# Comparative Study of Three Steganographic Methods Using a Chaotic System and Their Universal Steganalysis Based on Three Feature Vectors

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed Chaotic System

#### Description of the Cat Map Used

## 3. Enhanced Steganographic Algorithms

#### 3.1. Enhanced EALSBMR (EEALSBMR)

#### 3.1.1. Insertion Procedure

- Step 1:
- Capacity estimation
- To estimate the insertion capacity, we arrange the cover image into a 1D vector V, and we divide its content into non-overlapping embedding units (blocks) consisting of two consecutive pixels $\left(\right)$. Following this, we calculate the difference between the pixels of each block, and we increase by one the content of the vector-difference $VD$ of 31 elements $t\in \left(\right)open="\{"\; close="\}">1,2,3,\dots ,31$, in which each element contains $\left(\right)$ number of blocks where $EU\left(t\right)$ is a set of pixel pairs whose absolute differences are greater than or equal to t, as shown below:$$EU\left(t\right)=\left(\right)open="\{"\; close="\}">\left(\right)open="("\; close=")">{p}_{i},{p}_{i+1}\ge t,\forall \left(\right)open="("\; close=")">{p}_{i},{p}_{i+1}$$
- For a given secret message M of size $\left|M\right|$ bits, the threshold T used in the embedding process is determined by the following expression and pseudo-code (Algorithm 1):$$T=argma{x}_{t}\left(\right)open="\{"\; close="\}">2\ast \left(\right)open="|"\; close="|">EU\left(t\right)$$
**Algorithm 1**Pseudo-code determining the value of the threshold T- 1:
**procedure**- 2:
- $number\_\phantom{\rule{4pt}{0ex}}pixels$ = 0;
- 3:
**for**t = 31:-1:1**do**- 4:
- $number\_\phantom{\rule{4pt}{0ex}}pixels$ = $number\_\phantom{\rule{4pt}{0ex}}pixels$ + $VD\left(t\right)$;
- 5:
**if**(2*$number\_\phantom{\rule{4pt}{0ex}}pixels$ > = |M|)**then**- 6:
- $T=t$;
- 7:
- break;
- 8:
**end if**;- 9:
**end for**;- 10:
**end procedure**

- Step 2:
- Embedding process
- The embedding process is achieved as follows: we divide the cover image into two sub-images; one includes the odd columns, and the other includes the even columns.
- Following this, the chaotic system chooses a pixel position ($Ind$) from the odd sub-image; the second pixel position of the corresponding block must have the same $Ind$ in the even image. If a pair of pixel units $\left(\right)$ satisfies Equation (8), then a 2 bit-message can be hidden (one bit by pixel); otherwise, the chaotic system chooses another $Ind$.$$\left(\right)\ge T,\forall \left(\right)open="("\; close=")">{p}_{i},{p}_{i+1}$$
- For each unit $\left(\right)$, we perform data-hiding based on the following four cases [42]:
- Case 1:
- if $LSB\left({p}_{i}\right)={m}_{i}$ and $f({p}_{i},{p}_{i+1})={m}_{i+1}$$\to ({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})=({p}_{i},{p}_{i+1})$
- Case 2:
- if $LSB\left({p}_{i}\right)={m}_{i}$ and $f({p}_{i},{p}_{i+1})\ne {m}_{i+1}$$\to ({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})=({p}_{i},{p}_{i+1}+r)$
- Case 3:
- if $LSB\left({p}_{i}\right)\ne {m}_{i}$ and $f({p}_{i}-1,{p}_{i+1})={m}_{i+1}$$\to ({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})=({p}_{i}-1,{p}_{i+1})$
- Case 4:
- if $LSB\left({p}_{i}\right)\ne {m}_{i}$ and $f({p}_{i}-1,{p}_{i+1})\ne {m}_{i+1}$$\to ({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})=({p}_{i}+1,{p}_{i+1})$

In the above equations, ${m}_{i}$ and ${m}_{i+1}$ are the ${i}^{}$th and ${(i+1)}^{}$th secret bits of the message to be embedded; r is a random value belonging to $\left(\right)$, and $({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})$ denotes the pixel pair after data-hiding. The function f is defined as follows:$$f(a,b)=LSB(\left(\right)open="\lfloor "\; close="\rfloor ">\frac{a}{2}+b)$$ - Readjustment if necessary: After hiding, $({p}_{i}^{{}^{\prime}},{p}_{i+1}^{{}^{\prime}})$ may be out of range [0, 255] or the new difference value $\left(\right)$ may be less than the threshold T. In these cases, we need to readjust ${p}_{i}^{{}^{\prime}}$ and ${p}_{i+1}^{{}^{\prime}}$, and the new readjusted values, ${p}_{i}^{\u2033}$ and ${p}_{i+1}^{\u2033}$, are calculated as follows [3]:$$(\phantom{\rule{4pt}{0ex}}{p}_{i}^{\u2033},{p}_{i+1}^{\u2033})=argmi{n}_{({e}_{1},{e}_{2})}\left(\right)open="\{"\; close="\}">\left(\right)open="|"\; close="|">{e}_{1}-{p}_{i}^{{}^{\prime}}$$$$\left(\right)open="\{"\; close>\begin{array}{c}{e}_{1}={p}_{i}^{{}^{\prime}}+4{k}_{1}\\ {e}_{2}={p}_{i+1}^{{}^{\prime}}+2{k}_{2}\end{array}$$$$0\le {e}_{1},{e}_{1}\le 255\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}\left(\right)open="|"\; close="|">{e}_{1}-{e}_{2}$$$$\begin{array}{c}\hfill {p}_{i}^{{}^{\u2033}}={e}_{1}\\ \hfill \phantom{\rule{1.em}{0ex}}{p}_{i+1}^{{}^{\u2033}}={e}_{2}\end{array}$$The sequence follows as such for each new block position.
- Finally, we embed the parameter T of the stego image into the first five pixels or the last five pixels, for example.

