1 Introduction

With the rapid development of Internet technology, it has provided great convenience for the dissemination of digital information products. On the other hand, copyright protection has become increasingly important. Information hiding and digital watermarking as an important method of intellectual property protection have attracted more and more attention.

In machine learning, in pattern recognition, and in image processing, feature extraction starts from an initial set of measured data and builds derived values (features) intended to be informative and non-redundant, facilitating the subsequent learning and generalization steps, and in some cases, leading to better human interpretations. Feature extraction is related to dimensionality reduction.

When the input data to an algorithm is too large to be processed and it is suspected to be redundant (e.g., the same measurement in both feet and meters, or the repetitiveness of images presented as pixels), then it can be transformed into a reduced set of features (also named a feature vector). Determining a subset of the initial features is called feature selection [1]. The selected features are expected to contain the relevant information from the input data, so that the desired task can be performed by using this reduced representation instead of the complete initial data.

Feature extraction involves reducing the amount of resources required to describe a large set of data. When performing analysis of complex data, one of the major problems stems from the number of variables involved. Analysis with a large number of variables generally requires a large amount of memory and computation power; also, it may cause a classification algorithm to over fit to training samples and generalize poorly to new samples. Feature extraction is a general term for methods of constructing combinations of the variables to get around these problems while still describing the data with sufficient accuracy. Many machine learning practitioners believe that properly optimized feature extraction is the key to effective model construction.

One very important area of application is image processing, in which algorithms are used to detect and isolate various desired portions or shapes (features) of a digitized image or video stream. It is particularly important in the area of optical character recognition.

Results can be improved using constructed sets of application-dependent features, typically built by an expert. One such process is called feature engineering. Alternatively, general dimensionality reduction techniques are used such as (1) independent component analysis, (2) Isomap, (3) kernel PCA, (4) latent semantic analysis, (5) partial least squares, (6) principal component analysis, (7) multifactor dimensionality reduction, (8) nonlinear dimensionality reduction, (9) multilinear principal component analysis, (10) multilinear subspace learning, (11) semidefinite embedding, and (12) autoencoder.

Space technology [2, 3] and transform domain technology [4, 5] are the main methods of digital image hiding and embedding. From the airspace algorithm point of view, the LSB (least significant bits) method is a typical algorithm [6]. The method converts the spatial pixel values of the original carrier image from binary to decimalism and replaces each bit of information in the binary with the least significant bit in the corresponding carrier, and finally, the binary data containing secret information is converted to decimal pixel, in order to regain the secret image. Although the method has good concealment, so that the human eye is difficult to detect, it has poor robustness.

The transform domain method is relatively stable, and the image hidden by the transform domain method has a certain resistance to image compression, filtering, rotation, shearing, and noise. Discrete cosine transform (DCT) [7], discrete Fourier transform (DFT) [8], and discrete wavelet transform (DWT) [9, 10] are the main transform domain methods in recent years. We construct a unified form of the boundary value solving problem by constructing the matching function, and finally, using the conventional spectral method to solve is an important method in the domain of transform domain [10]. In the process of exploring image-hiding algorithms, some scholars have proposed an iterative hybrid image-hiding encryption algorithm [11]. The main idea of the algorithm is to embed an image into another image. The image is repeatedly embedded into a set of image, accordingly to adjust the iteration parameters to achieve the invisibility of the image. The algorithm provides a new way for better selection of the mixing parameters, but it shows weak robustness of the system due to the multiple product amplification of the mixed image change value. Due to the weak robustness caused by the error multi-amplification in [12], Urvoy [13] proposed an iterative hybrid algorithm for image modification that uses a partial image that does not contain a watermark to modify the image containing the watermark. Thus, we can eliminate the error from the image. However, in the face of random attacks such as Gaussian noise, the quality of the watermark we extract is poor. In order to improve the stability of the watermarking system, the literature [14, 15] proposed JADE blind separation watermarking algorithm. This method used an iterative hybrid method to embed the image and used the hidden image and the carrier image as different signals. By separating the matrix to determine whether there is hidden information, there is no need to know the exact location of the embedded image. When the two signals that need to be separated have a strong correlation, the method cannot separate the related signals well.

