Chapter 5 Authentication and Recovery of an Image by Sharing and Lattice-embedding
5.4.3 Image quality and our advantage
Firstly, the method of Varsaki et al.[72] was implemented, and the results are shown in Fig. 5.8. Fig. 5.8(a) is the 512×512 watermarked color image Lena which is 40.88 dB. As shown in (b) of the figure, the embedded recovery data is the size-reduced 128×128 gray-level version of the rotated host, the clockwise rotation is 180degree, as suggested by Varsaki et al.[72]. This 128×128 gray-level rotated version is embedded in the 128×128 blocks of the host image, and each block is 4×4. So, the recovery data of the rightmost bottom 4×4 block is embedded in the leftmost top 4×4 block, and the recovery data of the Southwest quadrant is embedded in the Northeast quadrant, and so on. Unfortunately, when the central 192×192 pixels of the watermarked Lena are cropped (the tampered image is shown in (c)), the recovery data extracted from the non-cropped area is as shown in (d). It can be seen that the tampered region still cannot be recovered because the recovery data of the central 192×192-pixels box was embedded earlier in the box itself. In other words, the recovery data of the cropped box is also cropped. This is very different from ours. As shown in Fig. 5.4, when the central 192×192 pixels of our watermarked Lena are cropped, the cropped region can still be recovered. This should come as no surprise because, according to Table 5.1, when M=20, a moderate-size-area’s tampering can be tolerated (up to 16.7% of the whole image’s blocks can be tampered).
Next, because Tsai and Chien’s method[16]used scaled versions of the originals as messages, then embedded the messages in the frequency domain, and also because they provided experimental results about the resistance to JPEG compression and Gaussian noise, we compare their method with ours.
Fig. 5.9 shows the experiment. The results of Tsai and Chien[16] are shown in (a-d), and ours are shown in (e-h). Notably, the PSNR value of their watermarked image Jet in (a) is 30.8 dB while ours in (e) is 32.29 dB. The PSNR value of the recovered image is 29.3 dB (Fig.
5.9(d)) for theirs, and 31.89dB (Fig. 5.9(h)) for ours. Notably, (d′) and (h′) show the details of (d) and (h), respectively. It is observed that, between the lower-middle and lower-left of the
104
image, there are some artifacts in their recovered snow area below the mountains, as shown in (d′). Therefore, our recovered image is better.
Fig. 5.10 shows the second experiment. The watermarked image Peppers are under both tampered attack and JPEG compression. The results of Tsai and Chien’s[16] method are shown in (a-d). Ours are shown in (e-h). (a) is their 30.6 dB watermarked image. (b) is their tampered image (inserting a sub-image compressed-decompressed by JPEG (QF=80, and the compression ratio is 4.3). (e) is our 32.24 dB watermarked image. (f) is our tampered image (inserting a sub-image, then use JPEG (QF=80, and the compression ratio is 5.4) to compress/decompress the mixed image). Having compared the two recovered images (d) and (h), it can be seen that ours (Fig. 5.10(h)) has better visual quality (because Fig. 5.10(d) has some noisy dots). Details are shown in (d′) and (h′).
Fig. 5.11 shows the third experiment. The watermarked image Peppers is tampered with;
and then the damaged image is attacked by adding Gaussian noises. The results of Tsai and Chien[16] are shown in (a-d), and ours are shown in (e-h). Having compared the two recovered images (d) and (h), again, it can be seen that ours (Fig. 5.11(h)) still has a better visual quality (because Fig. 5.11(d) has some noisy dots). Details are shown in (d′) and (h′).
(a) (b) (c) (d)
(e) (f) (g) (h)
105
(i) (j) (k) (l)
(m) (n) (o) (p)
(q) (r) (s) (t)
Fig. 5.4. Robustness test of the proposed method. (a-d): Our four watermarked images Lena, Peppers, Jet, and Scenery. (e-f): The four cropped images. (i-l): The corresponding verification results, after doing a JPEG compression on (e), adjusting brightness of (f), adding noise to (g), and adding white bar to (h). (m-p): The recovered images. (q-t): Close-up versions around the recover area of the recovered images (m-p).
(a) (b) (c) (d)
Fig. 5.5. Cut-and-paste attack. (a): Watermarked image, (b):Tampered image, (c):Verification result, (d): Recovered image.
106
(a) (b) (c) (d)
(e)
Fig. 5.6. Collage attack. (a): First watermarked image Boat, (b): Second watermarked image House, (c):Collaged image in which the car in (b) is copied-and-pasted to the same place as (a), (d):
Verification result, (e): Recovered image.
