q_{r} < {\bar{p}}_{r λ b}^{2} \\ 1, q_{r} = {\bar{p}}_{r λ b}^{2} \\ \frac{{\bar{p}}_{r λ b}^{3} - q_{r}}{{\bar{p}}_{r λ b}^{3} - {\bar{p}}_{r λ b}^{2}}, {\bar{p}}_{r λ b}^{2} < q_{r} \leq {\bar{p}}_{r λ b}^{3} \\ 0, otherwise \end{matrix}$ (21a)

and ${\underline{μ}}_{{\tilde{F}}_{r λ b}} (q_{r}) = {\begin{matrix} γ_{r λ b} \cdot \frac{q_{r} - {\underline{p}}_{r λ b}^{1}}{{\underline{p}}_{r λ b}^{2} - {\underline{p}}_{r λ b}^{1}}, {\underline{p}}_{r λ b}^{1} \leq q_{r} < {\underline{p}}_{r λ b}^{2} \\ γ_{r λ b}, q_{r} = {\underline{p}}_{r λ b}^{2} \\ γ_{r λ b} \cdot \frac{{\underline{p}}_{r λ b}^{3} - q_{r}}{{\underline{p}}_{r λ b}^{3} - {\underline{p}}_{r λ b}^{2}}, {\underline{p}}_{r λ b}^{2} < q_{r} \leq {\underline{p}}_{r λ b}^{3} \\ 0, otherwise \end{matrix}$ (21b)where ${\bar{p}}^{1}$ , ${\bar{p}}^{2}$ , ${\bar{p}}^{3}$ , ${\underline{p}}^{1}$ , ${\underline{p}}^{2}$ , and ${\underline{p}}^{3}$ indicate the positions of three vertices for the upper and the lower asymmetric triangular membership function, respectively; γ is the magnitude of the lower membership satisfying 0.5 ≤ γ ≤ 1 to avoid the invalid result. The triangular shape makes them lower computational effort [21]. In order to avoid unreasonable interval type-2 fuzzy membership function, the following conditions should be constrained by ${\bar{p}}^{1} \leq {\bar{p}}^{2} \leq {\bar{p}}^{3}$ , ${\underline{p}}^{1} \leq {\underline{p}}^{2} \leq {\underline{p}}^{3}$ , ${\bar{p}}^{1} \leq {\underline{p}}^{1}, {\bar{p}}^{3} \geq {\underline{p}}^{3}, {\bar{p}}^{1} + γ ({\bar{p}}^{2} - {\bar{p}}^{1}) \leq {\underline{p}}^{2} \leq {\bar{p}}^{3} - γ ({\bar{p}}^{3} - {\bar{p}}^{2})$ . The asymmetric membership functions are not only more flexible than symmetric ones but also provide the advantage of achieving the same performance with fewer rules [1 , 20].

Receptive-field Space: Each location of fuzzification space corresponds to an area in this space. By using t-norm product, the receptive-field space is defined as

$h_{λ b} = [{\underline{h}}_{λ b} {\bar{h}}_{λ b}] = [\prod_{r = 1}^{2} {\underline{φ}}_{r λ b} \prod_{r = 1}^{2} {\bar{φ}}_{r λ b}] .$ (22)

Note that the outputs of fuzzification space are interval sets, the outputs of receptive-field space are interval sets as well. Accordingly, the output of this space consists of its lower bound ${\underline{h}}_{λ b}$ and upper bound ${\bar{h}}_{λ b}$ .

Weight Space: Each location of receptive-field space corresponds to a particular adjustable value in the weight space. The weight space is expressed as $w_{λ b} = [{\underline{w}}_{λ b} {\bar{w}}_{λ b}] = [c_{λ b} - s_{λ b} c_{λ b} + s_{λ b}]$ (23)where $\underline{w}$ and $\bar{w}$ are the lower and upper bounds of w; c and s indicate the center and spread of w, respectively.

Output Space: The output of the T2FCMAC is the algebraic sum of the receptive-field space and weight space. Finally, the learning rate for BSS is $η_{i} (t) = \frac{1}{2} (\sum_{b = 1}^{n_{B}} \sum_{λ = 1}^{n_{λ}} {\underline{h}}_{λ b} {\underline{w}}_{λ b} + \sum_{j = 1}^{n_{B}} \sum_{λ = 1}^{n_{λ}} {\bar{h}}_{λ b} {\bar{w}}_{λ b}) .$ (24)

4.3 Particle swarm optimization for T2FCMAC

In order to enhance the performance of the BSS with T2FCMAC-based learning rates adaptation, the T2FCMAC system is optimized by the particle swarm optimization (PSO). The PSO algorithm is evolutionary algorithm of swarm intelligence techniques [11–13 , 16], which is motivated by social behavior of organisms such as bird flicking and fish schooling. It is a population-based algorithm that exploits a population of individuals to probe for promising regions in a search space.

