High Imperceptible Data Hiding Method Based on Combination Theory for AMBTC Compressed Images

Abstract

The rapid development of digital multimedia has resulted in a massive storm of large-scale data. The data compression method reduces data size and lowers transmission costs. As a result, data-hiding research in the compression domain poses significant challenges to researchers. This work proposes a highly imperceptible data-hiding scheme for a compressed absolute moment block truncation coding (AMBTC) image. The proposed method addresses the shortcomings of the current block-based AMBTC data-hiding scheme, with an incredibly complex block as non-embedding. This is because embedding in complex blocks causes substantial distortions in the image, resulting in low imperceptibility performance. Combination theory was implemented to ensure the high imperceptibility of the modified compressed images. The experimental findings proved that the proposed method achieved high imperceptibility and high hiding capacity by modifying one pixel in a block to embed confidential bits.

1. Introduction

A massive storm of large-scale data has come from the accelerated development of digital multimedia. Various applications produce images, audio, and videos daily. The security of critical data and the gigantic volume of data are the two major issues this trend has brought forth. To enhance the confidentiality of data transmission, the sender encrypts the data using conventional cryptographic techniques or imperceptibly transfers data using data-hiding techniques [1,2,3]. At the same time, the best way to minimize the data size and lower transmission costs is to use a data compression technique [4]. Inevitably, data-hiding research in the compression domain presents a immense challenge to academics because it is essential to balance the hiding capacity with compressed image quality.

Data hiding is defined as an imperceptible method of hiding confidential data inside other media i.e., digital images, videos, and audio. An image-data-hiding scheme can be categorized into three domains [5] namely the spatial domain, the frequency domain, and the compressed domain. In order to embed confidential data, the spatial domain method directly modifies the image pixel [6,7,8]. It causes low imperceptibility. On the other side, the frequency domain method hides the confidential data into transformed coefficients. Lastly, the compressed domain method manages the confidential messages in the compressed codes of digital images. Both the frequency domain and the compressed domain perform better imperceptibility than the spatial domain.

In general, two metrics are used to assess the performance of a data-hiding technique. The first is hiding capacity. Hiding capacity is identified as the quantity of confidential data concealed in a cover media. The second metric is imperceptibility. Imperceptibility is defined as the level of image quality after modification. The primary issue in data-hiding research is that the higher the hiding capacity, the lower the imperceptibility.

Along with the big data age, where the volume and variety of data are abundant, compression technology is experiencing advancements. Data compression technology is efficient in data transmission. This is due to the fact that computing is simple and inexpensive. There are several widely used compression methods, including vector quantization (VQ) [9,10,11,12], block truncation coding (BTC) [13,14,15,16], AMBTC [17,18], and JPEG compression [19,20,21], beginning with Delp and Mitchell [13] in 1979, who proposed block truncation coding (BTC) for image compression, and then further developed in 1984 by Lema and Mitchell [17], whose method is known as absolute moment block truncation coding (AMBTC). AMBTC offers simpler computation than BTC and acquires better image quality.

Due to the benefits of the compression technique, data-hiding research is evolving into compression domain data hiding. Researchers have used the AMBTC compressed image-data-hiding method [22,23,24,25,26,27,28,29,30,31]. In 2020, Horng et al. published a data-hiding scheme for AMBTC compressed images utilizing quotient value differencing (QVD) and least significant bit (LSB) replacement [23]. In the same year, Lee et al. [22] presented a reversible data-hiding method based on Turtle Shell for AMBTC-compressed images. The following year, in 2021, Lin et al. [23] proposed a data-hiding scheme that employs the difference value of higher and lower quantizers. The three methods are block classification based on data hiding. After performing AMBTC encoding, this method divides the sub-blocks into smooth or complex ones based on a pre-defined threshold. Complex blocks are non-embedding blocks that cannot embed the secret bits. Otherwise, smooth blocks are defined as embeddable blocks in which a certain number of secret bits can be embedded. These methods provide lower computation complexity and achieved accepted image quality.

Moreover, these methods benefit smooth images because they achieve higher embedding capacity. However, these methods are not applicable to complex images because they cannot embed confidential messages in a complex image. Embedding in the complex block distorts the image and results in low imperceptible performance for the data-hiding method.

Based on the concerns above, this research proposes an innovative data-hiding scheme for AMBTC-compressed images based on combination theory. The paper’s main contributions are highlighted in the following:

i.