In this paper, we proposed the image extraction method based on iterative learning algorithm, which is applied to the full estimation of time-varying parameters in finite time intervals under certain convergence conditions. For the repeated mixed images, a new image extraction method is proposed in this paper. Iterative learning identification technology is used to reconstruct the image information signals. For the class of chaotic map and iterative hybrid encryption image system, an iterative learning identification law is constructed. Under the conditions of given learning law and initial state of the system, the sufficient conditions for learning gain convergence are deduced, and the convergence of the system is proved. Using iterative learning identification method to estimate the time-varying parameters completely in finite time interval, we can achieve completely reconstruction of image information in digital image watermarking system.

Remarks: Here are some general mathematical symbols used in this paper. L2(Ω) (or short in L2) represents a kind of Ω function space consisted by all measureable functions and it is bounded, satisfying \( {u}_p={\left\{\underset{\Omega}{\int }{\left|u(x)\right|}^p dx\right.}^{1/p}<\infty \kern0.5em \left(1\le p\le \infty \right) \). Lp(Ω)is Banach space, L2(Ω) is Hilbert space.

For the n dimensional vector \( u=\left({u}_1^T,{u}_2^T,\cdots, {u}_i^T\right) \), where the norm of definition is\( \left\Vert u\right\Vert ={\left(\sum \limits_{i=1}^n{u}_i^2\right)}^{1/2} \). If ui(x) ∈ L2, i = 1, 2, ⋯, n then Q(x) = (Q1(x), Q2(x), L, Qn(x)) ∈ Rn ∩ L2 and \( {\left\Vert Q\right\Vert}_{L^2}={\left\{\underset{\Omega}{\int }{\left({Q}^T(x)Q(x)\right)}^2 dx\right\}}^{1/2} \).

For the function f(x, t) : Ω × [0, T] → Rn, f(gt) ∈ Rn ∩ L2, t ∈ [0, T], define its (L2, λ) norm as follows \( {\left\Vert f\right\Vert}_{\left({L}^2,\lambda \right)}=\underset{0\le t\le T}{\sup}\left\{\left({\left\Vert f\right\Vert}_{L^2}^2\right){e}^{-\lambda t}\right. \).

2 Digital image encryption

Chaotic cryptology includes two integral opposite parts: chaotic cryptography and chaotic cryptanalysis. Chaotic cryptography is the application of the mathematical chaos theory to the practice of the cryptography, the study or techniques used to privately and securely transmit information with the presence of a third party or adversary. The use of chaos or randomness in cryptography has long been sought after by entities wanting a new way to encrypt messages. However, because of the lack of thorough, provable security properties and low acceptable performance, chaotic cryptography has encountered setbacks [1, 16,17,18].

In order to use chaos theory efficiently in cryptography, the chaotic maps should be implemented such that the entropy generated by the map can produce required confusion and diffusion. Properties in chaotic systems and cryptographic primitives share unique characteristics that allow for the chaotic systems to be applied to cryptography [19]. If chaotic parameters, as well as cryptographic keys, can be mapped symmetrically or mapped to produce acceptable and functional outputs, it will make it next to impossible for an adversary to find the outputs without any knowledge of the initial values. Since chaotic maps in a real life scenario require a set of numbers that are limited, they may, in fact, have no real purpose in a cryptosystem if the chaotic behavior can be predicted. One of the most important issues for any cryptographic primitive is the security of the system. However, in numerous cases, chaotic-based cryptography algorithms are proved unsecure [17, 20,21,22]. The main issue in many of the cryptanalyzed algorithms is the inadequacy of the chaotic maps implemented in the system [23, 24].

The concept of chaos cryptography or in the other words chaos-based cryptography can be divided into two major groups: the asymmetric [25, 26] and symmetric [27,28,29] chaos-based cryptography. The majority of the symmetric chaos-based algorithms are based on the application of discrete chaotic maps in their process [27, 30].