(a) (b) (c)
Fig. 5.7. Vector quantization (VQ) attack. (a): Original image, (b): VQ-attack result of (a), (c): Verification result indicates that the whole image (b) is fake everywhere.
(a) (b) (c) (d)
107
Fig. 5.8. Cropping test for Varsaki et al.’s[72] method. (a): Watermarked image Lena, (b):
Recovery data embedded in (a), (c): When (a) is cropped, (d): Recovery data extracted from the support of the non-cropped area.
(a) (b) (c) (d)
(e) (f) (g) (h)
(d′) (h′)
Fig. 5.9. An experiment to compare our method with that of Tsai and Chien[16]. (a): Their 30.8 dB watermarked image Jet, (b): Their tampered image, (c): Their verification result, (d): Their 29.3 dB recovered image, (e): Our 32.29 dB watermarked image, (f): Our tampered image, (g): Our verification result, (h): Our 31.89dB recovered image, (Notably, (d′) and (h′) show the details of (d) and (h) respectively. There are some artifacts in (d′) on the recovered snow.)
108
(a) (b) (c) (d)
(e) (f) (g) (h)
(d′) (h′)
Fig. 5.10. Second experiment to compare our method with that of Tsai and Chien[16]. (a): Their 30.6 dB watermarked image Peppers, (b): Tampering with (a), followed by JPEG compression with QF=80, (c): Their verification result, (d): Their recovered image, (e): Our 32.24 dB watermarked image Peppers, (f): Tampering with (e), followed by a JPEG compression with QF=80, (g): Our verification result, (h): Our recovered image. (Notably, (d′) and (h′) show the details of (d) and (h), respectively.)
(a) (b) (c) (d)
109
(e) (f) (g) (h)
(d′) (h′)
Fig. 5.11. The other experiment to compare our method with that of Tsai and Chien[16]. (a): Their 30.6 dB watermarked image Peppers, (b): Tampering with (a), followed by adding Gaussian noises with σ2=12, (c): Their verification result, (d): Their recovered image, (e): Our 32.24 dB watermarked image Peppers, (f): Tampering with (e), followed by adding Gaussian noises with σ2=12, (g): Our verification result, (h): Our recovered image. (Notably, (d′) and (h′) show the details of (d) and (h), respectively.).
Table 5.1. PSNR quality of watermarked image and attack-tolerance (for various quantization step value M). The host images are Lena (L), Peppers (P), Jet (J), and Scenery (S).
Watermarked image Attack-tolerance
Quantiz.
step value (M) used in our algorithm
PSNR (dB) of
watermarked image
(1 －β), i.e.
percentage of area can be cropped or replaced
JPEG with Quality
Factor (QF) not less than thresholds shown here
Gaussian noise (σ2)
Range of
“brightness adjustment”
Lena;
Pepper
Jet;
Scene
2 53.37-53.40 1/8=12.5% 100 0 [-35,40];
[-10,40]
[-55,30];
[-20,25]
4 48.13-48.15 1/8 94 0 [-40,50];
[-15,50]
[-70,35];
[-30,30]
110
6 44.83-44.85 1/8 90 0 [-40,50];
[-15,55]
[-70,35];
[-30,30]
8 42.40-42.48 1/8 86
0-1 (0 for J;
1 for L&P&S)
[-40,50];
[-15,55]
[-70,35];
[-30,30]
10 40.52-40.54 1/8 82
2-3
(2 for J&P&S;
3 for L)
[-40,50];
[-15,55]
[-70,35];
[-30,30]
12 38.95-39.01 1/8 78 4 [-40,50];
[-20,55]
[-75,35];
[-30,30]
14 37.67-37.72 1/8 75
5-6
(5 for L&S; 6 for J&P)
[-40,55];
[-20,55]
[-75,35];
[-30,30]
16 36.56-36.61 1/8 71 7 [-40,55];
[-20,55]
[-75,35];
[-30,30]
18 35.55-35.62 1/8 67 10 [-40,55];
[-25,55]
[-75,35];
[-30,30]
20 34.63-34.80 1/6=16.7% 63 13 [-45,60];
[-25,60]
[-85,35];
[-35,35]
22 33.84-33.97 1/6 59 16 [-45,60];
[-25,60]
[-85,35];
[-35,35]
24 33.10-33.24 1/6 55 19 [-45,60];
[-25,60]
[-85,40];
[-40,40]
26 32.43-32.60 1/6 51 22 [-45,60];
[-25,65]
[-85,40];
[-40,40]
28 31.80-32.00 1/6 48 26 [-45,60];
[-25,65]
[-85,40];
[-40,40]
30 31.24-31.42 1/6 45 30 [-45,65];
[-25,65]
[-85,40];
[-40,45]
32 30.70-30.87 1/6 42 34 [-45,65];
[-25,65]
[-85,40];
[-40,45]
5.5 Comparison with other studies
In this section, the proposed method is compared with other studies. Firstly, the two studies[11, 12] are authentication methods without considering the issue of recovery, but ours is equipped with both authentication and recovery abilities. As for other semi-fragile watermarking methods[13-17] with both authentication and recovery abilities, to describe the difference between ours and those methods, each watermarking algorithm is divided into major sub-steps (from the perspective of methodology and system design), and then a comparison is made. The differences are described below.