#### 3.1.2. Extraction Procedure

- Extract the parameter T from the stego image.
- Divide the stego image into two sub-images; one includes the odd columns, and the other includes the even columns.
- Generate a pseudo-chaotic position (using the same secret key K), as done in the insertion procedure, to obtain the same order of pixel unit position as the odd sub-image. The second pixel block has the same $Ind$ in the even image.
- Verify if $\left(\right)open="|"\; close="|">{p}_{i}^{s}-{p}_{i+1}^{s}$ and then extract the two secret bits of M$\left(\right)$ as follows:$${m}_{i}=LSB\left({p}_{i}^{s}\right);\phantom{\rule{1.em}{0ex}}{m}_{i+1}=f({p}_{i}^{s},{p}_{i+1}^{s})$$Otherwise, the chaotic system chooses another pseudo-chaotic position. The sequence follows as such for each unit position until all messages have been extracted.
- Example of insertion:The cover image is this image of “peppers” as in Figure 4:The embedded message appears as follows in 40 × 40 pixels as shown in Figure 5:The corresponding sequence of the bits message has been given as follows:$$M=10001000100011001000110001100111001001111010010110$$$$11101011000110101011101000000110100010110010\dots $$The length of the binary message is 13,120 bits.Capacity estimation produces the threshold $T=12$Suppose that the pseudo-chaotic positions of a block to embed the two bits message ${m}_{1}=1$ and ${m}_{2}=0$ are (354, 375) and (354, 376) that correspond to the 141 and 129 gray values (see Figure 6).Hiding the message bits:$$LSB\left(141\right)=1={m}_{1}=1$$$$f({p}_{1},{p}_{2})=LSB(\left(\right)open="\lfloor "\; close="\rfloor ">\frac{{p}_{1}}{2}+{p}_{2})$$We are in Case 2:$$LSB\left({p}_{i}\right)={m}_{i};f({p}_{i},{p}_{i+1})\ne {m}_{i+1}$$Therefore, the new pixel values are as follows:$$({p}_{1}^{{}^{\prime}},{p}_{2}^{{}^{\prime}})=({p}_{1},{p}_{2}+r)=(141,130)\phantom{\rule{2.em}{0ex}}\mathrm{with}\phantom{\rule{2.em}{0ex}}r=1$$The difference between the new pixel values is:$${d}^{{}^{\prime}}=\left(\right)open="|"\; close="|">{p}_{1}^{{}^{\prime}}-{p}_{2}^{{}^{\prime}}$$Then we need to adjust the new pixel values:We test the values $-50<{k}_{1}<50$ and $-50<{k}_{2}<50$ until we obtain the smallest difference between the initial values ${p}_{1}^{{}^{\prime}}$ and ${p}_{2}^{{}^{\prime}}$ and the corresponding obtained values ${e}_{1}$ and ${e}_{2}$ by using Equations (12) and (13). In our example, we find ${k}_{1}=0$ and ${k}_{2}=-1$ and then: ${p}_{1}^{\u2033}=141$, ${p}_{2}^{\u2033}=128$.
- Extraction of the bits message in the previous insertion example:The extraction is performed using the following equation:$${m}_{1}=LSB\left({p}_{1}^{\u2033}\right)=LSB\left(141\right)=1$$$${m}_{2}=f({p}_{1}^{\u2033},{p}_{2}^{\u2033})=LSB(\left(\right)open="\lfloor "\; close="\rfloor ">\frac{{p}_{1}^{\u2033}}{2}+{p}_{2}^{\u2033})$$

#### 3.2. Enhanced DCT Steganographic Method (EDCT)

#### 3.2.1. Insertion Procedure

- Read the cover image and the secret message.
- Convert the secret message into a 1-D binary vector.
- Divide the cover image into 8 × 8 blocks. Then apply the 2D DCT transformation to each block (from left to right, top to bottom).
- Use the same chaotic system to generate a pseudo-chaotic $Ind$.
- Replace the LSB of each located DCT coefficient with the one bit of the secret message to hide.
- Apply the 2D Inverse DCT transform to produce the stego image.

#### 3.2.2. Extraction Procedure

- Read the stego image.
- Divide the stego image into 8 × 8 blocks and then apply the 2D DCT to each block.
- Use the same chaotic system to generate pseudo-chaotic $Ind$.
- Extract the LSB of each pseudo-located coefficient.
- Construct the secret image.

#### 3.3. Enhanced DWT Steganographic Method (EDWT)

#### 3.3.1. Insertion Procedure

- Read the cover image and the secret image.
- Transform the cover image into one level of decomposition using Haar Wavelet.
- Permute the secret image in a pseudo-chaotic manner.
- Fuse the DWT coefficients $\left(H\right)$ of the cover image and the permuted secret image $PSI$ as follows [45]:$$\begin{array}{c}{X}^{{}^{\prime}}=\alpha X+\beta \times PSI\hfill \\ \alpha +\beta =1;\phantom{\rule{1.em}{0ex}}\alpha \gg \beta \hfill \end{array}$$In the above equations, ${X}^{{}^{\prime}}$ is the modified DWT coefficient $\left(H\right)$; X is the original DWT coefficient $\left(H\right)$. $\alpha $ and $\beta $ are the embedding strength factors; they are chosen such that the resulting stego image has a large $PSNR$. In our experiments, we tested some values of $\beta $, and the best value was found to be approximately 0.01.
- Apply Inverse Discrete Wavelet Transform (IDWT) to produce the stego image in the spatial domain.

