High-Capacity Image Steganography based on Discrete Hadamard Transform

High capacity and high imperceptibility are the primary targets for ideal image steganography. For the traditional transform-based schemes, the main challenge is to balance the imperceptibility, hiding capacity, and running efficiency. To obtain a high-quality image in dense embedding, existing high-capacity schemes usually sacrifice running efficiency and security. These disadvantages make the schemes less appealing. In this paper, based on a lightweight transform Discrete Hadamard Transform (DHT), we introduce a simple but high-performance image steganographic model. In the case of only stego-image passed, the experiment results demonstrate that the proposed scheme achieves high imperceptibility and security even in the dense embedding of 8 BPP (Bits Per Pixel). Furthermore, the proposed scheme withstands various tests and shows desirable robustness. The comparative analyses are demonstrated that our scheme is efficient and feasible image steganography.

For the decade years, many image steganography has been introduced and developed. Generally, these schemes can be mainly divided into two categories: spatial domain and transform domain. The common spatial domain approaches include Least Significant Bit (LSB) [7][8][9], Histogram Shifting (HS) [10], and Pixel Value Differencing (PVD) [11][12][13]. In addition, other spatial approaches such as bit flipping are also proposed and developed recently [14,15].
In the spatial domain schemes, secret data is directly hidden over the pixel values of the cover image. For LSB methods, the least significant bits of pixels in the cover image are discarded and replaced by secret data. And in PVD based methods, the secret message is concealed in the difference value of the consecutive pixels group.
The advantages of spatial domain methods include easy implementation and fast running speed. However, these methods are usually less reliable in robustness and security [16]. It is found that the classic LSB method is vulnerable to RS attack [17]. To increase security, LSB matching (LSBM) is proposed [18]. Unlike the simple LSB substitution, in the LSB matching, the pixel values are increased or decreased randomly by one to match the messages to be hidden. The LSBM reduces the imbalance in the embedding distortion and thus shows higher security in resisting steganographic attacks. Recently, related improved works include dual-layer LSB matching [19], modified LSB matching combined with multi stego-medium [20] and pixel difference [21]. In addition, chaotic systems can also be combined with steganography to achieve better performance. For instance, scheme [22] encrypts the secret message with chaos encryption technology before embedding; scheme [23] utilizes a modified chaotic system to determine embedding locations; scheme [24] and scheme [9] combine the chaotic map system with the optimization algorithms to locate the optimal position and to minimize the distortion of stegoimages.
For PVD based methods, the existence of the Fall Off Boundary Problem (FOBP) has attracted the researcher's attention. Scheme [12] avoids the fall off boundary problem and reduces the distortion of the stego-image by utilizing the modulus function with pixel readjustment. Scheme [13] exploits the usage of multi-directional pixel value difference. The scheme improves the performance in Hiding capacity and image quality while avoiding the FOBP and incorrect extraction problem (IEP).
Unlike the spatial domain methods, transform based technology embedding the confidential data in the transformed coefficients. The most commonly used transforms include the Discrete Cosine Transform (DCT) [25,26] and the Discrete Wavelet Transform (DWT) [27,28]. Recently, more transform methods including Integer Wavelet Transform (IWT) [29,30], Complex Wavelet Transform (CWT) [31], and the Dual-Tree Complex Wavelet Transform (DT-CWT) [32,33] has attracted the researchers' attention. Generally, the existing approaches use different ways to boost efficiency. To improve security, chaotic systems has also been used in transform domain methods [34,35]. Scheme [34] utilize the modified 3D sine chaotic map to generate 3D embedding position used in color image steganography; Scheme [35] combines the DCT with chaotic map and proposes a hybrid method namely Randomly Chaotic Value Differencing (RCVD).
For both spatial based methods and transform based methods, the real challenge is the dense embedding will greatly degrade the similarity of stego-image with coverimage visually and statistically. To achieve high hiding capacity and higher imperceptibility, adaptive steganography is introduced and developed. The core strategy of adaptive steganography is to locate the optimal embedding position and thus produce better quality stego-images. The popular tools that search optimal embedding positions include Genetic algorithm (GA) [36][37][38], Particle Swarm Optimization (PSO) [29], and Ant Colony Optimization (ACO) [39]. Though the adaptive schemes improve the visual quality of the stego-image. the optimization procedure is usually time-consuming and generates extra supporting data such as location position or substitution matrix. For reversibility, supplementary data is required on the receiver side. The scheme [27] utilizes the strategy that matches the most similar cover image coefficient blocks with secret image coefficient blocks. This method achieves high visual quality and offers a good recovery of secret data. And the main disadvantage is that position data is quite considerable. Scheme [33] owns the properties of high capacity and high visual quality. however, it requires the duplicate of the cover image in the extraction stage. The original cover images (images without embedded data) are either sent through a secure channel or are already on the receiver side.
Although the above-mentioned schemes have achieved good results in hiding capacity and image quality, there is still room for improvement, especially in the condition of dense embedding. In addition, most of the methods still suffer from deficiencies in terms of operational efficiency, robustness, and security, especially most of them do not pass the common robustness tests and security tests. The scheme achieves high capacity, high imperceptibility, high robustness, high running efficiency, and without sending supporting data is rarely reported.