Bourbakis and Alexopoulos [31] in 1991 proposed supposedly the earliest fully intended digital image encryption scheme which was based on SCAN language. Later on, with the emergence of chaos-based cryptography, hundreds of new image encryption algorithms, all with the aim of improving the security of digital images, were proposed [21]. However, there were three main aspects of the design of an image encryption that was usually modified in different algorithms (chaotic map, application of the map, and structure of algorithm). The initial and perhaps the most crucial point was the chaotic map applied in the design of the algorithms [32,33,34,35,36]. The speed of the cryptosystem is always an important parameter in the evaluation of the efficiency of a cryptography algorithm; therefore, the designers were initially interested in using simple chaotic maps such as tent map, and the logistic map [19, 37]. However, in 2006 and 2007, the new image encryption algorithms based on more sophisticated chaotic maps proved that application of chaotic map with higher dimension could improve the quality and security of the cryptosystems [2, 3, 35, 38, 39].

In this paper, we use logistic encryption method to encrypt the digital image. The logistic chaotic map is described as follows:

$$ x\left(t+1\right)=\upmu x(t)\left(1-x(t)\right),x(t)\in \left(0,1\right) $$
(1)

when 3.5699456⋯ ≤ μ ≤ 4, the unpredictability of the sequence x(t) generated by logistic chaotic maps. If the same initial value is given, a random sequence will be generated, under the mapping of the parameter A. If a different initial value is given, different data sequences will be generated, but the correlation between the data sequences is almost zero. The original image G is marked as θ(t). In order to achieve the invisibility of the image, we superpose and mix the known image sequence with the parameter μ, namely μ(t) = λ + θ(t); at the same time, the Eq. (1) can be written as

$$ x\left(t+1\right)=\left(\uplambda +\uptheta (t)\right)x(t)\left(1-x(t)\right),x(t)\in \left(0,1\right) $$
(2)

Thus, the chaotic sequence {x(t), t = 1,2,3,...} is the desired encrypted image G′, denoted here as x(t).

3 Method—multiple image blends

Assume that the image F is a pair of M × N digital images, G′ needs to be hidden images, then a superimposed mixed image S can be described as

$$ \mathrm{S}=\alpha \mathrm{F}+\left(1-\alpha \right){\mathrm{G}}^{\prime } $$
(3)

α is adjustable mixing parameter, and F is carrier image. S is an image sequence produced after a superposition. According to Eq. (3), we know the 1 − α tends to 0, when mixing parameter α tends to 1, then the produced image sequence S=F. However, when the adjustable parameter α tends 0, the image sequence of superposition S will tend the hidden image G′. From the Eq. (3), we also know that when the value of the parameter α is closer to 1, the image G′ is hidden as much as possible. It is not easy to be detected, but it also increases the difficulty of the extraction.

Assume that the carrier images Fi(i = 1, 2, ⋯, n) are different M × N digital images and the mixing parameters are αi ∣ αi ∈ [0, 1], i = 1, 2, ⋯, n. According to the hybrid algorithm of image, firstly, we mix the image F1, G′, and α1 to get S1 = α1F1 + (1 − α1)S1; secondly, we mix the image F2, G′, and α2 to get S2 = α2F2 + (1 − α2) G′, and so on. Finally, mix the images to get Sn = αnFn + (1 − αn)Sn − 1; Sn is called the digital image group which is a mixture of N images.

The mixed image satisfies the following relation:

$$ \left\{\begin{array}{c}{\mathrm{S}}_1(t)={a}_1{\mathrm{F}}_1(t)+\left(1-{a}_1\right)x(t)\\ {}{\mathrm{S}}_2(t)={a}_2{\mathrm{F}}_2(t)+\left(1-{a}_2\right){\mathrm{S}}_1(t)\\ {}\vdots \\ {}{\mathrm{S}}_n(t)={a}_n{\mathrm{F}}_n(t)+\left(1-{a}_n\right){\mathrm{S}}_{n-1}(t)\end{array}\right. $$
(4)

so:

$$ {\mathrm{S}}_n={\alpha}_n{\mathrm{F}}_n+{\beta}_n{\alpha}_{n-1}{\mathrm{F}}_{n-1}+\cdots +{\beta}_n{\beta}_{n-1}\cdots {\beta}_{n-i}{\alpha}_{n-i}{\mathrm{F}}_{n-i}+\cdots +{\beta}_n{\beta}_{n-1}\cdots {\beta}_2{\beta}_1{\mathrm{G}}^{\prime } $$

where βi = 1 − αi, i = 1, 2, ⋯, n. When αi tends to 1, the better the effect of watermark embedding, the worse the effect of watermark extraction. We use the logistic map to produce these iterative parameters. Selected parameter μ′ and initial value α1.

$$ {\alpha}_{i+1}={\upmu}^{\prime }{\alpha}_i\left(1-{\alpha}_i\right) $$
(5)

According to Eq. (5), we can get a chaotic sequence αi and use αi as the parameter sequence for each image superposition and mixing. In order to avoid the correlation between the mixed images, the initial value α1 of the parameter μ′ in Eq. (5) cannot be the same as the parameter μ(t) and the initial value x(0) in Eq. (1).

