Chapter 3 Multi-Threshold Progressive Image Sharing With
3.2 Proposed method
3.2.2 Decoding
When any k1 of the n JPEG stego codes are received, the k1 shadows can be extracted from the k1 JPEG stego codes by inverse hiding. For each x=1,2,…,k1, the x’th shadow is partitioned to yield the x’th share of each code si (1 ≤ i ≤ 4). The k1
shares of the code s1 are used to reconstruct the low-quality JPEG code s1 using the Lagrange interpolation method. The reconstructed JPEG code s1 is then decompressed to yield the low-quality JPEG image q1, which is an approximate version of the original important image.
When k2 (or k3) JPEG stego codes are available, the reconstruction process is similar to that described earlier, and the reconstructed image q2 (or q3) will be of medium (or high) quality. Last, if at least k4 JPEG stego codes are received, the k4
shadows can also be extracted from the k4 JPEG stego codes by inverse hiding. Then, for each x=1,2,…,k4, the x’th shadow is divided to generate the x’th share of each code si (1 ≤ i ≤ 4). The k4 shares of the code s4 are used in inverse sharing to reconstruct the
Huffman code s4 by Lagrange interpolation, and the reconstructed code s4 is then decompressed to generate the difference image q4. Adding the difference image q4 to the image q3 yields the error-free important image.
3.2.3 Example of Sharing and Inverse-Sharing Processes based on GF(256)
An example of the sharing and inverse-sharing processes based on GF(256) and P(X)=X8+X4+X3+X+1 is presented. To partition ki=2 numbers 100 and 200 of 8 bits each into n=3 shares, Eq. (3.7) is used to compute the three shares: g(1) ≡ 100+200×1
≡ 100+200 ≡ 172 [over GF(256)]; g(2) ≡ 100+200×2 ≡ 100+139 ≡ 239 [over GF(256)]; and g(3) ≡ 100+200×3 ≡ 100+67 ≡ 39 [over GF(256)]. In obtaining the two shares g(1)=172 and g(3)=39, the two numbers 100 and 200 are revealed by Lagrange interpolation, as g(z) ≡ g(1)×(z–3)/(1–3) + g(3)×(z–1)/(3–1) ≡ 172×(z+3)/(1+3) + 39×(z+1)/(3+1) ≡ 172×(z+3)×141 + 39×(z+1)×141 ≡ 100+200z [over GF(256)].
3.3 Experiments and Comparisons
3.3.1 Experimental Results
The inequalities (k1=3)<(k2=4)<(k3=5)<(k4=6) and the irreducible polynomial P(X)=X8+X4+X3+X+1 are used to generate n=6 shadows of the important image. The JPEG source code used in the experiments is taken from the fourth public release of the Independent JPEG Group's free JPEG software [43]. The quality of an image is measured by the PSNR.
In the first experiment, the 512×512 grayscale important image Lena, displayed in Figure 3.2, is encoded using JPEG with three quality factors QFL=5, QFM=25, and QFH=85. The four codes s1, s2, s3, and s4 have lengths 5,750, 13,787, 45,972, and 115,458 bytes, respectively. The six cover images Peppers, Jet, Boat, Lake, Baboon,
and Zelda, shown in Figure 3.3, are all encoded using JPEG with QF=75 to hide the six shadows, and therefore generate the six JPEG stego codes. Figure 3.4 displays the n=6 images decompressed from our six JPEG stego codes without any extraction of the hidden shadows, and the PSNRs of the six decompressed images are 37.42, 37.41, 36.40, 34.39, 32.73, and 38.67 dB, respectively. When different numbers of JPEG stego codes are received, the reconstructed versions q1, q2, q3, and q4 of Lena are as plotted in Figure 3.5, and the respective PSNRs are 27.32, 33.67, 39.35, and ∞ dB.
In the second experiment, the important image is the 512×512 grayscale image Tiffany. The six shadows are generated and then remain hidden in the six JPEG codes generated in the first experiment. The PSNRs of the decompressed images from the six JPEG stego codes are 37.75, 37.70, 36.65, 34.61, 32.97, and 38.96 dB, respectively. (These values are a little better than the 37.42, 37.41, 36.40, 34.39, 32.73, and 38.67 dB values obtained in the first experiment.) For Tiffany, the PSNRs of the versions reconstructed using any three, four, and five JPEG stego codes are 28.37, 34.12, and 39.79 dB, respectively (these values are a little better than those, 27.32, 33.67, and 39.35 dB, for Lena). When six JPEG stego codes are collected, the reconstructed Tiffany is identical to the original Tiffany.
