Chin-Chen Chang, Thai-Son Nguyen, and Chia-Chen Lin

1. Introduction

Today, with the rapid development of multimedia technology and computer science, most information and multimedia data are exchanged by computers and the Internet without geographic limitations at any time. This leads to the wide transmission of huge amounts of information and multimedia (i.e., digital images, text, audio,and video), which means that malicious users can easily catch important content from the Internet without authorization. Therefore, ensuring the secrecy and security of data transmission is of great importance. Recently,researchers have proposed several protection algorithms,such as cryptography[1],[2]and data hiding[3]–[6]. In the cryptography, the sender will convert secret data into an unrecognizable form. Then, only an authorized receiver can reconstruct this secret data into their original form by using a secret key known only by the authorized sender and receiver. However, the meaningless form of secret data will be significantly drawn by malicious users, who may attempt to decrypt messages to get information. By contrast, data hiding is a better method to guarantee the security of data transmission by avoiding the attention of malicious attackers.In data hiding, the secret information is hidden into cover digital multimedia without arousing the attention of malicious attackers. Because of the original nature of the cover, the multimedia remains even though it conveys the secret data.

Data hiding can be divided into two basic types. The first is irreversible data hiding, which hides secret data into a cover image. Many irreversible data hiding techniques[4],[6]have been proposed. One primary advantage of irreversible data hiding is that high hiding capacity is transmitted through the Internet. However, a weakness in irreversible data hiding is that the cover image may be damaged and cannot be reconstructed after the secret data are extracted.

The second type of data hiding is reversible data hiding[3],[5](also known as lossless data hiding and distortion free data hiding), which has a reversibility feature to deal with the weakness of irreversible data hiding. This reversibility enables the cover image to be recovered correctly after extraction of secret information. This property makes reversible data hiding useful for real-time applications in medical and military areas, where the visual quality of reconstructed image is essential.

Recently, the interest of researchers has increased significantly in high dynamic range (HDR) images, which contrast to low dynamic range (LDR) images. HDR images have become popular in several fields such as computer graphics, remote sensing, digital photography, movie,computer game, medical imaging, etc. Over last three years,some data hiding schemes[7],[8]in HDR images have been proposed. In 2009, Cheng and Wang[7]proposed the first steganography approach for HDR images. They used a two-sided algorithm that is modified from Chang and Tseng’s scheme[9]and developed their own L-sided algorithm, inspired from Zhang and Wang[10]. This scheme presents the high visual quality of a stego image when a large amount of secret data is embedded. However, image distortion is unavoidable and the original image cannot be reconstructed exactly after secret data are extracted. In 2011,Yuet al.[8]proposed a new distortion-free data hiding technique involving HDR images encoded by the radiance red-green-blue-exponent (RGBE) format[11]. In this scheme,the secret data are embedded by depending on the advantage of some homogeneous representations inherent in the radiance RGBE encoding format. This scheme achieves the exact identity between the tone-mapped cover image and tone-mapped stego image. However, the average embedding capacity of Yuet al.’s scheme is approximately 0.12 bpp.To further improve the embedding capacity, in this paper,we propose a distortion-free data embedding scheme that depends on the Cartesian product algorithm to try using the homogeneous representations of pixels in HDR images flexibly in data embedding. The experimental results confirm that our scheme is superior to Yuet al.’s scheme in terms of the embedding capacity while guaranteeing that the tone-mapped cover image and stego-image are identical.

The rest of the paper is organized as follows. Section 2 reviews Yuet al.’s scheme[11]. Then, the details of our proposed scheme are demonstrated in Section 3. Section 4 illustrates experimental results. Finally, some conclusions are given in Section 5.

2. Related Work

In 1991, Ward[11]introduced the format of HDR images,also known as the 32-bit radiance RGBE format, since then which has found widespread utilization in graphics communication. In the HDR image encoded with the radiance RGBE format, the pixel value is represented by three primary channels and one exponent channel. Each channel is in the range of 0 to 255. This means that each channel will use 8 bits to store its value. Fig.1 shows the HDR pixel encoded in the 32-bit radiance RGBE format.