Image encryption and its embedded process are shown in Fig. 1.

Fig. 1
figure 1

Image encryption and its embedded process

Full size image

4 The iterative learning control for image hidden with multi-blending

We use the image’s multiple hybrids embedding technology to embed image information into the time-varying parameters of the digital image system and establish a mathematical model for the digital image system. The image information is regarded as a finite time, and the iterative learning identification method is applied to the image system. Variable parameters are estimated to achieve complete reproduction of image information.

We mark the initial image G as θ(t), then the encryption image G′ as x(t) sequence. Carrier image group Fi(i = 1, 2, ⋯, n) is ωi and the hybrid image Si is y(t); the system is described as

$$ \left\{\begin{array}{c}x\left(t+1\right)=f\left(x(t),\uptheta (t),t\right)\\ {}{y}^1(t)={h}_1\left(x(t),t\right)\\ {}{y}^2(t)={h}_2\left({y}^1(t),t\right)\\ {}\cdots \\ {}{y}^n(t)={h}_n\left({y}^{n-1}(t),t\right)\end{array}\right. $$
(6)

where t ∈ {0, 1, 2, ⋯, N}, x(t) ∈ Rn, θ(t) ∈ R1, y1(t) ∈ R1, y(t) ∈ R1 are nonlinear function. f(x(t), θ(t), t) is the function of initial encryption image. The nonlinear function h1(x(t), t) represents an iterative mixed function of the encrypted image and carrier image, and hn(yn − 1(t), t) represents the mixed function of n iterations.

The iterative learning control system for estimating θ(t) is:

$$ \left\{\begin{array}{c}{x}_k\left(t+1\right)=f\left({x}_k(t),{\uptheta}_k(t),t\right)\\ {}{y}_k^1(t)={h}_1\left({x}_k(t),t\right)\\ {}{y}_k^2(t)={h}_2\left({y}_k^1(t),t\right)\\ {}\cdots \\ {}{y}_k^n(t)={h}_n\left({y}_k^{n-1}(t),t\right)\end{array}\right. $$
(7)

where k is iterative times and we use the same initial value. We suppose there exist f partial derivative Ak(t), Bk(t) about x, θ, h1 partial derivative Ck(t) about x and hi partial derivative Dki(t) about y.

We use the D-type learning law:

$$ \left\{\begin{array}{c}{\uptheta}_k(t)=\mathrm{sat}\left({\vartheta}_k(t)\right)\\ {}{\vartheta}_{k+1}(t)=\mathrm{sat}\left({\vartheta}_k(t)\right)+L(t)\left[{e}_k\left(t+1\right)-{e}_k(t)\right.\end{array}\right. $$
(8)

where sat(∙) is the saturation function, L(t) is the learning gain, and \( {e}_k(t)={y}^n-{y}_k^n \).

Theorem 1 For the system (7) and learning law, if there exist

$$ \left\Vert 1\right.-L(t){\prod}_{i=2}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){B}_k(t)\left\Vert \le \right.\uprho <1 $$
(9)

then when k → ∞ , θ(t) converges to θk(t) interval {0, 1, ⋯, N}.

Proof

Remark δxk(t) = xd(t) − xk(t), δθk(t) = θd(t) − θk(t)

$$ {\displaystyle \begin{array}{l}\updelta {x}_k(t)=f\left({x}_d\left(t-1\right),{\uptheta}_d\left(t-1\right),t-1\right)-f\left({x}_k\left(t-1\right),{\uptheta}_k\left(t-1\right),t-1\right)\\ {}+f\left({x}_d\left(t-1\right),{\uptheta}_k\left(t-1\right),t-1\right)-f\left({x}_d\left(t-1\right),{\uptheta}_k\left(t-1\right),t-1\right)\end{array}} $$