B. RELATED WORK AND MOTIVATION
In the proposed scheme, DHT is adopted for obvious reasons. Compared with other common transforms, DHT is more running efficiently: its operation type only contains addition and subtraction [40][41][42]. Furthermore, compared with DFT, DCT, and DWT, the distortion of cover images caused by embedding is less in DHT [43].
In the existing literature, there are many DHT based watermarking and DHT based image steganography is rarely reported [43][44][45][46]. These watermarking achieve high robustness but poor hiding capacity. In the existing DHT based image steganography, the work [42] exploits the usage of DHT in the image steganography and introduces several schemes that hide the message in the DHT coefficients. In the scheme, the cover image is split into 8 8  nonoverlapping blocks and the designed methods use different strategies to hide 1 bit in each block. In the scheme, only predefined 8 coefficients in the 8 8  blocks are considered to be the possible embedding position. It makes the hiding capacity very poor (1024 bits for 256 256  grayscale image). Besides, of all the schemes proposed in the work [42], only scheme (d) and scheme (e) are justified and tested.
To improve the hiding capacity, we consider using more coefficients for improving the hiding capacity. In the proposed scheme, all the coefficients are utilized in the maximum hiding capacity case. Due to the fact of DHT coefficients contains both positive and negative value. For the purpose of extracting the hidden data in a simple and uniform way, we process the embedding process differently depending on the positivity and negativity of the coefficients. It makes it possible to extract the hidden data at the receiver side without receiving any data even in a different hiding payload. For less distortion and better recovery, Unlike the scheme coding the secret data as a bitstream, we use a predefined value to quantize the embedded message and the secret data are embedded as float numbers instead of integer values. The simulation results demonstrate the strategy produces high-quality stego-images while preserving the good similarity of embedded data.
The main contributions of this paper are: 1) this paper introduces an efficient steganographic model. The proposed scheme shows high performance in important criterion includes hiding capacity, robustness, high imperceptibility; 2) compared with other schemes, the proposed scheme improves efficiency by increasing hiding capacity, ease of implementation, and high running efficiency.
The remainder of this paper is organized as follows. In Section II, the Discrete Hadamard Transform is introduced. In Section III, the proposed algorithm is presented. The performance analysis and a comprehensive comparison are reported in Section IV and Section V, respectively. Finally, the conclusions are drawn in Section VI.