#### 3.3.2. Extraction Procedure

- Read the stego image.
- Transform the stego image into one level of decomposition using Haar Wavelet.
- Apply inverse fusion transform to extract the permuted secret image as follows:$$PSI=({X}^{{}^{\prime}}-\alpha X)/\beta $$The extraction procedure is not blind, as we need the cover image to extract the permuted secret message.
- Apply the inverse permutation procedure using the same chaotic system to obtain the secret image.

## 4. Experimental Results and Analysis

- -
- The Entropy E, given by the following relation:$$E=-\sum _{0}^{{2}^{L}-1}p\left({P}_{i}\right)lo{g}_{2}\left(p\left({P}_{i}\right)\right)$$
- -
- The Redundancy R is usually represented by the following formula:$$R=\frac{{E}_{max}-E}{E}$$$$IR=\frac{{\sum}_{i=1}^{L}\left(\right)open="|"\; close="|">{R}_{i}-{R}_{opt}}{}{R}_{opt}({2}^{L}-1)+(S-{R}_{opt})$$Called Image Redundancy ($IR$) with:
- S being the size of the image under test;
- ${R}_{i}$ being the number of occurrences of each pixel value;
- ${R}_{opt}$ being the optimal number of occurrences that each pixel value should have to get a non-redundant image.

In the following section, we present and compare the performance of the three implemented steganographic methods.

#### 4.1. Enhanced EALSBMR

#### 4.2. Enhanced DCT Steganographic Method

#### 4.3. Enhanced DWT Steganographic Method

#### 4.4. Performance Comparison of the Three Steganographic Methods

#### 4.5. Performance Using Parameters E, R and $IR$

## 5. Universal Steganalysis

#### 5.1. Multi-Resolution Wavelet Decomposition

#### 5.2. Feature Vector Extraction

#### 5.2.1. Method 1: Feature Vectors Extracted from the Empirical Moments of the PDF-Based Multi-Resolution Coefficients and Their Prediction Error

#### 5.2.2. Method 2: Feature Vectors Extracted from Empirical Moments of CF-Based Multi-Resolution

^{st}level decomposition on the error image.

#### 5.2.3. Method 3: Feature Vector Extracted from Empirical Moments Based on the FC and the PDF of Image Prediction Error and Its Different Sub-Bands of the Multi-Resolution Decomposition

#### 5.3. Classification

#### 5.3.1. FLD Classifier

- Learning processThe learning process involves optimizing the following expression:$$J\left(W\right)=\frac{|{M}_{cp}-{M}_{sp}{|}^{2}}{{S}_{cp}+{S}_{sp}}$$$${M}_{cp}=\frac{1}{{N}_{1}}\sum _{{Z}_{p}\in {Z}_{cp}}{Z}_{p}=\frac{1}{{N}_{1}}\sum _{Z\in {Z}_{c}}{W}^{t}Z={W}^{t}{M}_{c}$$$${M}_{c}=\frac{1}{{N}_{1}}\sum _{Z\in {Z}_{c}}Z$$The mean feature vector of stego class after projection is represented as follows:$${M}_{sp}=\frac{1}{{N}_{2}}\sum _{{Z}_{p}\in {Z}_{sp}}{Z}_{p}=\frac{1}{{N}_{2}}\sum _{Z\in {Z}_{s}}{W}^{t}Z={W}^{t}{M}_{s}$$$${M}_{s}=\frac{1}{{N}_{2}}\sum _{Z\in {Z}_{s}}Z$$The scatter matrix of the cover class after projection has been shown as follows:$${S}_{cp}=\sum _{{Z}_{p}\in {Z}_{cp}}{({Z}_{p}-{M}_{cp})}^{2}=\sum _{Z\in {Z}_{c}}{({W}^{t}Z-{W}^{t}{M}_{c})}^{2}=\sum _{Z\in {Z}_{c}}{W}^{t}(Z-{M}_{c}){(Z-{M}_{c})}^{t}W={W}^{t}{S}_{c}W$$$${S}_{c}=(Z-{M}_{c}){(Z-{M}_{c})}^{t}$$The scatter matrix of the projected samples of a stego class has been shown as follows:$${S}_{sp}=\sum _{{Z}_{p}\in {Z}_{sp}}{({Z}_{p}-{M}_{sp})}^{2}=\sum _{Z\in {Z}_{s}}{({W}^{t}Z-{W}^{t}{M}_{s})}^{2}=\sum _{Z\in {Z}_{s}}{W}^{t}(Z-{M}_{s}){(Z-{M}_{s})}^{t}W={W}^{t}{S}_{s}W$$$${S}_{s}=(Z-{M}_{s}){(Z-{M}_{s})}^{t}$$The within-class scatter matrix after projection is defined as follows:$${S}_{cp}+{S}_{sp}={W}^{t}({S}_{c}+{S}_{s})W={W}^{t}{S}_{w}W$$$${S}_{w}={S}_{c}+{S}_{s}$$$${({M}_{cp}-{M}_{sp})}^{2}={({W}^{t}{M}_{c}-{W}^{t}{M}_{s})}^{2}={W}^{t}({M}_{c}-{M}_{s}){({M}_{c}-{M}_{s})}^{t}W={W}^{t}{S}_{B}W$$$${S}_{B}=({M}_{c}-{M}_{s}){({M}_{c}-{M}_{s})}^{t}$$$$J\left(W\right)=\frac{{W}^{t}{S}_{B}W}{{W}^{t}{S}_{w}W}$$$${W}_{opt}={S}_{w}^{-1}({M}_{c}-{M}_{s})$$
- Testing processThe testing process (classification step) is conducted as follows:Let Z be the matrix containing the feature vectors of covers and stegos.The projection of Z on the orientation vector ${W}_{opt}$ gives all projected values ${Z}_{p}$.$${Z}_{p}\left(j\right)=\sum _{i=1}^{9}{W}_{opt}\left(i\right)\times Z(i,j)+b\phantom{\rule{1.em}{0ex}}j=1,2,\dots ,N$$$$b=0.5\times ({M}_{cp}+{M}_{sp})$$$$\begin{array}{c}\hfill {M}_{cp}={W}_{opt}^{t}\times {M}_{c}\\ \hfill {M}_{sp}={W}_{opt}^{t}\times {M}_{s}\end{array}$$In the above equations, ${W}_{opt}^{t}$ is the transposed of ${W}_{opt}$.The result ${Z}_{p}\left(j\right)$, $j=1,\dots ,N$ determines the cover or stego class of every test image.Indeed, if ${Z}_{p}\left(j\right)\ge 0$, then the image under test is cover; otherwise, it is stego.