According to the differential mean value theorem, we have

$$ {\displaystyle \begin{array}{l}f\left({x}_d\left(t-1\right),{\uptheta}_d\left(t-1\right),t-1\right)-f\left({x}_k\left(t-1\right),{\uptheta}_k\left(t-1\right),t-1\right)={A}_k\left(t-1\right)\updelta {x}_k(t)f\\ {}\left({x}_d\left(t-1\right),{\uptheta}_d\left(t-1\right),t-1\right)-f\left({x}_d\left(t-1\right),{\uptheta}_k\left(t-1\right),t-1\right)={B}_k\\ {}\left(t-1\right){\updelta \uptheta}_k(t)\updelta {x}_k(t)={A}_k\left(t-1\right)\updelta {x}_k\left(t-1\right)+{B}_k\left(t-1\right){\updelta \uptheta}_k\left(t-1\right)\end{array}} $$
(10)

Taking the norm of Eq. (9)

$$ \left\Vert \updelta \right.{x}_k(t)\le {c}_A^t\left\Vert \updelta \right.{x}_k(0)\left\Vert +\right.{c}_A^{t-1}{c}_B\left\Vert \updelta \right.{\uptheta}_k(0)\left\Vert +\right.{c}_A^{t-2}{c}_B\left\Vert \updelta \right.{\uptheta}_k(1)\left\Vert +\right.\cdots +{c}_B\left\Vert \updelta \right.{\uptheta}_k\left(t-1\right) $$
(11)

Because the initial state is the same at each iteration, δxk(0) = 0. The (10) can be written

$$ \left\Vert \updelta \right.{x}_k(t)\left\Vert \le \right.{\sum}_{j=0}^{t-1}{c}_A^{t-j-1}{c}_B\left\Vert \updelta \right.{\uptheta}_k\left.(j)\right\Vert $$
(12)

Multiplied by the two sides \( {c}_A^{-\uplambda t} \), yields

$$ {\left\Vert \updelta {x}_k(t)\right\Vert}_{\uplambda}\le \sup {\sum}_{j=0}^{t-1}\left({c}_B{c}_A^{t-j-1}{c}_A^{-\uplambda t}\left\Vert {\updelta \uptheta}_k(j)\right\Vert \right)\le \frac{c_B}{c_A\left({c}_A^{\uplambda -1}-1\right)} $$
(13)

Suppose \( c>\max \left\{{c}_A,\overline{1}\right\} \), yields

$$ {\left\Vert \updelta {x}_k(t)\right\Vert}_{\uplambda}\le \frac{c_B}{c_A\left({c}^{\uplambda -1}-1\right)}{\left\Vert {\updelta \uptheta}_k(t)\right\Vert}_{\uplambda} $$
(14)

By (8), we have

$$ {\displaystyle \begin{array}{l}\updelta {\vartheta}_{k+1}(t)={\updelta \uptheta}_k(t)-L(t)\left({e}_k\left(t+1\right)-{e}_k(t)\right)={\updelta \uptheta}_k(t)-L(t)\\ {}\left\{\left[{h}_n\left({y}^{n-1}\left(\left(t+1\right.,t+1\right.-{h}_n{y}_k^{n-1}\left(t+1\right),t+1\right.\right.-{h}_n\left({y}^{n-1}\left((t),t\right)-{h}_n{y}_k^{n-1}(t),t\right)\right.\end{array}} $$
(15)

Remark

$$ {A}_k(t)=\frac{\partial f\left({x}_k(t),{\uptheta}_k(t),t\right)}{\partial {x}_d(t)}\mid {x}_k(t)={\upxi}_k(t) $$

where ξk(t) = (1 − σ3)x(t) + σ3xk(k), 0 < σ3 < 1

$$ {B}_k(t)=\frac{\partial f\left({x}_k(t),{\uptheta}_k(t),t\right)}{\partial {\uptheta}_k(t)}\mid {\uptheta}_k(t)={\upeta}_k(t) $$

where ηk(t) = (1 − σ4)x(t) + σ4xk(k), 0 < σ4 < 1

\( {D}_{kn}(t)=\frac{\partial {h}_n\left(y,t\right)}{\partial y}\mid y={\upxi}_{kn}(t) \)