111
a). Embedding location (i.e. where to embed?) and our advantage
In methods[13-15], the recovery data of each block is embedded in another block. For example, the recovery data of block A0 is embedded in block A1, the recovery data of block A1
is embedded in block A2, and so on. In the verification and recovery phase, if block A0 is judged as “tampered”, then the recovery data in block A1 is extracted to recover block A0, and so on. In this example, if blocks A0 and A1 are both tampered, then block A0 cannot be recovered. In Tsai and Chien’s method[16, 17], although the processing domain is the discrete wavelet domain, the recovery data in low-frequency bands still needs to find some other location in high-frequency domain to undertake embedding. Therefore, if both locations are attacked, a similar recovery-disabled problem exists, although it is less severe.
However, in our method, as long as the number of valid blocks reaches a threshold, our inverse operation of the two-layer sharing can always decode the recovery data, so there is no need to consider the case that a block A0 and the block A1 storing its recovery data are simultaneously tampered. We only have to consider the percentage of the damaged area occupied in the whole image. As long as the damaged blocks occupy less than, say, 1/6=16.7% of the whole image’s blocks, recovery can always be undertaken. In general, it is hard to predict in advance which part would be tampered. There is no way to predict the trace of tampering, and worrying about “the percentage of blocks (in the whole image) being tampered” is simpler than worrying about “how to predict the actual location of the tampered area”.
Fig.5.12. Diagram of the 1-deimensional parity-check quantization used in many research works.
−2M −M 0 M 2M
0 1 0 1 0
p
112
Fig. 5.13. Diagram to explain a two-dimensional case of parity-check quantization. Here, two host pixel values (p1, p2) are replaced by one of the centers for the purpose of embedding a two-bit data.
b). Embedding method (i.e. how to embed?) and our advantage
Parity-check quantization (or similar works) is used in a great many research works[13, 14, 16, 17]. (As for Ref. [15], it embeds the data in Least Significant Bits (LSB) of the host image. Notably, 1-bit LSB embeds one bit per pixel, so two bits are embedded in two pixels, and the largest distortion for a pair of stego-pixels is 2 ≈1.414, but in our lattice embedding, when M=2 in Fig.5.1, our largest distortion for two host pixels is
155 . 1 3 / 2 3
/ = ≈
M ; hence smaller.) As shown in Fig. 5.12, in 1-dimensional parity-check quantization, the host values are divided into many regions like [−3M/2, −M/2), [−M/2, M/2), [M/2, 3M/2), [3M/2, 5M/2), and so on, with M is called the quantization level or step size.
Each region corresponds to a binary value 0 or 1. If a bit is to be embedded in a host value p, then just find the nearest region of p so that the nearest region corresponds to the bit value to be embedded. Then output the central coordinate (0, or ±M, or ±2M, or …) of the picked region as the stego-value that replaces p. To extract the hidden secret bit, just locate the region which contains the stego-value, then output the corresponding bit value 0 or 1. Our method is different, since we use lattice embedding as the embedding method. As shown in Fig. 5.1, the space of the host pair-values (p1, p2) is divided into many hexagonal regions, and the center of each region corresponds to a ternary value 0, 1 or 2. (In Fig. 5.1, small rectangles, triangles, and circles are used to represent the three values, respectively.) If a ternary value is to be embedded in the host pair (p1, p2), the nearest hexagon-center is found (i.e. small rectangle,
p1
p2
M
*
M*
* *
113
triangle, or circle) which corresponds to the ternary value to be embedded. Then the 2-dimensional coordinate of the hexagon-center is output. Later, to decode the embedded value, just locate the hexagonal region which contains the stego-pair (p1, p2), then output the corresponding value 0, 1, or 2 of the region. The major reason for using lattice embedding is that the distortion of the embedding is smaller, because the hexagon-centers are denser in the plane than the square-centers. To see this, first let us inspect Fig. 5.13, which explains the two-dimensional case of parity-check quantization, i.e. it explains what happens when two pixels (p1, p2) are modified by parity-check quantization in order to embed two bits (00=0, 01=1, 10=2, or 11=3) of the data. According to the location of (p1, p2), the nearest square-center, whose class-reading in {0, 1, 2, 3} must coincide with the given two-bit data, is picked and the value of (p1, p2) is replaced by the coordinate of the square-center. The quantization step size M in Fig. 5.13 is the same as in Fig. 5.1. Having compared Figs. 5.1 and 5.12, it can be seen that the distance between the hexagon-centers is smaller than the distance between the square-centers (each hexagon can be contained by a square box of size M-by-M).