II. DISCRETE HADAMARD TRANSFORM
The Hadamard Transform is a generalized form of Fourier Transform. The Hadamard transform (HT) is a nonsinusoidal, orthogonal transformation that decomposes a signal into a set of orthogonal, rectangular waveforms [47]. Due to the computation is comprise of addition and subtraction, Hadamard Transform is more efficient than the Fast Fourier Transform [48].
To perform DHT on a N N  matrix where 2 n N  , the N N  Hadamard unitary matrix is generated by the following rule: where n H represents Hadamard unitary matrix of order n ， and  denotes Kronecker product of two matrices, and When n 2  , the Eq. (1) can be transformed: The orthogonality of the Hadamard unitary matrix is demonstrated as follows: Thus, the transform core for forward and inverse Hadamard Transform is the same, which simplifies the calculation. The forward Discrete Hadamard Transform for an N N  matrix X is defined as follows: And the Inverse Discrete Hadamard Transform is calculated by the equation: DHT is adopted for the following reasons: 1) easy implementation; 2) simplicity and high running efficiency; 3) moderate energy compacting. The features such as easy implementation and high running efficiency make the DHT more feasible in the application, especially in resourcelimited situations [42].
The moderate energy compacting of DHT is shown in Fig. 1: most of the DHT coefficients are in the range of [-10,10]. In the instance, a 512 512  test image Peppers is converted to DHT coefficients and only the coefficients with absolute values larger than 10 are retained. The retained coefficients are approximately the top 20 percent of the amplitude of all coefficients. There is no noticeable difference between the two images by the naked human eye. The characteristic of DHT coefficients includes 1) the coefficients include both positive and negative values; 2) the most coefficients are small.

III. THE PROPOSED ALGORITHM
In the proposed scheme, DHT is applied to decompose the carrier image, the transformed coefficients are modified to embed the scaled secret image pixel value. The proposed algorithm consists of two phases: the embedding procedure and the extraction procedure.

A. PSEUDOCODE OF EMBEDDING ALGORITHM
The cover image is expected to be in RGB image format, either grayscale or color. Without loss of generality, suppose the cover image is a N N  grayscale image C and the secret data is a w h  grayscale image S (provided w N  and h N  ). For color images, the proposed method is applied in each channel separately. Besides, the cover images of size m n  ( m n  ) are also applicable for the algorithm, and can be sliced, trimmed or padded to make the image size meet the requirements. The pseudocode for embedding algorithm is detailed in Algorithm 1. Fig. 2 plots a flowchart of the embedding algorithm.

1) APPLYING DHT ON THE COVER IMAGE
According to Eq. (5), the cover image C is converted to a N N  Hadamard coefficients matrix HC .

2) ADJUSTING THE EMBEDDED DATA
To reduce the distortion caused by embedding, the embedded value should be scaled to a possible range of the coefficient value. The embedded value range is defined as . To reduce the round-off error, the chosen IF value is usually smaller than the theoretical max value MAX IF . The influence of parameters IF and Th values are discussed in Section IV.

3) EMBEDDING PROCESS
The secret data should be hidden in the coefficient values, and since the secret data are positive and the Hadamard coefficients contain both positive and negative values, the embedding process needs to be done in different ways depending on the positive and negative values. Suppose ( , ) HC i j represents the coefficients in the matrix HC at the position of the i-th row and j-th column.
After embedding, the coefficient values still maintain their original positivity and negativity. For all the modified coefficients ( , ) HC i j , the maximum absolute difference with the original coefficient values is within Th.

4) TRANSFORM THE EMBEDDED COEFFICIENTS TO SPATIAL DOMAIN
The stego-image matrix ' S is obtained by applying IDHT on the matrix HC . After converting all pixel values in matrix ' S into integers. The stego-image ' S is produced. This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2022.3181179

B. PSEUDOCODE OF EXTRACTING ALGORITHM
In the extraction procedure, stego-image ' S , Threshold value (Th), and Insert Factor (IF) are required. The secret image size w h  is provided unless the secret image size is same with cover image ( w N  , h N  ). The pseudocode for extracting algorithm is detailed in Algorithm 2. Fig. 3 plots a flowchart of the extracting algorithm.

1) APPLYING DHT ON THE STEGO-IMAGE
The stego-image ' S size is N N  , and it is converted to a N N  Hadamard coefficients matrix HC by Eq. (5).

2) EXTRACTING PROCESS
In the extracting process, the embedded data is extracted from DHT coefficients. The first step of the process is to convert all coefficients to absolute values. In the original coefficient values, coefficient values less than or equal to Th are replaced by secret data. Thus, for the coefficients

3) RECOVERING THE EXTRACTED DATA
The retrieved data should be scaled into the range of S i j into integers. The recovered image S is produced.