where ξkn(t) = (1 − σn)yn − 1(t) + σnyn − 1(t), 0 < σn < 1

$$ {\displaystyle \begin{array}{l}{h}_n\Big({y}^{n-1}\left(\left(t+1\right),t+1\right)-{h}_n\left({y}_k^{n-1}\left(t+1\right),t+1\right)={D}_{kn}\left(t+1\right)\\ {}\Big({h}_{n-1}\left({y}^{n-2}\left(\left(t+1\right),t+1\right)-{h}_{n-1}\left({y}_k^{n-2}\left(t+1\right),t+1\right)\right.\end{array}} $$
(16)
$$ {h}_n\left({y}^{n-1}(t),t\right)-{h}_n\left({y}_k^{n-1}(t),t\right)={D}_{kn}(t)\Big({h}_{n-1}\left({y}^{n-2}\left(\left(t+1\right),t+1\right)-{h}_{n-1}\left({y}_k^{n-2}(t),t\right)\right. $$
(17)
$$ {\displaystyle \begin{array}{l}\updelta {\vartheta}_{k+1}(t)={\updelta \uptheta}_k(t)-L(t)\Big[{\prod}_{i=2}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){B}_k(t)\\ {}\left(f\left({x}_d(t),{\uptheta}_d(t),t\right)-f\left({x}_k(t),{\uptheta}_k(t),t\right)-{\prod}_{i=2}^n{D}_{ki}(t){C}_k(t)\updelta {x}_k(t)\right.\end{array}} $$
(18)

By definition of Ak(t) and Bk(t), (18) can be written

$$ {\displaystyle \begin{array}{l}\updelta {\vartheta}_{k+1}(t)=\left[1-L(t){\prod}_{i=1}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){B}_k(t)\right]{\updelta \uptheta}_k(t)-L(t)\\ {}\left[{\prod}_{i=1}^n{D}_{ki}(t){C}_k(t)-{\prod}_{i=1}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){A}_k(t)\right]\updelta {x}_k(t)\end{array}} $$
(19)

From Eq. (7), we have

$$ \updelta {\vartheta}_{k+1}(t)={\uprho \updelta \uptheta}_k(t)+L\left[{\prod}_{i=1}^n{D}_{ki}(t){C}_k(t)-{\prod}_{i=1}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){A}_k(t)\right.\updelta {x}_k(t) $$
(20)

Remark

\( {M}_k(t)=L\left[\prod \limits_{i=1}^n{D}_{ki}(t){C}_k(t)-\prod \limits_{i=1}^n{D}_{ki}\left(t+1\right){C}_k\left(t+1\right){A}_k(t)\right. \)

So (19) can be written

$$ \updelta {\vartheta}_{k+1}\left(\mathrm{t}\right)={\uprho \updelta \uptheta}_k(t)+{M}_k(t)\updelta {x}_k(t) $$
(21)

Suppose Mk(t) ≤ M, we take λ norm of (21).

$$ {\left\Vert \updelta {\overline{\vartheta}}_{k+1}(t)\right\Vert}_{\uplambda}\le \uprho {\left\Vert {\updelta \uptheta}_k(t)\right\Vert}_{\uplambda}+M{\left\Vert \updelta {x}_k(t)\right\Vert}_{\uplambda} $$
(22)

Introducing (14) into (22), yields

$$ {\left\Vert \updelta {\overline{\vartheta}}_{k+1}(t)\right\Vert}_{\uplambda}\le \uprho {\left\Vert {\updelta \uptheta}_k(t)\right\Vert}_{\uplambda}+\frac{Mc_B}{c_A\left({c}^{\uplambda -1}-1\right)}{\left\Vert {\updelta \uptheta}_k(t)\right\Vert}_{\uplambda} $$
(23)

If there exist enough λ, then

\( \underset{k\to \infty }{\lim }{\left\Vert {\updelta \uptheta}_k(t)\right\Vert}_{\uplambda}=0 \)

The steps of using an iterative learning method to reproduce the hidden images are as follows:

  1. 1.

    We find the superimposed mixed image y(t), and the initial estimate θ0(t) with the given arbitrary image sequence of multiple carrier wi(t) (i = 1,2,…,n), where t ∈ {1, 2, ⋯, N}.

  2. 2.

    Set initial state xk(0) = x0, substituting θk(t) into the iterative learning control system, and obtain Sk(t) by solving.