Non-block classification. In the proposed scheme, all blocks are embeddable. The threshold value is not established because there is no specification for a smooth and complex block.

ii.

High embedding capacity in the complex block. By utilizing the concept of combination theory, the proposed scheme can embed at least four secret bits resulting in higher embedding capacity in the complex block compared with the related works.

iii.

High imperceptible AMBTC data hiding. By applying the concept of combination theory, the proposed scheme only needs to change 1 bit of pixel to embed 4 secret bits. Hence, it has advantages for complex images.

The remainder of this work is structured as follows. Section 2 provides an overview of the related study. Our proposed data-hiding method is described in Section 3. Section 4 contains the findings of the experiments. Section 5 finally presents the conclusions.

2. Related Works

2.1. Absolute Moment Block Truncation Coding (AMBTC)

In 1984, AMBTC was introduced by Lema and Mitchell [17]. The main steps of AMBTC consist of encoding and decoding procedures and are explained in detail as follows:

The Encoding Procedure

Step 1. Segment the images into a collection of t × t non-overlapping blocks.

Step 2. Calculate the mean μ and the standard deviation α by using (1) and (2), respectively. Assume that $p_{ij}$ is the value of pixel in the block’s position (i, j).

$$\mu =\frac{1}{t\times t}{\sum }_{i=1}^t{\sum }_{j=1}^t{p}_{ij}$$

(1)

$$\alpha =\sqrt{\frac{{\sum }_{j=1}^{t\times t}\left|{p_{ij}}^2-{\mu }^2\right|}{t\times t}}$$

(2)

Step 3. Create the bitmap value $bm=\left\{m_{ij}|m_{ij}\in \left\{\mathrm{0,1}\right\},1\le i,j\le t\right\}$ by using (3):

$$m_{ij}=\left\{\begin{array}{c}1,if{p}_{ij}\ge \mu \\ 0,if{p}_{ij}<\mu \end{array}\right.$$

(3)

The Decoding Procedure

Step 1. Calculate two quantizers, i.e., high quantizer hq and low quantizer lq, using (4) and (5), respectively.

$$hq=\mu -\alpha \sqrt{\frac{q}{t\times t-q}}$$

(4)

$$lq=\mu +\alpha \sqrt{\frac{t\times t-q}{q}}$$

(5)

where q is the quantity of $p_{ij}$ which is greater than or equal to the mean value μ.

Step 2. Create the reconstructed block $rm=\left\{r_{ij}|r_{ij}\in \left\{hq,lq\right\},1\le i,j\le k\right\}$ by using (6):

$$r{m}_{ij}=\left\{\begin{array}{c}hq,if{m}_{ij}=1\\ lq,if{m}_{ij}=0\end{array}\right.$$

(6)

In order to provide a detailed understanding of the AMBTC compression method, we provide a numerical example as follows. Figure 1 depicts the illustration of AMBTC. First, Figure 1a depicts the original image block size of 4 × 4. Next, we computed the block means μ and standard deviation α by using (1) and (2), respectively. In this example, the value of μ = 78 and α = 19. Next, we generate the bitmap in the encoding procedure using (3). The result is shown in Figure 1b. Finally, we calculate the higher and lower quantization levels, hq and lq, using (4) and (5), respectively. In this example, hq = 105 and lq = 62. Finally, the reconstructed image block generated in the decoding procedure using (6) is shown in Figure 1c.

2.2. Combination Theory

Wu et al. [32] introduced a novel data-hiding method for binary images using combination theory in 2017. In their method, senders can hide more than one bit of confidential data by modifying one pixel at a maximum in one block. The experimental result presented by Wu et al.’s scheme [32] increases the embedding capacity and reduces the distortion of the modified image.

The main procedure of Wu et al.’s scheme [32] is as follows:

Step 1. Calculate the block maximum payload by (7):

$$b=⌊{\log }_2(u\times v+1⌋$$

(7)

Step 2. Develop a hidden location matrix P as described in [32].

Step 3. Produce the new computed hidden location matrix P’ by P ⊗ Gi.

Then, using Algorithm 1, calculate the total number T(Sr, r = 1, 2,…, b) of each bit of confidential data that exists in P’.