Therefore, when two host pixel values (p1, p2) are modified to embed the data, the distortion of the Lattice embedding is smaller. (As shown in Figs. 5.9-5.11, our watermarked images have a better image quality.)
Of course, when Fig. 5.1 is adopted to replace Fig. 5.13, the price is that the data embedded in a hexagon can only have a data value of 0, 1 or 2, but not 3. In other words, we sacrifice the size of the embedding to get a smaller distortion. However, this difficulty is overcome because a sharing technique is used in the proposed method to reduce the amount of data to be embedded, along with the second benefit of increasing the recovery ability from scattered large–area tampering. In general, the size of each share is only a small portion of the original size of the data. (As shown in Figs. 5.9-5.11, although our watermarked images have a better image quality, our recovery ability is still very competitive.)
5.6 Conclusions
This chapter proposes an authentication-recovery method. The watermarked image can be moderately altered by doing a JPEG compression, adding a Gaussian noise, or adjusting the brightness. Certain security tests, such as a cut-and-paste attack, a collage attack, and a VQ attack, are also tested. In our design, the recovery data is embedded in DCT coefficients using lattice embedding to reduce distortion. The recovery data is dispersed into many blocks by two-layer sharing. Compared with previously reported methods, our specialty is that the
114
tampered region can be recovered as long as the percentage of the tampered blocks does not exceed a pre-defined threshold, say, 16.66%. Notably, as stated in Part (a) of Sec. 5.5, it is hard to predict in advance which part of a watermarked image will be cropped or replaced by attackers. The traditional mapping-sequence strategy for finding locations to hide recovery data is not a suitable strategy. This dilemma is avoided in the proposed method by using sharing. After all, worrying about “the percentage of blocks being tampered” is simpler than worrying about “how to predict the actual locations of tampered area”.
115
Chapter 6
Conclusions and Future works
6.1 Conclusions
In this dissertation, some technologies are proposed to protect digital images. The technologies are Flip Visual Cryptography (Ch. 2), weighted secret image sharing (Ch. 3), data hiding (Ch. 4), and semi-fragile watermarking (Ch. 5).
In Chapter 2, Opaque-oriented FVC and non-opaque-oriented FVC schemes were introduced. We proved that both schemes satisfy perfect security and they are conditionally optimal in contrast. The generated transparencies do not lead to any expansion in size. The experimental results show the revealing of double secrets via flipping and stacking the transparencies together. Due to the double secrets feature of the proposed method, one of the applications is the double checking of ownership for personality identification. Since the size is non-expanded, the space needed to carry a transparency to a meeting is economical (the size is the same as the space needed to carry an original image).
In Chapter 3, a fast weighted secret image sharing method with a (t, n) threshold was proposed. This method shares the secret image among weighted participants, and the secret image can be losslessly recovered if the sum of the weights of the participants is greater than or equal to the threshold t. Additionally, the execution time in the weighted secret image sharing phase is improved by using properties of GF(2r). As shown in Fig. 2.5, our execution time is better than that of Thien and Lin whenwi > . The executives of a company can use 1 our method to share secret images.
In Chapter 4, an embedding method based on a weighted sum function was proposed. As shown in our figures and tables, this method has a wide range of embedding rates (0.5−4.0 bpp), and has a competitive PSNR over the entire range. The predicted PSNR values (PSNRest by Eq. (3.14)) are also extremely close to the actual PSNR values. Therefore, embedding errors can be predicted even before the actual embedding. With this PSNR-prediction property (Table 3.2), for each secret data the customer gives us, we can determine the necessary size of a host image if the customer also specifies the minimal PSNR value he can tolerate. This determines a set of host images for that secret data. Sec. 3.4 proved that
116
Modulus-based method [8] and LSB matching methods [9, 10] are special cases for us. The worst-case PSNR discussed in Item V of Sec. 3.5 also shows that, even if some very strange data (data artificially made by picky users and quite unnatural) was to be embedded, our method is still competitive with others.