#### 5.3.2. SVM Classifier

- Linear Kernel:$$K({Z}_{i},{Z}_{j})={Z}_{i}^{T}{Z}_{j}$$
- Polynomial Kernel:$$K({Z}_{i},{Z}_{j})={(\gamma {Z}_{i}^{T}{Z}_{j}+r)}^{d}\phantom{\rule{1.em}{0ex}}\gamma >0$$
- RBF Kernel:$$K({Z}_{i},{Z}_{j})=exp(-\gamma {\left(\right)}^{{Z}_{i}}2)$$
- Sigmoid Kernel:$$K({Z}_{i},{Z}_{j})=\mathrm{tan}\mathrm{h}(\gamma {Z}_{i}^{T}{Z}_{j}+r)$$

## 6. Experimental Results of Steganalysis

- Poor agreement = Less than 0.20
- Fair agreement = 0.20 to 0.40
- Moderate agreement = 0.40 to 0.60
- Good agreement = 0.60 to 0.80
- Very good agreement = 0.80 to 1.00

#### 6.1. Classification Results Applied to the Steganographic Method EEALSBMR

#### 6.2. Classification Results Applied to the Steganographic Method EDCT

#### 6.3. Classification Results Applied to the Steganographic Method EDWT

#### 6.4. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 9.**(

**a**) Cover image, (

**b**) Stego image with embedding rate of 5%, (

**c**) Stego image with embedding rate of 40%.

**Figure 10.**(

**a**) Cover image, (

**b**) Stego image with embedding rate of 5%, (

**c**) Stego image with embedding rate of 40%.

**Figure 11.**(

**a**) Cover image, (

**b**) Stego image with embedding rate of 5%, (

**c**) Stego image with embedding rate of 40%.

Embedding Rate | Cover Image | $\mathit{PSNR}$ | $\mathit{IF}$ | $\mathit{SSIM}$ |
---|---|---|---|---|

5% | Baboon | 68.3810 | 0.9999 | 0.9999 |

Lena | 68.1847 | 0.9999 | 0.9999 | |

Peppers | 67.7160 | 0.9999 | 0.9999 | |

10% | Baboon | 65.5986 | 0.9999 | 0.9999 |

Lena | 65.2821 | 0.9999 | 0.9999 | |

Peppers | 64.7763 | 0.9999 | 0.9999 | |

20% | Baboon | 62.3551 | 0.9999 | 0.9999 |

Lena | 62.3559 | 0.9999 | 0.9996 | |

Peppers | 61.7066 | 0.9999 | 0.9995 | |

30% | Baboon | 60.6902 | 0.9998 | 0.9999 |

Lena | 60.5630 | 0.9998 | 0.9990 | |

Peppers | 59.9585 | 0.9998 | 0.9992 | |

40% | Baboon | 59.4245 | 0.9997 | 0.9999 |

Lena | 59.2608 | 0.9997 | 0.9985 | |

Peppers | 58.6662 | 0.9997 | 0.9988 |

Embedding Rate | Cover Image | $\mathit{PSNR}$ | $\mathit{IF}$ | $\mathit{SSIM}$ |
---|---|---|---|---|

5% | Baboon | 71.2372 | 0.9999 | 0.9999 |

Lena | 71.1769 | 0.9999 | 0.9999 | |

Peppers | 70.4866 | 0.9999 | 0.9999 | |

10% | Baboon | 64.8846 | 0.9999 | 0.9999 |

Lena | 64.9487 | 0.9999 | 0.9998 | |

Peppers | 64.1426 | 0.9999 | 0.9998 | |

20% | Baboon | 59.6895 | 0.9997 | 0.9999 |

Lena | 59.6225 | 0.9997 | 0.9992 | |

Peppers | 58.9535 | 0.9997 | 0.9993 | |

30% | Baboon | 57.4212 | 0.9995 | 0.9998 |

Lena | 57.3421 | 0.9995 | 0.9989 | |

Peppers | 56.7406 | 0.9995 | 0.9988 | |

40% | Baboon | 56.3421 | 0.9994 | 0.9997 |

Lena | 56.2265 | 0.79994 | 0.9987 | |

Peppers | 55.4876 | 0.9994 | 0.9985 |

Embedding Rate | Cover Image | $\mathit{PSNR}$ | $\mathit{IF}$ | $\mathit{SSIM}$ |
---|---|---|---|---|