Algorithm 1: Confidential Data Calculation

FOR p = 1 :u; q = 1 : v

IF P’x,y hasSr

T(Sr) = T(Sr) +1

END

END

Step 4. Calculate the residual value S’r for the total amount of instances of each bit Sr by modulo as in (8):

$${S\prime }_r=T\left(S_r\right)mod2,\hspace{1em}\hspace{1em}\hspace{1em}wherer=\mathrm{1=},2,\dots ,b$$

(8)

Step 5. Investigate the original confidential data Sr to the feature value S’_r by (9), which consists of S_r such that Sr ≠ S’_r

$$S^{″}=S_r\odot (S_r\oplus {S\prime }_r)$$

(9)

Step 6. Conceal the confidential data into a block Gi by changing only one pixel of Gi according to Algorithm 2.

Algorithm 2: Confidential Data Embedding

FOR p = 1: u; q = 1: v

IF Px,y= S”

IF Gi (p,q) = 1

Gi (p,q) = 0

ELSE

Gi (p,q) = 1

END

END

END

2.3. Block-Classification-Based RDH for AMBTC-Compressed Images

Lee et al. [22], Horng et al. [23], and Lin et al. [24] presented a block-based data-hiding scheme for AMBTC-compressed images. The classification is applied in the sub-block of the image by utilizing a pre-defined threshold. The threshold is used to classify whether the sub-block is smooth or complex. When the sub-block is smooth, then it is called an em-beddable block. This means the secret bits can be embedded in the block. Otherwise, when the block is complex, it is called a non-embedding block. This means that the secret bits cannot be embedded in the block. The threshold value is varies depending on the needs of the user. When the user needs to embed more secret bits, a high threshold value is set. However, the higher embedding capacity affects the image quality. The way of carrying out block classification of three related works is presented in

Table 1. The rules for block classification.

	Lee et al.’s Scheme [22]	Horng et al.’s Scheme [23]	Lin et al.’s Scheme [24]
Complex Block	difference value D = |hq − lq| > TH	difference value D = ||hq − lq| > TH	difference value D = ||hq − lq|≤ 2m
Smooth Block	difference value D = ||hq − lq|≤ TH	difference value D = ||hq − lq|≤ TH	difference value D = ||hq − lq| > 2m

hq is the high quantizer, lq is the low quantizer, and m is the length of the binary representation of difference value D.

.

3. The Proposed Method

Inspired by Wu et al.’s scheme [32], in this paper, we propose an innovative data-hiding scheme for AMBTC image compression based on combination theory. Figure 2 shows the flowchart of the proposed data-hiding scheme, consisting of image compression and data embedding. The proposed scheme involves embedding and extracting procedures. Section 3.1 provides the embedding procedures, while Section 3.2 presents the extracting procedures.

3.1. The Embedding Procedures

In this part, we present the procedures of concealing confidential data into the compressed image using a hidden position matrix which was generated by combination theory. Our proposed embedding procedure is explained in depth as follows.

Input: An original image TI, with R width and S height, and confidential data S.

Output: Compressed stego image TI’.

Given an original image TI, with R width and S height, segment into the n×n non-overlapping block image.

Step 1. Compress the original image block to obtain the bitmap value, Bm, according to Section 2.1.

Step 2. Embed the confidential data S into the bitmap Bm according to Section 2.2.

An example is given to further describe the proposed scheme for better understanding.

Input: Suppose we have a block size of 4 × 4 pixels of an image. The block consists of the original pixel values (44, 61, 72, 58, 74, 65, 79, 62, 127, 93, 75, 56, 112, 117, 103, and 60}). The confidential data stream S that must be concealed is (1101)₂.

Step 1. Compress the original image block by using the AMBTC method. Figure 3 is the AMBTC encoding of the given pixel value.

Step 2. Embed the confidential data bits into a bitmap by implementing combination theory as explained in Section 2.1. Figure 4 depicts the secret matrix generation which a multiplication operation between bitmap and secret position matrix P. Therefore, Figure 5 shows the secret matrix calculation of each A, B, C and D in the new secret position matrix P’. For every sum value of A, B, C and D, which are 3, 3, 4 and 6 respectively the mod 2 operation is managed for resulting S’_r as S’_{r =} (1100).

Moreover, as Sr ≠ S’_r, 1101 ≠ 1100 then S” = (ABCD) ⊙ ((1101) ⊕ (1100)) = (ABCD) ⊙ (0001) = D. Figure 6 depicts the final result of the embedding procedures, which consisting the original bitmap, stego bitmap, reconstructed original image and reconstructed stego image.