IV. PERFORMANCE ANALYSIS
To validate the proposed scheme, simulation experiment results and corresponding analysis are presented in this section. The experiments are conducted with MATLAB version R2019a on a desktop computer with 8G RAM and Intel (R) Core (TM) i7-7700 CPU (3.6GHZ). In our test, a famous large-scale dataset BOSSBase 1.01 [49] is used. Besides, to facilitate comparison with other well-known methods, the 512 512  pixels gray-scale images including Lena, Baboon, Lake, Jet, House, Peppers, Goldhill, and Boat are selected as test images. As shown in the Fig. 4, all these images quite popular in research community [7,29,36,45,50]. In all tests, the retrieved images are Median filtered in the last step of extraction process.
The performance metrics mentioned in this paper include Hiding Capacity (in BPP) and image quality (PSNR and SSIM).
The Hiding Capacity (HC) evaluates the amounts of bits that can be embedded in the cover image. It can be represented by Bits Per Pixel (BPP) and calculated:

return S
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.
Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Measure (SSIM) evaluate the similarity of two images. These metrics are usually used to measure the distortion in stego-image caused by the embedding process. When the PSNR value is above 36 dB, the stego-image is indistinguishable from the original image for Human Visual System (HVS) [51]. For an 8-bit grayscale image, the PSNR is calculated: where Mean Squared Error (MSE) is defined as: (9) where N denotes the number of pixels in the cover image; x y xy In the proposed algorithm, the parameters include Th and IF. The scenarios of different parameters are conducted on 8 classical images.

. 8 classic test images. (a) Lena; (b) Baboon; (c) Lake; (d) Jet; (e) House; (f) Peppers; (g) Goldhill; (h) Boat.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.

. PSNR and SSIM values on different IF values. (a) PSNR vs IF; (b) SSIM vs IF. (a) Stego-Lena (b) Stego-Baboon (c) Stego-Lake (d) Stego-Jet (e) Recovered from (a) (f) Recovered from (b) (g) Recovered from (c) (h) Recovered from (d) (i) Stego-House (j) Stego-Peppers (k) Stego-Goldhill (l) Stego-Boat (m) Recovered from (i) (n) Recovered from (j) (o) Recovered from (k) (p) Recovered from (l) FIGURE 7. Stego-image and Retrieved image for 8 classic test images.
This article has been accepted for publication in IEEE Access. This is the author's version which has not been fully edited and content may change prior to final publication.  Fig. 8. As shown in the figure, the average PSNR and average SSIM values of the stego-imges and cover images both decrease with the hiding capacity increasing from 1 BPP to 8 BPP. It can justify that the imperceptibility of the stego-images is not significantly degraded in the case of high hidden capacity. Even in the max hiding capacity (8 BPP), the PSNR value still maintains above 36 dB (approximately 37 dB). To maximize the use of payload, 8 BPP is recommended in the application.
To further evaluate the influence of different carriers, we conduct the proposed method in a large-scale image dataset: BOSSbase 1.01. The simulation results performed on 1000 images are shown in Table II. As shown in the Table, the proposed method maintains a stable high performance in the large-scale tests. Table III demonstrates the comparison with other highcapacity schemes. As is shown in the Table, the proposed scheme shows more stability and better performance in both Hiding Capacity (in BPP) and image quality (in PSNR).
To show the proposed method is also applicable for color images, we test the classical color images and the simulation results are shown in Fig. 9. It can be observed that, as with the results of the grayscale images, the color stego-images also show no visible distortion to the naked eye. The PSNR and SSIM values of the stego-image are