5% | Baboon | 59.1876 | 0.9999 | 0.9999 |

Lena | 58.7673 | 0.9997 | 0.9999 | |

Peppers | 58.1699 | 0.9997 | 0.9999 | |

10% | Baboon | 56.2224 | 0.9997 | 0.9999 |

Lena | 55.8085 | 0.9994 | 0.9999 | |

Peppers | 55.2086 | 0.9993 | 0.9999 | |

20% | Baboon | 53.3463 | 0.9988 | 0.9999 |

Lena | 52.8205 | 0.9988 | 0.9999 | |

Peppers | 52.2269 | 0.9987 | 0.9999 | |

30% | Baboon | 52.0465 | 0.9984 | 0.9999 |

Lena | 51.6471 | 0.9984 | 0.9999 | |

Peppers | 51.0509 | 0.9983 | 0.9999 | |

40% | Baboon | 51.3450 | 0.9982 | 0.9999 |

Lena | 50.9536 | 0.9981 | 0.9999 | |

Peppers | 50.3417 | 0.9980 | 0.9999 |

Embedding Rate | Cover Image | E | R | $\mathit{IR}$ |
---|---|---|---|---|

5% | Baboon | 7.3586 | 0.0802 | 0.3805 |

Lena | 7.4455 | 0.0693 | 0.3261 | |

Peppers | 7.5715 | 0.0536 | 0.2975 | |

10% | Baboon | 7.3586 | 0.0802 | 0.3805 |

Lena | 7.4456 | 0.0693 | 0.3261 | |

Peppers | 7.5715 | 0.0535 | 0.2976 | |

20% | Baboon | 7.3585 | 0.0802 | 0.3805 |

Lena | 7.4457 | 0.0693 | 0.3261 | |

Peppers | 7.5717 | 0.0535 | 0.2977 | |

30% | Baboon | 7.3584 | 0.0802 | 0.3805 |

Lena | 7.4457 | 0.0693 | 0.3261 | |

Peppers | 7.5718 | 0.0535 | 0.2975 | |

40% | Baboon | 7.3578 | 0.0803 | 0.3806 |

Lena | 7.4454 | 0,0693 | 0.3260 | |

Peppers | 7.5722 | 0.0535 | 0.2973 |

Embedding Rate | Cover Image | E | R | $\mathit{IR}$ |
---|---|---|---|---|

5% | Baboon | 7.3585 | 0.0802 | 0.3804 |

Lena | 7.4456 | 0.0693 | 0.3261 | |

Peppers | 7.5716 | 0.0536 | 0.2976 | |

10% | Baboon | 7.3585 | 0.0802 | 0.3805 |

Lena | 7.4456 | 0.0693 | 0.3262 | |

Peppers | 7.5717 | 0.0535 | 0.2976 | |

20% | Baboon | 7.3585 | 0.0802 | 0.3804 |

Lena | 7.4457 | 0.0693 | 0.3263 | |

Peppers | 7.5725 | 0.0534 | 0.2973 | |

30% | Baboon | 7.3584 | 0.0802 | 0.3802 |

Lena | 7.4459 | 0.0693 | 0.3261 | |

Peppers | 7.5730 | 0.0534 | 0.2969 | |

40% | Baboon | 7.3578 | 0.0803 | 0.3806 |

Lena | 7.4462 | 0,0692 | 0.3257 | |

Peppers | 7.5734 | 0.0533 | 0.2973 |

Embedding Rate | Cover Image | E | R | $\mathit{IR}$ |
---|---|---|---|---|

5% | Baboon | 7.3581 | 0.0802 | 0.3805 |

Lena | 7.4455 | 0.0693 | 0.3261 | |

Peppers | 7.5715 | 0.0536 | 0.2975 | |

10% | Baboon | 7.3580 | 0.0802 | 0.3806 |

Lena | 7.4456 | 0.0693 | 0.3261 | |

Peppers | 7.5717 | 0.0535 | 0.2974 | |

20% | Baboon | 7.3580 | 0.0802 | 0.3806 |

Lena | 7.4456 | 0.0693 | 0.3261 | |

Peppers | 7.5718 | 0.0535 | 0.2975 | |

30% | Baboon | 7.3580 | 0.0802 | 0.3805 |

Lena | 7.4456 | 0.0693 | 0.3261 | |

Peppers | 7.5718 | 0.0535 | 0.2974 | |

40% | Baboon | 7.3580 | 0.0803 | 0.3806 |

Lena | 7.4457 | 0,0693 | 0.3261 | |

Peppers | 7.5721 | 0.0533 | 0.2973 |

Cover Image | E | R | $\mathit{IR}$ |
---|---|---|---|

Baboon | 7.3585 | 0.0802 | 0.3805 |

Lena | 7.4455 | 0.0693 | 0.3261 |

Peppers | 7.5715 | 0.0536 | 0.2976 |

H0: Stego Image | H1: Cover Image | |||
---|---|---|---|---|

Test outcome | Test outcome positive | True Positive $TP$ | False Positive $FP$ | Positive predictive value ($PPV$), or Precision $Pr=\frac{TP}{TP+FP}$ |

Test outcome negative | False Negative $FN$ | True Negative $TN$ | Negative predictive value ($NPV$) $NPV=\frac{TN}{TN+FN}$ | |

True positive rate ($TPR$), or, Sensitivity ($Se$), $Se=\frac{TP}{TP+FN}$ | True negative rate ($TNR$), or Specificity($Sp$), $Sp=\frac{TN}{TN+FP}$ | Accuracy ($Ac$), $Ac=\frac{TP+TN}{TP+FN+FP+TN}$ |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2744 | 0.2714 | $Pr$ = 0.5027 |