Each particle moves to a new positionx₁ (t + 1) according to the new velocity v₁ (t + 1) which includes its previous velocity v_l (t), and the moving vectors according to the particle’s past best solution P_best,1 (t + 1) and global best solution G_best,1 (t + 1). The particle position updates laws are [12, 15] $\begin{matrix} v_{l} (t + 1) = w \cdot v_{l} (t) + c_{1} \cdot {rand}_{1 l} \cdot [P_{best, l} (t) - x_{l} (t)] \\ + c_{2} \cdot {rand}_{2 l} \cdot [G_{best, l} (t) - x_{l} (t)] \end{matrix}$ (25) $x_{l} (t + 1) = x_{l} (t) + v_{l} (t + 1)$ (26)where w, c₁, c₂ are constant parameters, and rand_1l, rand_2l are random values in [0, 1].

Subsequently, we select the second-order correlation measure D to be the fitness function (or objective function) of optimization. As the properties of the measures D and HD, the output states are almost independent to each other if both D and HD are sufficiently small. The separations are worse if either D or HD is large (or small enough). On the other hand, the output is correlated with another output when the D and HD are not small enough. Since D_i (t) and HD_i (t) perform the dependence measure of separated signals, they present the information of the learning rate selection in the BSS. In order to obtain the suitable learning rate η_i (t), both D_i (t) and HD_i (t) are adopted to be the inputs of the learning rate selection system and we can obtain the better learning rates for the BSS through the optimization process.

Herein, the PSO algorithm is utilized to optimize the T2FCMAC parameters according to the evaluation of second-order correlation measure D. Therefore, the particle denotes the adjustable T2FCMAC parameters, ${\bar{p}}^{1}$ , ${\bar{p}}^{2}$ , ${\bar{p}}^{3}$ , ${\underline{p}}^{1}$ , ${\underline{p}}^{2}$ , ${\underline{p}}^{3}$ , γ, c, and s. Thus, consider the T2FCMAC having n_λ layers and n_B blocks, the dimensions of particle are n_λ × n_B × 2 ×7 for fuzzification space and n_λ × n_B × 2, for weight space.

5 Computer simulation results

In this section, the simulation results of image encryption based on the BSS principle using the T2FCMAC-based learning rate is introduced. To illustrate the underdetermined problem in the BSS, there are two examples to be compared with. One is to encrypt the original images with the key images when there are no key images in the decrypted model which is categorized as the underdetermined problem. Another one is to encrypt the original images with the key images; this means that it has the key images in the decrypted model.

Herein, two original images (Lena and kitten) shown in Fig. 5(a) are used for the illustrated the example. Both of the images are 128×128 pixels and will be processed under gray scale in range [0–255]. The key images, as the same number of the original images which are generated randomly in range [0–255], shown in Fig. 5(b). In order to compare the results in the two examples, we set the same initial state here. The mixing matrix A is given as

$A = [\begin{matrix} 0.1355 & 0.0038 \\ 0.6518 & 0.2875 \end{matrix}]$ (27)and β= 15, the encrypted images are shown in Fig. 5(c).

To process the images, they are turned 2-D into 1-D. The horizontal scanning with line by line is adopted. The specifications of computer for the simulation are, Operating System (OS): Microsoft Windows XP, Central Processing Unit (CPU): i5-2500K, and Random Access Memory (RAM): 3.24GB.

Herein, we use the EASI algorithm to develop the proposed BSS-based encryption system. The EASI algorithm with fixed learning rate (denoted EASI) and T2FCMAC-based adaptive learning rate adjusted by PSO are introduced to illustrate the performance of our proposed approach. The fixed learning rate of the traditional EASI is chosen as η = 0.2. For the T2FCMAC, the initial parameters, ${\bar{p}}^{1}$ , ${\bar{p}}^{2}$ , ${\bar{p}}^{3}$ , ${\underline{p}}^{1}$ , ${\underline{p}}^{2}$ , ${\underline{p}}^{3}$ , and γ, are selected randomly in the own block but restricted and 0.5 ≤ γ ≤ 1 for the interval type-2 triangular asymmetric fuzzy membership function. The parameters c and s in the weight space are initialized in the range of [0, 1]. Since n_λ = 4 and n_B = 3, the partial representation has 192 parameters. The particle number of T2FCMAC is half number of the partial representation (96). In PSO, the parameters are given by w = 0.35, c₁ = 0.75, and c₂ = 0.75, respectively.