Basically, the color of the HDR pixel is a floating point value. Therefore, it can be derived by using the floating point conversion which is defined as

where the color pixel (R,G,B) represents the floating point.Assume thatP(r,g,b,e) indicates the pixel encoded with the radiance RGBE format, with the three primary color channelsr,g,bandeshown as an exponent channel,respectively.

Likewise, the color pixel with the floating point (R,G,B) can be converted into the radiance RGBE format (r,g,b,e) by using integer conversion as shown in (2):

It is clear that with the exponent channelein the radiance format more than one representation can be used to show one color pixel. By using the division operation with the divisor 2 for each primary channel and adding 1 to the exponent channel, or by using the multiplication operation with the multiplier 2 for each primary channel and subtracting 1 from the exponent channel, we can get the new representation of pixels that can provide nearly the same floating point color value and also give the identical color value with the original pixel when a tone mapping scheme is applied. For example, assume that an original pixel isP(r,g,b,e). By using division or multiplication, the new representations are produced asP(r/2,g/2,b/2,e+1) orP(r×2,g×2,b×2,e–1), respectively. However, the new representationP(r/2,g/2,b/2,e+1) must satisfy that each channel still has an integer value. For the new representationP(r×2,g×2,b×2,e–1), each channel must be in the legal range from 0 to 255. By using the above mentioned process,each pixel contains a number of different representations. In other words, each pixel has a set of different representations,known as the homogeneous representation group (RG). In the RG group, each element can describe the same color pixel as the original oneP(r,g,b,e). We define the number of elements in the group RG asN. Table 1 shows the detailed example of the sorted RG group with the number of elementsN.

By exploiting the RG group of each pixel in an HDR image, Yuet al.[8], in 2011, proposed a new distortion-free data hiding algorithm that can embed secret messages into the HDR images. The advantage of Yuet al.’s scheme is that it can convey the secret message to generate the stego HDR image. Moreover, when the tone mapping technique is processed, there is no distortion between the original and stego images. In this scheme, a RG group of pixels, as shown in Table 1, is explored to embed the secret bits. The homogeneous index table is generated in advance to support embedding data, as shown in Table 2.

Fig.1. HDR pixel in 32-Bit of radiance RGBE format.

Table 1: Detail example of RG group with N=3

Table 2: Homogeneous index table used to embed secret bits

The embedding algorithm is shown as follows.

Input: the HDR cover imageI, homogeneous index table HIT, and secret messageB.

Output: HDR stego imageI′.

Step 1. Read a pixelPfrom the imageI.

Step 2. Determine the sorted homogeneous representation group RGPand the number of elementsNof group RGP.

Step 3. IfN≤1, it means that no secret bits can be embedded, go back to Step 1. Otherwise, compute the number of bits, |bP|, of the secret bitsbPthat can be embedded by

Then, the secret bitsbPis read from the secret messageB.

Step 4. Depending onNand secret bitsbP, the suitable indexdis found in HIT.

Step 5. Then the stego pixelP′will be generated by choosing the corresponding representation in the RG group which has the homogeneous index equal tod.

Step 6. Repeat Steps 1 to 5 until all pixels in the imageIare processed completely.

After completing the embedding phase, the sender sends the stego imageI′to the decoder. HIT is also sent to the receiver to support the extracting process. To better illustrate the embedding algorithm, an example is presented as follows. Assume that the original pixelP(32, 24, 80, 128)is given. Then the group RGPis determined as shown in Table 3. The number of elementsNof group RGPis 5.Therefore, from (3), the number of secret bits that can be embedded into pixelPis 2. Suppose that two secret bits,bp,are read as “11”. Then, according to the number of elements of group RGP,N=5, andbp=“11”, the homogeneous indexd=2 will be selected from HIT. Then the stego pixelP′is the element in the group RGP,which has the corresponding homogeneous indexd=2. Therefore,for this case, the stego pixelP′is (16, 12, 40, 129).