C. ROBUSTNESS ANALYSIS
With steganography, robustness refers to the ability to extract hidden data from corrupted stego-files. Unlike watermarking, robustness is not the primary goal of steganography. For watermarking, the stego-files must be able to withstand various deliberate attacks such as rotation, sharpening, etc. For steganography, however, active attack scenarios are not a consideration [6]. Although high robustness to active attacks is not a mandatory requirement for steganography. However, in the real world, the stego-images may encounter some unintended attacks during transmission. The critical secret data may be lost during transmission, thus steganography should maintain robustness against various possible image attacks [32]. We test the proposed method with those possible scenarios such as compression attacks, noise attacks, cropping attacks. In the test, image Peppers is embedded in Lena, the hiding capacity is 8 BPP.
In the experiments, three common noises: Gaussian White noise, Salt and Pepper noise, and Speckle noise are added to the stego-image, respectively. The simulation results of noise attacks are displayed in Fig. 10 Fig. 10 indicates that the proposed scheme owns desirable resistance against various noise attacks.
The retrieved images from stego-images with JPEG compression at different Quality Factors (QF) are plotted in Fig. 11. The PSNR values are 22.15 dB, 16.30dB, and 13.94 dB, respectively. As shown in the picture, the retrieved images still hold a certain degree of similarity at QF=85. This indicates that the proposed method can resist compression attacks to a certain extent.
Stego-image with center cropped is depicted on Fig.12  The stego-image with corner cropping is shown in Fig.12  (g)-(i). The cropping size is 32 32  , 64 64  , and 128 128  . Fig.12 (j)-(l) shows the influence of corner cropping. The PSNR values are 33.52 dB, 29.66 dB, and 23.76 dB, respectively. As is shown in Fig.12 (l), in the situation of 128 128  size cropped ( 6.25% data loss), we successfully retrieve the secret image. The simulation results demonstrate that the proposed scheme owns desirable resistance against the cropping attack.

D. SECURITY ANALYSIS
For steganography, the term "security" indirectly refers to undetectability. Hence a steganographic scheme is considered secure as long as the hidden data is not detectable by statistical means [6]. Those means, also known as steganalysis, refer to the method that aims to detect the presence of secret data based on the modification traces of stego-media [52].
To demonstrate the security of the proposed scheme, we have performed two well-known steganalysis technologies on the proposed scheme: RS attack, and Chi-square attack. In the test, we randomly choose the test image Peppers as the secret image and six test images as the cover images: Lena, Baboon, Lake, Jet, House, and Peppers. Besides, the 1-bit LSB is utilized for comparison. For the LSB method, the secret data is randomly generated and is embedded sequentially. For the proposed scheme, the secret image is resized to meet the requirements of the experimental condition.
As an efficient steganalysis against LSB steganography, RS analysis not only can detect the existence of secret bits but also can estimate the embedding payload [17]. In RS analysis, all the pixels of the image will be divided into disjoint groups. The security of the proposed scheme against RS analysis is shown in Fig.13. The masks used in the test:

 
, m m S S  increase with the percentage of pixels embedded increasing, which means the LSB method is easily detected. In contrast, for the proposed scheme, the difference between m embedded pixels is higher. Thus, the LSB scheme is easily detected. In contrast, for the proposed scheme, the expected probability is consistent with zero in most cases. In a few exceptional cases, the expected probability is higher than the case of 0% embedded data. These cases include the test image Baboon with a modified percentage of 30% and 70%. However, 100% payload is the most commonly used in the application. Hence, we can still conclude that the proposed scheme owns high resistance to the Chi-square attack.
The steganalysis methods are used as the classifier to distinguish the stego-images from normal images (without any data hidden). To better show the performance in security, we draw the Receiver Operating Characteristic (ROC) curves in Fig. 15. ROC curve is a performance measurement for classification problems at various threshold settings. the AUC (Area Under the Curve) represents the degree of differentiability. the higher the AUC, the better ability of the classifier in prediction. When the AUC is 0.5, it means that the model does not have any category separation capability.
For both ROC curves plotted in Fig. 15, the AUC is relatively small (close to 0.5). This indicates that both methods do not classify effectively, i.e., it is difficult to distinguish stego-images from the normal images. Thus, the proposed scheme shows high security against both attacks.