  3. 3.

    Calculate the output error ek(t).

  4. 4.

    Using the iterative learning law to calculate θk + 1(t).

  5. 5.

    The given error \( {J}_k={\sup}_{t\in \left\{0,1,2,\cdots N\right\}}\left\Vert \updelta {\uptheta}_k(t)\right\Vert \), if ek(t) < Jk, then the system will stop iteration; otherwise, set k = k + 1, go to step 2).

The image information is hidden in time-varying parameter θk(t) obtained by the iterative learning method. A set of image sequences is known, and then the hidden images are restored according to the pixel ratio of the image.

5 Results and discussions

For mixed images and restored hidden images, we can use the peak signal-to-noise ratio PSNR and normalized cross-correlation coefficient NC to measure the objective fidelity of the image. The peak signal-to-noise ratio of the original carrier image F and image S containing hidden information is measured.

PSNR is:

$$ \mathrm{PSNR}=101\mathrm{g}\left(\frac{M\times N\times {255}^2}{\sum_{i=1}^M{\sum}_{j=1}^N{\left(\mathrm{F}\left(i,j\right)-\mathrm{S}\left(i,j\right)\right)}^2}\right) $$
(24)

The peak signal-to-noise ratio PSNR is used as a measure of the objective fidelity of the image. The larger the value, the higher the fidelity of the image mixture.

Normalized cross correlation (NC) measures the degree of similarity between the extracted image and the original image. It is defined as:

$$ \mathrm{NC}=\frac{\sum_{i=1}^{M\times M}x(t)\times \widehat{x}(t)}{\sum_{i=1}^{M\times M}x{(t)}^2} $$
(25)

where x(t) and \( \widehat{x}(t) \) denote pixel value sequence of the original image and pixel value sequence of extracted image.

In this experiment, we use Camera (256*256) and Lena (256*256) as the two carrier images, and the binary image “true” (32*32) as the image to be hidden.

The state of the digital image embedding system is as follows:

\( \left\{\begin{array}{c}x\left(t+1\right)=\left(\lambda +\uptheta (t)\right)x(t)\left(1-x(t)\right)\\ {}{y}_1(t)={\alpha}_1{w}_1(t)+\left(1-{\alpha}_1\right)x(t)\\ {}{y}_2(t)={\alpha}_2{w}_2(t)+\left(1-{\alpha}_2\right){y}_1(t)\\ {}y(t)={y}_2(t)\end{array}\right. \)

Example 1. In the experiment, the hidden image “true” is first embedded in the carrier image Lena. Then, the mixed image is embedded into the carrier image Camera again, and finally we obtain the mixture images. Taking λ = 3.568, the initial xk(0) = 0.5. Original image θ(t) is binary image, the carrier image w1(t) and w2(t) are two pixel points of the same gray image, and the images obtained after two times of superposition and mixing are y (t).

Taking λ = 3.568, α1 = 0.52, according to the chaotic sequence generated by the Eq. (5), {αi}, we choose the required experimental parameter sequence {α1,α2}., the given learning gain is:

\( {L}_k(t)=\frac{1}{\left({\beta}_1{\beta}_2\right){x}_k(t)\left(1-{x}_k(t)\right)} \)

where βi = 1 − αi, i = 1, 2

Figure 2 is the hidden image “true” used during the experiment, and Fig. 3 is the carrier image Camera and Lena. Firstly, the hidden image “true” needs to be embedded in the first carrier image Lena, and then the resulting mixed image is further embedded into the second carrier image Camera. Figure 4 is the final mixed image Camera and extracted hidden image using an iterative learning method. Experimental results show that this method can completely recover hidden images.

Fig. 2
figure 2

Hidden image

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Fig. 3
figure 3

Carrier image

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Fig. 4
figure 4

Embedded carrier image and extracted hidden image

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To verify the performance of the algorithm, we define the index function as

$$ {J}_k=\underset{t\in \left\{1,2,\cdots, N\right\}}{\sup}\left|{\updelta \uptheta}_k(t)\right| $$

We reconstruct the error convergence process of hidden image “true” by iterative learning algorithm. From Fig. 5, it can be seen that when the iterative times reach to 3000, the identified image and original image error has reached 10−12. The experimental results show that the method effectively reconstructs the finite length information.