Last, Table 3.1 shows the bit rates [bits per pixel (bpp)] of the JPEG-Q75 codes created using JPEG with QF=75 before and after hiding our shadows.The bit rate will increase significantly after hiding a large-size secret. However, the bit rate of our JPEG stego code still falls in the reasonable range of JPEG, i.e., the bit rate of our JPEG stego code is smaller than that of the JPEG-Q95 code generated using JPEG with QF=95, as shown in Table 3.1. This alleviates the problem of code length. [The reason using QF=95 as the upper bound to examine the bit rate of our JPEG stego codes is that, as stated in Kim et al.’s work [54], the general quality factors (QFs) used in digital cameras are between 90 and 95.]
3.3.2 Comparisons
Table 3.2 compares other sharing schemes [21,25-30] with ours in terms of shadow-size expansion, progressive ability, and lossless reconstruction ability. Each shadow in two of the related works [25,27] is four times larger than the original important image, indicating that size expansions occur. In contrast, each shadow in all of the associated works [21,26,28-30] and ours is smaller than the original important image. Although Thien and Lin [21], Tso [26], and Hung et al. [30] all shared the image without size expansion, Thien and Lin [21] and Tso [26] could not reconstruct the image progressively, whereasHung et al. [30] could not reconstruct the image in an error-free manner. Only Chen and Lin [28], Wang and Shyu [29], and ourselves have achieved reconstruction with all three desired characteristics.
Among these three methods, as presented in Table 3.3, each shadow size herein (12.89% of 512×512 bytes) is smaller than those in Chen and Lin’s method [28]
(22.22%) and Wang and Shyu’s [29] (50%). Therefore, the transmission time in the proposed method is less, and the survival rate in an unfriendly environment, in which the network connection time is unstable among the n channels used to store the n shadows, is increased. Equivalently, in the proposed method, the storage space in a distributed storage system is most reduced. The smaller size of the shadows also facilitates the hiding of shadows in stego media.
The construction of Table 3.3, which compares the shadow sizes among non-expanded schemes, is explained in the following. For fairness of comparison, the shadow sizes in Table 3.3 are all measured before hiding: all are shadow sizes, and none is a stego media size. This action eliminates the size-altering effects of particular post-processing (hiding) approaches. Assume that the important image is the 512×512 grayscale image Lena, and the (largest) threshold value is set to six for all schemes, except that Hung et al.’s scheme [30] uses five as the largest threshold value because
their scheme did not provide a version with a threshold value being six. For each x=1,2,…,n, the four x’th shares are combined to form the x’th shadow, and therefore each shadow in the proposed method has size 4
1 i /
i
s k
∑
= i= (5,750/3) + (13,787/4) + (45,972/5) + (115,458/6) = 33,802 bytes (which is 12.89% of the size of the 512×512 grayscale image Lena).According to Table 3.3, the shadow sizes in the proposed method and Hung et al.’s [30] are more economic than those in the related works [21,26,28,29]. However, Hung et al.’s method [30] is not lossless when all shadows are collected. In fact, if the original important image can be satisfactorily reconstructed with some loss, then our step 1b can be omitted, such that no Huffman code s4 is generated. Then, each of our shadows can be reduced to 3
1 i /
i
s k
∑
= i= (5,750/3) + (13,787/4) + (45,972/5) = 14,559 bytes (which is 5.55% of the size of the 512×512 grayscale image Lena). Restated, the size of each shadow in this lossy version is about half of that in Hung et al.’s scheme.Moreover, in this lossy version, the total shadow size is 14,559×6=87,354 bytes, which is still smaller than 30,723×5=153,615 bytes obtained by Hung et al. [30].
When the five shadows are collected, the 39.35-dB Lena [identical to that in Figure 3.5(c)] is reconstructed, better than the 37.04-dB Lena revealed by Hung et al. [30].
Last, the size of each shadow in the proposed [(k1, k2, k3, k4), n] threshold scheme
∑
= i. Therefore, it is suggested that the readers set the largest threshold k4 to n to save storage space. However, if the readers want to have more freedom, they may use their own choice of a threshold k4 being less than n, at the price of wasting space for the shadows. When k4 is less than n, a simulation is done in the following. Assume that n, the number of cover images, is at least six. In general, the smallest threshold k1cannot be one because the purpose of sharing is that no participant alone can be trusted. Therefore, {1<k1=2<k2=3<k3=4<k4=5<n=6} is used to generate n=6 shadows,
the size of which is then compared to those in the image sharing schemes [21,26,28-30] when the (largest) threshold value is set to five for all these schemes.
The comparison results are shown in Table 3.4. It is observed that each shadow in the proposed method is still smaller than those in the related works [21,26,28,29], and each shadow in our lossy approach is also smaller than that in Hung et al.’s lossy approach [30].