After the embedding phase is processed completely, the sender sends the HDR stego imageI′to the decoder. When the receiver gets the HDR stego imageI′, the following extraction algorithm is used to extract secret messageB.

Input: the HDR stego imageI′and homogeneous index table HIT.

Output: the secret messageB.

Step 1. Read the pixelP′from the imageI′.

Step 2. Determine the group RGPandNof RGP.

Step 3. IfN≤1, it means that no secret bits can be extracted, and go back to Step 1. Otherwise, compute the number of bits of secret bitsbPthat can be extracted by using (3).

Step 4. Determine the suitable homogeneity indexdof stego pixelP′from the group RGP.

Step 5. By depending onNand the homogeneity indexd, the extracted secret bitsbPis found from the table HIT.Then,bPis sent to the secret messageB.

Step 6. Repeat Steps 1 to 5 until all pixels in imageI′are processed completely.

After the above six steps are processed completely, the secret messageBis extracted. Here is an example to explain our extracting phase in detail. Suppose that the stego pixelP′is (16, 12, 40, 129). Now, to extract the secret bitsbPfrom the pixelP′, the group RGPofP′is determined as Table 3, and the number of elementNof group RGPequals 5. It is easy to see that the homogeneity index of pixelP′isd=2 in the group RGP. Therefore, throughNand the homogeneity indexd, secret bitsbP=“11” are found in the HIT table, which are the extracted secret bits. Then,bPis sent to the messageB.

3. Proposed Scheme

The proposed scheme is described in detail in this section. The cover image is the HDR imageIsizedW×H,and the secret messageBis denoted asB=(b0,b1, ··,br),wherebjis the secret bit generated randomly andbj∈{0, 1},0≤j≤r. The proposed distortion-free data hiding scheme can be divided into two phases, data embedding and data extracting, which are discussed in Subsection 3.1 and Subsection 3.2, respectively.

3.1 Data Embedding Phase

After carefully observing Yuet al.’s hiding scheme[8],we figure out that for each pixel, the number of secret bits islog2(N), which is embedded into each pixel of imageI,whereNis the number of elements of RG group. Therefore,the total embedding capacity of imageIsizedW×His

Table 3: Homogeneous representation group RGP of color pixel P(32, 24, 80, 128)

It is obvious to see that the summary oflog2aandlog2bwill be smaller or equal tolog2(a×b), whereaandbare integer values. For example, ifa=5 andb=13,respectively,log25+log213=5≤log2(5×13)=log265=6.Therefore, by exploiting the above mentioned idea, in this paper, we use a Cartesian product algorithm to further improve the embedding capacity of Yuet al.’s scheme while guaranteeing the high quality of the stego image.

Basically, a Cartesian product is the direct product of two setsAandB,which is denoted asA×BandAandBare the ordered set of all possible ordered pairs whose first component is a member ofAand whose second component is a member ofB, as given by

For example, the Cartesian product of 4-element setA,namely by {1, 2, 3, 4}, and 3-element setBgiven as {x,y,z}, is the 12-element set with all possible ordered pairs as{(1,x), (1,y), (1,z), (2,x),···, (4,z)}. The corresponding Cartesian product has 4×3=12 elements.

In general, a Cartesian product of two finite sets can be shown by a table, with one set as the rows and the others as the columns and creating the order pairs, the cells of the table, by selecting the set from the row and column.

Fig.2 shows the flowchart of our embedding algorithm,which involves four operations, namely determining the RG group for each pixel, computing the Cartesian productSof all RG groups, then selecting the suitable bit string for each element inS, and the embedding process.

Our proposed data embedding phase can be divided into four steps. The corresponding algorithm is shown in detail as below.

Input: the original HDR imageIsizedW×Hand secret messageB.

Output: the stego imageI′.

Step 1. For each pixelPi, determine the sorted homogeneous representation group RGiand the corresponding number of elementNiin RGi, whereiis in the range from 1 toW×H.

Step 2. Compute the Cartesian productSof all groups RGiby using (6), where the number of elements of setS, |S|,is computed by using (7).