H1 | 0.2256 | 0.2286 | $NPV$ = 0.5033 |

$Se$ = 0.5487 | $Sp$ = 0.4572 | $Ex$ = 0.5030 | |

$Kappa$ = 0.0060 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2690 | 0.2645 | $Pr$ = 0.5042 |

H1 | 0.2310 | 0.2355 | $NPV$ = 0.5048 |

$Se$ = 0.5380 | $Sp$ = 0.4710 | $Ex$ = 0.5045 | |

$Kappa$ = 0.0090 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2745 | 0.2459 | $Pr$ = 0.5275 |

H1 | 0.2255 | 0.2541 | $NPV$ = 0.5298 |

$Se$ = 0.5490 | $Sp$ = 0.5082 | $Ex$ = 0.5286 | |

$Kappa$ = 0.0572 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2612 | 0.2405 | $Pr$ = 0.5207 |

H1 | 0.2387 | 0.2595 | $NPV$ = 0.5208 |

$Se$ = 0.5225 | $Sp$ = 0.5190 | $Ex$ = 0.5208 | |

$Kappa$ = 0.0415 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2504 | 0.2448 | $Pr$ = 0.5057 |

H1 | 0.2496 | 0.2552 | $NPV$ = 0.5056 |

$Se$ = 0.5008 | $Sp$ = 0.5105 | $Ex$ = 0.5056 | |

$Kappa$ = 0.0112 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3191 | 0.1946 | $Pr$ = 0.6212 |

H1 | 0.1809 | 0.3054 | $NPV$ = 0.6280 |

$Se$ = 0.6382 | $Sp$ = 0.6108 | $Ex$ = 0.6245 | |

$Kappa$ = 0.2490 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2489 | 0.2476 | $Pr$ = 0.5013 |

H1 | 0.2511 | 0.2524 | $NPV$ = 0.5012 |

$Se$ = 0.4977 | $Sp$ = 0.5048 | $Ex$ = 0.5012 | |

$Kappa$ = 0.0025 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2559 | 0.2299 | $Pr$ = 0.5268 |

H1 | 0.2441 | 0.2701 | $NPV$ = 0.5253 |

$Se$ = 0.5117 | $Sp$ = 0.5403 | $Ex$ = 0.5260 | |

$Kappa$ = 0.0520 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2990 | 0.1985 | $Pr$ = 0.6010 |

H1 | 0.2010 | 0.3015 | $NPV$ = 0.6000 |

$Se$ = 0.5980 | $Sp$ = 0.6030 | $Ex$ = 0.6005 | |

$Kappa$ = 0.2010 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3438 | 0.3431 | $Pr$ = 0.5005 |

H1 | 0.1562 | 0.1569 | $NPV$ = 0.5011 |

$Se$ = 0.6876 | $Sp$ = 0.3137 | $Ac$ = 0.6870 | |

$Kappa$ = 0.0013 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4006 | 0.3977 | $Pr$ = 0.5018 |

H1 | 0.0994 | 0.1023 | $NPV$ = 0.5071 |

$Se$ = 0.8011 | $Sp$ = 0.2046 | $Ac$ = 0.5029 | |

$Kappa$ = 0.0057 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3251 | 0.3199 | $Pr$ = 0.5041 |

H1 | 0.1749 | 0.1801 | $NPV$ = 0.5074 |

$Se$ = 0.6503 | $Sp$ = 0.3602 | $Ac$ = 0.5052 | |

$Kappa$ = 0.0105 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2220 | 0.2188 | $Pr$ = 0.5037 |

H1 | 0.2780 | 0.2812 | $NPV$ = 0.5029 |

$Se$ = 0.4440 | $Sp$ = 0.5625 | $Ac$ = 0.5032 | |

$Kappa$ = 0.0065 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2189 | 0.2161 | $Pr$ = 0.5032 |

H1 | 0.2811 | 0.2839 | $NPV$ = 0.5024 |

$Se$ = 0.4377 | $Sp$ = 0.5678 | $Ac$ = 0.5028 | |

$Kappa$ = 0.0055 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2282 | 0.1999 | $Pr$ = 0.5330 |

H1 | 0.2718 | 0.3001 | $NPV$ = 0.5247 |

$Se$ = 0.4564 | $Sp$ = 0.6002 | $Ac$ = 0.5283 | |

$Kappa$ = 0.0566 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2275 | 0.2264 | $Pr$ = 0.5013 |

H1 | 0.2725 | 0.2736 | $NPV$ = 0.5010 |

$Se$ = 0.4550 | $Sp$ = 0.5472 | $Ac$ = 0.5011 | |

$Kappa$ = 0.0023 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2412 | 0.2380 | $Pr$ = 0.5034 |

H1 | 0.2588 | 0.2620 | $NPV$ = 0.5031 |

$Se$ = 0.4825 | $Sp$ = 0.5240 | $Ac$ = 0.5032 | |

$Kappa$ = 0.0065 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2922 | 0.2684 | $Pr$ = 0.5212 |

H1 | 0.2078 | 0.2316 | $NPV$ = 0.5271 |

$Se$ = 0.5844 | $Sp$ = 0.4632 | $Ac$ = 0.5238 | |

$Kappa$ = 0.0476 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2524 | 0.2454 | $Pr$ = 0.5070 |

H1 | 0.2476 | 0.2546 | $NPV$ = 0.5069 |

$Se$ = 0.5048 | $Sp$ = 0.5091 | $Ac$ = 0.5070 | |

$Kappa$ = 0.0139 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2617 | 0.2238 | $Pr$ = 0.5390 |

H1 | 0.2383 | 0.2762 | $NPV$ = 0.5368 |

$Se$ = 0.5234 | $Sp$ = 0.5524 | $Ac$ = 0.5379 | |

$Kappa$ = 0.0758 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3104 | 0.1719 | $Pr$ = 0.6436 |