Fig. 5
figure 5

Iterative output error

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Example 2. In the experiment 2, the “wrong” is hidden image. Cherry and Lena are carrier images, then we take the hidden image embeds into the carrier images and obtain the mixture images finally. Taking λ = 4.256, the initial xk(0) = 0. The rest of the conditions are the same as the Example 1.

By the chaotic sequence produced by Eq. (5), {αi}, the required experimental parameter sequences that we choose are {α1,α2} and the given learning gain is:

\( {L}_k(t)=\frac{1}{\left({\beta}_1{\beta}_2\right){x}_k(t)\left(1-{x}_k(t)\right)} \)

where βi = 1 − αi, i = 1, 2

Figure 6 is the hidden image “wrong” used during the experiment, and Cherry and Lena are the carrier images in Fig. 7. Firstly, we embed the “wrong” image into the Cherry and obtain a mixed image. Secondly, we embed the mixed image into the Lena image. Figure 8 is the final mixed image Cherry and extracting hidden image by the iterative learning way. Experimental results illustrate that the way that we use can completely reconstruct the hidden image.

Fig. 6
figure 6

Hidden image

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Fig. 7
figure 7

Carrier image

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Fig. 8
figure 8

Embedded carrier image and extracted hidden image

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We use the same index function as experiment 1:

$$ {J}_k=\underset{t\in \left\{1,2,\cdots, N\right\}}{\sup}\left|\updelta {\theta}_k(t)\right| $$

We use the same algorithm to reconstruct the convergence process of hidden image “wrong.” We know the iterative error has reached 10–14 with the iterative times reach to 3000 from Fig. 9. The simulation shows the algorithm can effectively reconstruct the finite length information.

Fig. 9
figure 9

Iterative output error

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The anti-attack results of the test algorithm are as follows:

  1. 1.

    Anti-attack test adding Gauss noise and salt and pepper noise

Adding Gauss noise and salt and pepper noise to the mixed image, the Gauss noise variance VGN is 0.001, 0.01, and 0.02, respectively. Noise intensity of salt and pepper (INSP) is 0.001, 0.01, and 0.02.

Figures 10 and 11 show the mixed image and the extracted image after the attack. The similarity NC between the original image and the extracted image after the attack are shown in Table 1. The peak signal-to-noise ratio PSNR of the original mixed image and the mixed image after the attack is shown in Table 2.

  1. 2.

    Anti-attack test with compressed

Fig. 10
figure 10

Attacked with Gaussian noise and extracted hidden image (Example 1)

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Fig. 11
figure 11

Attacked with salt and pepper noise and extracted hidden image (Example 1)

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Table 1 Adding the Gaussian noise attack with Example 1
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Table 2 Adding the salt and pepper noise attack with Example 1
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The JPEG image compression method is a compression attack for the mixed image. The smaller the compression factor Q, the higher the compression, that is, the greater the pixel loss. In Experiment 1 and Experiment 2, we separately select different compression factors Q to extract the hidden images respectively. When Q = 10, the compression ratio is already very large. From Figs. 10, 11, 12, and 13, it can be seen that the “true” image proposed by the iterative learning identification method still has a high correlation NC (0.78) and better PSNR. It shows that the algorithm has better robustness for JPEG compression attack. The robustness of algorithms for JPEG compression attack are shown in (Tables 3 and 4).

Fig. 12
figure 12

Attacked with Gaussian noise and extracted hidden image (Example 2)

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Fig. 13
figure 13

Attacked with salt and pepper noise and extracted hidden image (Example 2)

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Table 3 Adding the Gaussian noise attack with Example 2
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Table 4 Adding the salt and pepper noise attack with Example 2
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6 Conclusions

In this paper, we proposed the image extraction problem by using iterative learning algorithm. This method is to embed the hidden image into several different carrier images. Finally, the iterative learning is used to fully track the parameters of the image information, and the hidden image is reproduced. The parameters of the iterative learning law are updated with the change of time, which shows that the algorithm has better self-adaptability. Both experimental results show that when the mixed image with hidden information is attacked, the iterative learning recognition method can still recover the hidden image, and the ordinary reversible image solving method cannot restore the hidden image. The experimental data also proves the anti-attack of the algorithm.