Step 3. Compute the suitable secret bit stringrifor each element of setS.Here, the length ofriis calculated by

Step 4. To embed the secret messageB, the corresponding element inS, which has secret bit stringriequal toB, is found.Then this element is sent out as the stego imageI′.

Fig.2. Detailed flowchart of embedding algorithm.

Fig.3. Original image I and correspond RGi and Ni of each pixel.

Fig.4. Stego image I′.

After these four steps are processed completely, the stego imageI′is obtained, which is sent to the receiver without any extra information. This example explains the data embedding phase in detail. Assume that the original HDR image is given as Fig.3.

In Fig.3, it is clear to see that the first pixelP1has group RG1={x1,y1} and its number of elements of group RG1,N1=2. Then the Cartesian productSof the 4 groups RG1, RG2, RG3, and RG4is 36 possible elements as {(x1,x2,x3,x4), (x1,x2,x3,y4), (x1,x2,x3,z4), ··, (y1,z2,y3,z4)}.According to (8), the number of elements of setSwill be|S|=2×3×2×3=36 elements. For each element in setS, we match one secret bit stringriwhich has the length=log2|S|=log2|36|=5 bits (from “00000” to“11111”). This means that we can embed the secret messageBwhich has the same length as the secret bit stringri. Therefore, suppose that the secret messageB=“00010” is embedded. According to Step 4 in the embedding algorithm,we can get the stego imageI′(x1,x2,x3,z4), which is the element in the setS, have the secret bit stringrithat is the same as the secret messageB. Then, the stego imageI′is generated as presented in Fig.4.

Fig.5. Detailed flowchart of our data extracting phase.

Fig.6. Example of extracting phase.

3.2 Data Extracting Phase

After receiving the stego imageI′from the sender, the receiver can extract the secret messageBexactly by using the following steps in our data extracting phase. A detailed flowchart of our data extracting phase is provided in Fig.5.

Extracting algorithm:

Input: the stego imageI′.

Output: the extracted secret messageB.

Step 1. For each pixelPifrom the imageI′,determine the sorted homogeneous representation group RGiand the corresponding number of elementNiof group RGi, whereiis in the range from 1 toW×H.

Step 2. Compute the Cartesian productSof all groups RGiby using (6), where the number of element of setS, |S|,is determined by (7) .

Step 3. Select the suitable secret bit stringrifor each element of setS.Here, the length ofriis calculated by (8).

Step 4. To extract the secret messageB, find the element in the setSwhich has the same color value as the stego imageI′.Then the corresponding secret bit string of this element is the extracted secret messageB.

After these four steps, the secret messageBis extracted correctly. For example, suppose that the stego HDR imageI′is given, as shown in Fig.6. Then determine the corresponding group RGiand the number of elementsNifor each pixel of the stego image, as shown in Fig.6. The Cartesian productSof 4 groups RG1, RG2, RG3, and RG4has 36 possible elements as {(x1,x2,x3,x4), (x1,x2,x3,y4),(x1,x2,x3,z4),··, (y1,z2,y3,z4)}, and the number of element of setSis |S|=2×3×2×3=36 elements. For each element inS,the corresponding secret bit stringrihas the length=log2|S|=log2|36|=5 bits (from “00000” to“11111”). Then, to extract the secret message, the setSis searched to find the element which has the same value as the stego imageI′. When the match element is found, the secret messageBis extracted as the corresponding secret bit stringriof the match element. Thus, for this case, the stego imageI′equals to the third element (x1,x2,x3,z4).Therefore, the extracted secret messageBis the corresponding secret bit stringriof the third element,“00010”.

4. Experimental Results

To illustrate the performance of our proposed scheme and Yuet al.’s schemes[8], five HDR test images, “Church,”“Hall,” “Aspen,” “Bush,” and “Pine” presented in Fig.7 were used in our experiments. The size of the first four images is 720×480 pixels, and that of the last test image is 2000×1312 pixels. All computing was performed on a PC with a 2.1 GHz Intel(R) Core?2 CPU and a 1 GB RAM.The operating system was Windows 7 Professional and our algorithm was programmed by Microsoft Visual Studio 2005 C#.