H1 | 0.1896 | 0.3281 | $NPV$ = 0.6337 |

$Se$ = 0.6208 | $Sp$ = 0.6562 | $Ac$ = 0.6385 | |

$Kappa$ = 0.2770 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2548 | 0.2343 | $Pr$ = 0.5209 |

H1 | 0.2452 | 0.2657 | $NPV$ = 0.5200 |

$Se$ = 0.5095 | $Sp$ = 0.5314 | $Ac$ = 0.5205 | |

$Kappa$ = 0.0410 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3242 | 0.1893 | $Pr$ = 0.6313 |

H1 | 0.1758 | 0.3107 | $NPV$ = 0.6386 |

$Se$ = 0.6484 | $Sp$ = 0.6213 | $Ac$ = 0.6349 | |

$Kappa$ = 0.2697 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4409 | 0.0635 | $Pr$ = 0.8741 |

H1 | 0.0591 | 0.4365 | $NPV$ = 0.8807 |

$Se$ = 0.8817 | $Sp$ = 0.8730 | $Ac$ = 0.8773 | |

$Kappa$ = 0.7547 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2611 | 0.2499 | $Pr$ = 0.5110 |

H1 | 0.2389 | 0.2501 | $NPV$ = 0.5115 |

$Se$ = 0.5223 | $Sp$ = 0.5002 | $Ac$ = 0.5112 | |

$Kappa$ = 0.0225 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.2780 | 0.2136 | $Pr$ = 0.5655 |

H1 | 0.2220 | 0.2864 | $NPV$ = 0.5633 |

$Se$ = 0.5560 | $Sp$ = 0.5728 | $Ac$ = 0.5644 | |

$Kappa$ = 0.1288 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3739 | 0.1243 | $Pr$ = 0.7505 |

H1 | 0.1261 | 0.3757 | $NPV$ = 0.7487 |

$Se$ = 0.7478 | $Sp$ = 0.7514 | $Ac$ = 0.7496 | |

$Kappa$ = 0.4992 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.0653 | 0.0591 | $Pr$ = 0.5249 |

H1 | 0.4347 | 0.4409 | $NPV$ = 0.5035 |

$Se$ = 0.1307 | $Sp$ = 0.8817 | $Ac$ = 0.5062 | |

$Kappa$ = 0.0124 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.0848 | 0.0644 | $Pr$ = 0.5683 |

H1 | 0.4152 | 0.4356 | $NPV$ = 0.5120 |

$Se$ = 0.1695 | $Sp$ = 0.8712 | $Ac$ = 0.5204 | |

$Kappa$ = 0.0408 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.1734 | 0.0843 | $Pr$ = 0.6729 |

H1 | 0.3266 | 0.4157 | $NPV$ = 0.5600 |

$Se$ = 0.3469 | $Sp$ = 0.8314 | $Ac$ = 0.5891 | |

$Kappa$ = 0.1783 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3156 | 0.3138 | $Pr$ = 0.5014 |

H1 | 0.1844 | 0.1862 | $NPV$ = 0.5024 |

$Se$ = 0.6312 | $Sp$ = 0.3724 | $Ac$ = 0.5018 | |

$Kappa$ = 0.0036 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3572 | 0.3266 | $Pr$ = 0.5224 |

H1 | 0.1428 | 0.1734 | $NPV$ = 0.5485 |

$Se$ = 0.7145 | $Sp$ = 0.3469 | $Ac$ = 0.5307 | |

$Kappa$ = 0.0613 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4217 | 0.2220 | $Pr$ = 0.6551 |

H1 | 0.0783 | 0.2780 | $NPV$ = 0.7803 |

$Se$ = 0.8434 | $Sp$ = 0.5560 | $Ac$ = 0.6997 | |

$Kappa$ = 0.3994 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3053 | 0.3020 | $Pr$ = 0.5027 |

H1 | 0.1947 | 0.1980 | $NPV$ = 0.5042 |

$Se$ = 0.6107 | $Sp$ = 0.3960 | $Ac$ = 0.5033 | |

$Kappa$ = 0.0067 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3021 | 0.2924 | $Pr$ = 0.5082 |

H1 | 0.1979 | 0.2076 | $NPV$ = 0.5120 |

$Se$ = 0.6042 | $Sp$ = 0.4152 | $Ac$ = 0.5097 | |

$Kappa$ = 0.0194 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3264 | 0.2427 | $Pr$ = 0.5736 |

H1 | 0.1736 | 0.2573 | $NPV$ = 0.5971 |

$Se$ = 0.6528 | $Sp$ = 0.5147 | $Ac$ = 0.5837 | |

$Kappa$ = 0.1674 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4786 | 0.0150 | $Pr$ = 0.9695 |

H1 | 0.0214 | 0.4850 | $NPV$ = 0.9577 |

$Se$ = 0.9571 | $Sp$ = 0.9699 | $Ac$ = 0.9635 | |

$Kappa$ = 0.9270 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4941 | 0.0056 | $Pr$ = 0.9888 |

H1 | 0.0059 | 0.4944 | $NPV$ = 0.9882 |

$Se$ = 0.9882 | $Sp$ = 0.9888 | $Ac$ = 0.9885 | |

$Kappa$ = 0.9770 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4993 | 0.0005 | $Pr$ = 0.9990 |

H1 | 0.0007 | 0.4995 | $NPV$ = 0.9987 |

$Se$ = 0.9987 | $Sp$ = 0.9990 | $Ac$ = 0.9989 | |

$Kappa$ = 0.9977 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4048 | 0.0470 | $Pr$ = 0.8961 |