Herein, we have the information of key images in the decrypted model. The mixed model of the BSS is present as the Equation (24). The T2FCMAC-based BSS method for image encryption system without key images decrypting is shown in Fig. 1. For the image decryption, there is information of key images to make the BSS determined. The decrypted images, decrypted from encrypted images with key images in the BSS, are shown in Fig. 6.

Since there are four images, two encrypted images and two key images, as the input of the T2FCMAC-based BSS, then the outputs are four images. In the Fig. 7, the original images are presented in y₁ and y₃, and the key images are presented in y₂ and y₄. In addition, the decrypted images show the uncertainties of ICA. They not only present the uncertain permutation but also perform the uncertain scaling. The pixel values in decrypted images opposite to the values in original images, shown in y₁ and y₃. With the key images in decryption model, the underdetermined BSS problem is solved and the original image can be recovered from the encrypted images. The performance of the T2FCMAC-based BSS method for the image encryption system is shown in Fig. 7. The T2FCMAC-based learning rate for the BSS system shows the better performance than EASI with fixed learning rate method. This demonstrates the effectiveness and performance of our approach.

6 Conclusions

In this paper, the underdetermined BSS problem with adaptive learning rates technique was utilized for the image encryption to ensure the security of cryptosystem. With the information of key images, it is a determined system for BSS and the original images can be recovered well by the proposed method. The proposed adaptive learning rates approach was implemented by using the T2FCMAC. To enhance the performance of convergence, the T2FCMAC system is optimized by PSO algorithm. Simulation results have shown the effectiveness and performance of our approach.

Appendix: Calculation of Dependence Measure

The mutual information I (W) is a good measure of the dependence between the signals. Due to the unknown probability density function of the separated signals Y (t), it cannot be used to evaluate the dependence between each output component at the separation stage of the source signals.

Herein, we introduce the calculation of dependence measure D and HD. Assume that y_i (t) and y_j (t) are two different components of signals Y (t). According the results of [13 , 27], y_i (t) and y_j (t) can be classified with a second-order correlation measure and high order correlation measure. The second-order correlation coefficient is defined as

$\begin{matrix} r_{ij} = \frac{C_{ij}}{\sqrt{C_{ii} C_{jj}}} = \frac{cov [y_{i} (t), y_{j} (t)]}{\sqrt{cov [y_{i} (t)] cov [y_{j} (t)]}}, \\ for i, j = 1, 2, \dots, l, i \neq j \end{matrix}$ (28)where $cov [x (t), y (t)] = E {[x (t) - {\bar{m}}_{x}] [y (t) -$ ${\bar{m}}_{y}]}$ , $cov [x (t)] = E {{[x (t) - {\bar{m}}_{x}]}^{2}}$ , and ${\bar{m}}_{x} = E {x (t)}$ . Note that the calculations of the covariance and mean of signal y_j (t) is between 0 to t time instant, which means that ${\bar{m}}_{x} = E {x (t)}$ provides the mean value of [x (0) , … , x (t)]. According to the correlation coefficient r_ij, the second-order correlation measure can be defined as $D_{i} (t) \equiv D (y_{i} (t)) = \sqrt{\frac{1}{n - 1} \sum_{i = 1, j \neq i} {(r_{ij})}^{2}} .$ (29)

It is useful to adopt D_i (t) to measure the dependence between two Gaussian signals. However, a limitation of BSS is that there is at most one Gaussian source signal. Therefore, we consider the high order correlation measure to ensure the restriction of BSS. The high order correlation measure is defined as ${hr}_{ij} = \frac{{HC}_{ij}}{\sqrt{H_{ii} C_{jj}}} = \frac{cov [φ (y_{i} (t)), y_{j} (t)]}{\sqrt{cov [φ (y_{i} (t))] cov [y_{j} (t)]}}$ (30)where $φ (y_{i} (t)) = y_{i}^{2} (t) + y_{i}^{3} (t)$ is also a nonlinear transformation function in array power. As Equation (29), we have ${HD}_{i} (t) \equiv HD (y_{i} (t)) = \sqrt{\frac{1}{n - 1} \sum_{i = 1, j \neq i} {({hr}_{ij})}^{2}} .$ (31)

Footnotes

Acknowledgments

The authors would like to thank anonymous reviewers and Associate Editor for their insightful comments and valuable suggestions. This work was supported in part by the National Science Council, Taiwan, R.O.C., under contracts MOST-100-2221-E-005-093-MY2, MOST-102-2221 -E-005-95-MY2.

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