3.2. The Extraction Procedures

The extraction procedures are performed to retrieve confidential data. Our proposed extraction procedure is explained in depth below.

Step 1. Calculate the new matrix $P^{\prime }=P\odot {G\prime }_i$ where ${p\prime }_{x,y}=P_{x,y}\times {G\prime }_{i(x,y)}$ for all p = 1, 2,…, u, and q = 1, 2,…, v.

Step 2. Calculate the total amount of each confidential data bit $T\left(S_r\right),r=1,2,\dots ,k$ in the new matrix P’ using Algorithm 3.

Algorithm 3: Confidential Data Calculation

FOR p = 1: u ; q = 1: v

IF P’x,y hasSr

T(Sr) = T(Sr) +1

END

END

Step 3. Reassemble the confidential data bit $S_r=S_r\in \mathrm{0,1},r=\mathrm{1,2},\dots ,b,$ using (10).

$$S_r={T(S}_r)mod2$$

(10)

Here is an example of an extraction procedure to provide an in-depth understanding. The stego image generated from the embedding procedures in Section 3.1 consists of the pixel following values (62, 62, 62, 105, 62, 62, 105, 105, 105, 105, 62, 62, 105, 105, 105, and 62). Thus, the bitmap value ${G\prime }_i$ are (0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, and 0). Moreover, the known matrix P values are (A, B, C, D, AB, AC, AD, BC, BD, CD, ABC, ABD, BCD, ACD, ABCD, and A). Therefore, the step-by-step extraction is as follows:

Step 1. Calculate the new matrix $P^{\prime }=P\odot {G\prime }_i$. Figure 7 depicts the new matrix P’ calculation which an operation between bitmap and matrix P.

Step 2. Calculate the total amount of each confidential data bit $T\left(S_r\right),r=1,2,\dots ,k$ in the new matrix P’. Figure 8 shows the confidential data bit calculation which a sum operation of each A, B, C and D in the new matrix P’.

Step 3. Reassemble the confidential data bit.

$$S_1=3 mod 2=1$$

$$S_2=3 mod 2=1$$

$$S_3=4 mod 2=0$$

$$S_4=7 mod 2=1$$

Finally, the confidential data stream S extracted is (1101)₂.

4. Experimental Results

4.1. The Environment and Parameters

We performed extensive experiments to determine the efficacy of our proposed data-hiding technique. An assessment was conducted to determine the hiding capacity and imperceptibility. We utilized the common test grayscale images, known as airplane, baboon, boat, Lena, and peppers, as shown in Figure 9a–e, respectively. All of the test images are 512 × 512 pixels in size. We performed binary confidential data which were produced by a pseudo-random number generator. The experimental test runs on Windows 10 Home Single Language with an Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz 1.99 GHz. Furthermore, all programs were generated and executed in MATLAB 9.8 (R2020a). The hiding capacity is identified by the number of bits, while the imperceptibility is identified by image quality. To assess image quality, the commonly used parameter is the peak signal-to-noise ratio (PSNR) as defined in Equation (11).

$$PSNR=10{\log }_{10}\frac{{255}^2}{MSE}$$

(11)

The mean square error (MSE) is defined in Equation (12).

$$MSE=\frac{1}{N\times N}{\sum }_{i=1}^N{\sum }_{j=1}^N{({TI}_{ij}-{TI\prime }_{ij})}^2$$

(12)

where ${TI}_{ij}$ is the pixel in position of the i-th row and the j-th column of image $TI$, and ${TI\prime }_{ij}$ is the pixel in the i-th row and the j-th column image $TI\prime$

•  

Figure 9. Image testing: (a) airplane; (b) baboon; (c) boat; (d) Lena; and (e) pepper.

Figure 9. Image testing: (a) airplane; (b) baboon; (c) boat; (d) Lena; and (e) pepper.

4.2. Results and Discussion

The experimental findings are presented in

Table 2. Performance comparison with related works.