Table 4 shows the details of the five HDR test images.For example, the “Church” HDR image has 302482 pixels which have the number of elementN=1 in the group RG,and has 37640 pixels whenN=2, respectively.

Fig.7. Five HDR test images: (a) Church, (b) Hall, (c) Aspen (d)Bush, and (e) Pine.

Table 4: Detail characteristics of five HDR test images

Table 5: Embedding capacity of our scheme and Yu et al.’s scheme[8] for five HDR test images

Fig.8. Tone mapped cover and stego of three test images: (a) the tone mapped cover image “Hall” and the tone mapped stego image “Hall”, (b) the tone mapped cover image “Aspen” and the tone mapped stego image “Aspen”, and (c) the tone mapped cover image “Bush” and the tone mapped stego image “Bush”.

Table 5 presents the comparison of our scheme and Yuet al.’s in term of the embedding capacity. Obviously, the embedding capacity of our scheme is better in all HDR images. The average embedding rate of our scheme is 0.1355 bpp, and that of Yuet al.’s scheme is 0.1270 bpp.Yuet al.’s scheme is worse than ours. This is because Yuet al.’s scheme does not use some homogeneous representation to embed secret bits. For example, in line 7 of Table 2, when the number of elements in the group RG equals 6, Yuet al.’s scheme only used the first four homogeneous representations to embed two secret bits, and the last two representations have not been used. Besides, in Yuet al.’s scheme, the homogeneous index table HIT is needed as extra information that is sent to the receiver to support the extracting process. In contrast, our scheme applies the Cartesian product to try to use all homogeneous representations of each group RG,and no extra information is required.Thus, our scheme can achieve higher embedding capacity compared with Yuet al.’s scheme.

Fig.8 provides the tone mapping results of utilizing the Luminance HDR software for the HDR images “Hall,”“Aspen,” and “Bush.” Obviously, the tone mapped cover stego images are identical with different types of tone mapping algorithms. When the Reinhardet al.’s tone mapping scheme[12]is used with the parameters of gamma=1.000, brightness=-10.0, chromatic=1.00, and light=0.00, respectively, the tone mapped cover image“Hall” and the tone mapped stego image “Hall” are identical, as shown in Fig.8 (a). Fig.8 (b) presents the tone mapping result of the HDR image “Aspen” by utilizing the tone mapping method of Durand and Dorsey[13]with four parameters gamma=1.000, spatial=8.00, range=0.40, and contrast=5.00, respectively. The “Bush” HDR image using the tone mapping scheme of Mantuiket al.[14]is shown in Fig.8 (c), and three parameters, gamma, contrast, and saturation, are set as 1.000, 0.300, and 0.800, respectively.From Fig.8 the tone mapped cover image and tone mapped stego image are identical; therefore, we can conclude that our proposed scheme leads to a distortion-free data hiding scheme for HDR images encoded in the RGBE format.

5. Conclusions

In this paper, the distortion-free data hiding scheme for HDR image encoded in the 32-bits radiance RGBE format is proposed to further improve the performances of Yuet al.’s scheme[8]. In term of the embedding capacity, the average embedding rate of our proposed scheme is around 0.1355 bpp, which outperforms Yuet al.’s scheme (0.1270 bpp). This is because, in our scheme, all homogeneous representations are used to embed secret bits. Moreover, the experiment shows that the tone mapped cover and stego images are identical. It means that our scheme tends to distortion-free data embedding. In other words, when compared with Yuet al.’s scheme, our scheme achieves a better result in embedding capacity, while maintaining the good quality of the stego image. However, the embedding capacity is still low, for the average embedding capacity is smaller than 0.15 bpp. Thus, in the future, we intend to design a new distortion free data embedding algorithm for HDR images with higher embedding capacity by using all homogeneous representations of each pixel more efficiently.

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