V. COMPARISON WITH THE RELATED SCHEME
In this section, we provide a comprehensive comparison with related schemes. For comparison, we have considered a lot of related works in the spatial domain and transform domain. Table IV list a comparative review of the major features of these soft-of-the-art schemes with the proposed scheme. We embed secret data in some famous test images and compare it with other schemes in terms of hiding capacity and visual quality of Stego-image (in PSNR). Besides, other considered properties include robustness, security, and the channel's payload. The property "Channel's payload" refers to the data size that needs to be transferred through the channel to the receiver side. For a fair comparison, the hiding capacity is calculated by the information provided in some works. For the proposed scheme, both the cover image size and secret image size are 512 512  . Thus, the hiding capacity of the proposed scheme is 100% or 8 BPP.
In Ref. [36], a high-capacity scheme based on DWT is proposed. The scheme utilizes GA to search for an optimal mapping matrix. Base on the obtained substitution matrix, the secret data is embedded in the less significant coefficients. and Optimal Pixel Adjust Process (OPAP) is applied to reduce the difference error between stego-image and cover image. Ref. [38] introduce another high-capacity scheme based on GA. In this scheme, an optimal chromosome with seven genes is generated by GA, and the mapping rules are defined by the chromosome. Compared with the scheme [36], the search space is less and the scheme also achieves high capacity. Both schemes achieve a high hiding payload. As shown in Table IV, even in the much higher payload, the proposed scheme show superiority to the scheme in visual quality. Besides, the robustness and security are not proved in these methods but our method shows high robustness and high security.
Ref. [29] presents an adaptive steganographic model based on PSO. Based on the idea of the search optimal substitution matrix, the scheme utilizes the coefficients of  LH, HL, and HH band to conceal secret messages. Unlike the scheme, this approach conceals the substitution matrix in the predefined position. Thus, no supporting data needs to be sent via an extra channel. The scheme [29] shows robustness and security against various attacks. However, the disadvantage of the method lies in the time-consuming optimization. Besides, in the case of n=4, the payload is 256 256 3 4    and equal to 786432 bits. Considering the used image size is 512 512  . Thus, the maximum hiding payload of the scheme is only 37.5%. Compared with the 100% hiding payload of our methods, the payload is poor.
The above-mentioned schemes are all complex and timeconsuming. Ref. [7] introduces a simple and high-capacity LSB-based scheme. Different from the traditional LSBbased technologies, this model utilizes the Median Edge Detector (MED) to locate the complex area of the cover image. To reduce the sensitivity of HVS, fewer secret data is embedded in the flat area. Besides, OPAP is also applied to reduce image distortion. The proposed scheme shows an edge over the scheme [7] in the hiding capacity, visual quality, robustness, and security.
Ref. [33] introduce a scheme based on the edge detection over DT-CWT. The cover image is split into nonoverlapping blocks and the textured patch will be embedded more messages. The scheme could produce the high quality stego-images with high payload. However, the main disadvantage is that the duplicate of original cover image is required in the receiver side. Scheme [27] utilize Root Mean Square Method (RMSE) as the criteria to match the block of secret image with the block of cover image. However, though the used secret image size is the same with cover image, only the LL band coefficients is embedded and the hiding payload is only 25%. Even though the scheme is slightly better in terms of image quality, it is safe to said that our method is superior considering that our hiding capacity is four times higher. Moreover, the scheme needs very considerable retrieve position in the extraction process but our method requires nothing.
From the above discussion, we can conclude that the proposed scheme is efficient image steganography. Compared with the related scheme, the superiority of the proposed one is mainly reflected in the hiding capacity, visual quality, and payload in the channel.

VI. CONCLUSIONS
This paper presents an efficient image steganographic scheme. As a computationally efficient transform, DHT is adopted to convert the cover image. Based on the nature of DHT coefficients, an efficient embedding strategy based on substitution is utilized in the embedding procedure. In experiments conducted on over 1000 images, the proposed scheme produces high quality stego-images and retrieved images in the dense embedding of 100%. For robustness, the proposed scheme is tested with cropping attacks, JPEG compression, and three types of noise attacks. For security, the proposed scheme can resist RS attack and Chi-square attack. Besides, a comparative analysis shows that our scheme achieves good visual quality and higher hiding capacity than the other schemes. Thus, we can conclude that our scheme outperforms other former schemes.
In future work, we will investigate the DHT-based image steganography combined with edge detection technologies.