Image	Metrics	Lee et al.’s Scheme TH = 0 [22]	Horng et al.’s Scheme TH = 0 [23]	Lin et al.’s Scheme TH = 0 [24]	Proposed Scheme
Airplane	PSNR	33.63	31.65	31.95	38.25
	MSE	28.22	44.47	41.50	9.72
	Capacity	95,255	114,666	81,097	195,928
Baboon	PSNR	30.88	26.39	26.00	37.97
	MSE	53.12	149.30	163.33	10.36
	Capacity	83,060	126,600	64,882	104,284
Boat	PSNR	31.68	30.59	31.15	36.27
	MSE	44.11	56.76	49.89	15.34
	Capacity	83,240	115,448	78,178	158,440
Lena	PSNR	33.48	31.42	33.97	37.42
	MSE	29.15	46.78	26.06	11.76
	Capacity	91,190	124,877	75,128	201,364
Pepper	PSNR	32.67	31.72	33.40	35.89
	MSE	35.18	43.76	29.72	16.72
	Capacity	83,600	125,657	78,799	197,632

, where the PSNRs, MSEs, and hiding capacity are shown. For the proposed scheme, in terms of hiding capacity, the highest hiding capacity is 201,364 for the image of Lena. While the lowest hiding capacity is 104,284, as experienced by the image of the baboon. On average, the hiding capacity is 171,529. This means 60% of the test images exceed the average hiding capacity. This indicated that the proposed scheme proved to have a high hiding capacity.

Furthermore, regarding image quality, the experimental results provided excellent findings, whereby all the PSNR values of the test images are above 35.89 dB. The highest PSNR value was shown by the image of the airplane, 38.25 dB. It is widely accepted knowledge in image data-hiding that when the PSNR value of an image exceeds 30 db, image modifications remain undetectable. Therefore, the findings proved that the proposed scheme has high imperceptibility.

Further testing was performed by comparing the proposed scheme to those developed by Lee et al. [22], Horng et al.’s scheme, and Lin et al.’s scheme. The main reason for comparing these three schemes is to find the appropriate scheme for embedding a secret message in the complex block. That is why the predefined threshold is equal to 0. The performance comparison of the proposed scheme and others’ schemes can be seen in Table 2 and Figure 9.

As seen in Table 2, Lee et al.’s scheme’s PSNR value for all test images is lower than the proposed scheme. The difference is 5dB overall. This pattern is proportional to the value of the MSE. As the higher the MSE, the lower the PSNR is. Nevertheless, Lee et al.’s approach produces adequate image quality because all their PSNRs are more than 30 dB. The notable difference is in the number of embedded bits. As demonstrated in Figure 10, the proposed method has double the hiding capacity of Lee et al.’s approach. This demonstrates that the proposed approach is highly advantageous for complex images.

The second comparison is against Horn et al.’s scheme. As presented in Table 2, the most significant difference in the PSNR value is experienced by the baboon image, 26.39 dB and 37.97 dB for Lee et al.’s scheme and the proposed scheme, respectively. It has been proven that the imperceptibility of the proposed scheme is outstanding. Figure 10 depicts the strange trend of the MSE baboon image value, which corresponds to the PSNR value. This is because the PSNR value is relatively low, precisely 26 dB; therefore, the MSE is very high, 149.30. Moreover, Figure 10 illustrates the most exciting trend regarding hiding capacity in the baboon image. In terms of hiding capacity, as demonstrated in Figure 10, the proposed method has double the hiding capacity of Horng et al.’s approach.

The last comparison is to Lin et al.’s scheme. As seen in Table 2, the pattern of Lin et al.’s scheme is generally similar to Horng et al.’s scheme. However, regarding image quality, Lin et al.’s scheme is better than Horng et al.’s. Conversely, Horng et al.’s scheme can embed more secret bits than Lin et al.’s. Finally, from the above discussion, it can be proven that the proposed scheme performs better than the three other schemes for handling the complex block. The proposed scheme is highly invisible and has the most excellent hiding capacity.

5. Conclusions

With combination theory, this paper proposed a highly imperceptible data-hiding method in AMBTC-compressed images. The proposed scheme is a non-block-classification-based scheme, as no predefined threshold is set to classify the block into smooth and complex. All the blocks are embeddable. The embedding strategy utilizes combination theory. The proposed scheme can embed at least 4 bits in a block using combination theory. The proposed scheme can achieve higher hiding capacity even in complex images. Moreover, by only modifying 1 bit in a block, the proposed scheme proved to be highly imperceptible. This is a very fundamental requirement for data hiding in the compressed domain.

Further, our proposed scheme performs well with a practical and low-computation algorithm. The experimental results show that the proposed scheme satisfied the modified compressed image quality and had a high hiding capacity. Moreover, the proposed scheme performs better than related schemes, especially in complex images.

For further development, our proposed scheme can be very effective and adopted appropriately in medical imaging for securing electronic patient information because our proposed scheme induces invisible modification of image quality.