H1 | 0.0952 | 0.4530 | $NPV$ = 0.8263 |

$Se$ = 0.8095 | $Sp$ = 0.9061 | $Ac$ = 0.8578 | |

$Kappa$ = 0.7156 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4536 | 0.0311 | $Pr$ = 0.9358 |

H1 | 0.0464 | 0.4689 | $NPV$ = 0.9100 |

$Se$ = 0.9072 | $Sp$ = 0.9377 | $Ac$ = 0.9225 | |

$Kappa$ = 0.8450 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4753 | 0.0232 | $Pr$ = 0.9534 |

H1 | 0.0247 | 0.4768 | $NPV$ = 0.9508 |

$Se$ = 0.9507 | $Sp$ = 0.9535 | $Ac$ = 0.9521 | |

$Kappa$ = 0.9042 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3946 | 0.0650 | $Pr$ = 0.8587 |

H1 | 0.1054 | 0.4350 | $NPV$ = 0.8049 |

$Se$ = 0.7891 | $Sp$ = 0.8701 | $Ac$ = 0.8296 | |

$Kappa$ = 0.6592 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4394 | 0.0387 | $Pr$ = 0.9191 |

H1 | 0.0606 | 0.4613 | $NPV$ = 0.8839 |

$Se$ = 0.8789 | $Sp$ = 0.9227 | $Ac$ = 0.9008 | |

$Kappa$ = 0.8015 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4603 | 0.0321 | $Pr$ = 0.9348 |

H1 | 0.0397 | 0.4679 | $NPV$ = 0.9218 |

$Se$ = 0.9206 | $Sp$ = 0.9358 | $Ac$ = 0.9282 | |

$Kappa$ = 0.8564 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4770 | 0.0230 | $Pr$ = 0.9541 |

H1 | 0.0230 | 0.4770 | $NPV$ = 0.9541 |

$Se$ = 0.9541 | $Sp$ = 0.9541 | $Ac$ = 0.9541 | |

$Kappa$ = 0.9082 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4893 | 0.0058 | $Pr$ = 0.9883 |

H1 | 0.0107 | 0.4942 | $NPV$ = 0.9789 |

$Se$ = 0.9787 | $Sp$ = 0.9884 | $Ac$ = 0.9835 | |

$Kappa$ = 0.9670 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4984 | 0.0084 | $Pr$ = 0.9835 |

H1 | 0.0016 | 0.4916 | $NPV$ = 0.9967 |

$Se$ = 0.9968 | $Sp$ = 0.9832 | $Ac$ = 0.9900 | |

$Kappa$ = 0.9800 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3366 | 0.1658 | $Pr$ = 0.6700 |

H1 | 0.1634 | 0.3342 | $NPV$ = 0.6716 |

$Se$ = 0.6731 | $Sp$ = 0.6684 | $Ac$ = 0.6708 | |

$Kappa$ = 0.3415 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4107 | 0.1371 | $Pr$ = 0.7497 |

H1 | 0.0893 | 0.3629 | $NPV$ = 0.8024 |

$Se$ = 0.8213 | $Sp$ = 0.7257 | $Ac$ = 0.7735 | |

$Kappa$ = 0.5470 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4605 | 0.1175 | $Pr$ = 0.7967 |

H1 | 0.0395 | 0.3825 | $NPV$ = 0.9063 |

$Se$ = 0.9210 | $Sp$ = 0.7650 | $Ac$ = 0.8430 | |

$Kappa$ = 0.6859 |

5% | H0: Stego Images | H1: Cover Images | |

H0 | 0.3707 | 0.1108 | $Pr$ = 0.7699 |

H1 | 0.1293 | 0.3892 | $NPV$ = 0.7506 |

$Se$ = 0.7413 | $Sp$ = 0.7785 | $Ac$ = 0.7599 | |

$Kappa$ = 0.5198 | |||

10% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4332 | 0.0725 | $Pr$ = 0.8567 |

H1 | 0.0668 | 0.4275 | $NPV$ = 0.8649 |

$Se$ = 0.8665 | $Sp$ = 0.8550 | $Ac$ = 0.8608 | |

$Kappa$ = 0.7215 | |||

20% | H0: Stego Images | H1: Cover Images | |

H0 | 0.4672 | 0.0724 | $Pr$ = 0.8659 |

H1 | 0.0668 | 0.4276 | $NPV$ = 0.9288 |

$Se$ = 0.9345 | $Sp$ = 0.8552 | $Ac$ = 0.8949 | |

$Kappa$ = 0.7897 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Battikh, D.; El Assad, S.; Hoang, T.M.; Bakhache, B.; Deforges, O.; Khalil, M.
Comparative Study of Three Steganographic Methods Using a Chaotic System and Their Universal Steganalysis Based on Three Feature Vectors. *Entropy* **2019**, *21*, 748.
https://doi.org/10.3390/e21080748

**AMA Style**

Battikh D, El Assad S, Hoang TM, Bakhache B, Deforges O, Khalil M.
Comparative Study of Three Steganographic Methods Using a Chaotic System and Their Universal Steganalysis Based on Three Feature Vectors. *Entropy*. 2019; 21(8):748.
https://doi.org/10.3390/e21080748

**Chicago/Turabian Style**

Battikh, Dalia, Safwan El Assad, Thang Manh Hoang, Bassem Bakhache, Olivier Deforges, and Mohamad Khalil.
2019. "Comparative Study of Three Steganographic Methods Using a Chaotic System and Their Universal Steganalysis Based on Three Feature Vectors" *Entropy* 21, no. 8: 748.
https://doi.org/10.3390/e21080748