In Chapter 5, a semi-fragile method with recovery ability was proposed. The watermarked image can be moderately altered by JPEG compression, adding Gaussian noise, or adjusting the brightness. Certain security tests, such as a cut-and-paste attack, a collage attack, and a VQ attack, were also tested. In our method, the recovery data is embedded in DCT coefficients using lattice embedding to reduce distortion. The recovery data is dispersed into many blocks by two-layer sharing. The defining characteristic of our method is that unlike previously reported methods, tampered regions can be recovered as long as the percentage of the tampered blocks does not exceed a pre-defined threshold, say, 16.66%.
6.2 Future works
Based on the proposed methodologies in this dissertation, some further works can be studied.
1. Visual Cryptography with multiple secrets is an interesting study issue (e.g. circular VC methods [3, 19]). Based on the method proposed in Chapter 2, in the future we plan to design a circular VC method for multiple secrets (the number of secrets can be larger than 2), using perfect security (and optimal contrast, if possible).
2. A fast sharing algorithm under GF(2k) is proposed in Chapter 3. However, in our method, the calculation in the decoding uses an extended Lagrange polynomial equation (2.8), which involves matrix multiplication. Therefore, it needs Θ(t) to decode a secret digit ai. Creation of a fast decoding algorithm is one of our future works.
3. The method proposed in Chapter 5 is processed under the DCT domain. In recent years, wavelet transform has been widely used in image compression. In the future, we plan to design a semi-fragile method in the wavelet domain.
117
References
[1] M. Naor and A. Shamir, "Visual Cryptography," in Proceedings of Advances in Cryptology — EUROCRYPT'94, Perugia, Italy, vol. 950, pp. 1-12, May 1994.
[2] C. N. Yang, "New visual secret sharing schemes using probabilistic method," Pattern Recognition Letters, vol. 25, no. 4, pp. 481-494, 2004.
[3] S. H. Shyu, "Image encryption by random grids," Pattern Recognition, vol. 40, no. 3, pp. 1014-1031, 2007.
[4] A. Shamir, "How to Share a Secret," Communications of the Acm, vol. 22, no. 11, pp.
612-613, 1979.
[5] G. R. Blakley, "Safeguarding cryptographic keys," in Proceedings of the National Computer Conference, New York, vol. 48, pp. 313-317, 1979.
[6] M. O. Rabin, "Efficient Dispersal of Information for Security, Load Balancing, and Fault Tolerance," Journal of the ACM, vol. 36, no. 2, pp. 335-348, 1989.
[7] C. C. Thien and J. C. Lin, "Secret image sharing," Computers & Graphics, vol. 26, no.
5, pp. 766-770, 2002.
[8] C. C. Thien and J. C. Lin, "A simple and high-hiding capacity method for hiding digit-by-digit data in images based on modulus function," Pattern Recognition, vol. 36, no. 12, pp. 2875-2881, 2003.
[9] X. L. Li, B. Yang, D. F. Cheng, and T. Y. Zeng, "A Generalization of LSB Matching,"
IEEE Signal Processing Letters, vol. 16, no. 1-3, pp. 69-72, 2009.
[10] J. Mielikainen, "LSB matching revisited," IEEE Signal Processing Letters, vol. 13, no.
5, pp. 285-287, 2006.
[11] C. K. Ho and C. T. Li, "Semi-Fragile Watermarking Scheme for Authentication of JPEG Images," in Proceedings of the International Conference on Information Technology: Coding and Computing, Las Vegas, Nevada, vol. 1, pp. 7-11, April 2004.
[12] C. H. Lin, T. S. Su, and W. S. Hsieh, "Semi-Fragile Watermarking Scheme for Authentication of JPEG Images," Tamkang Journal of Science and Engineering, vol.
10, no. 1, pp. 57-66, 2007.
[13] C. Y. Lin and S. F. Chang, "Semi-fragile watermarking for authenticating JPEG visual content," in SPIE International Conference on Security and Watermarking of Multimedia Contents II, San Jose CA, USA, vol. 3971, pp. 140-151, Jan 2000.
[14] S. L. Hsieh, P. D. Wu, I. J. Tsai, and B. Y. Huang, "A Recoverable Semi-fragile Watermarking Scheme Using Cosine Transform and Adaptive Median Filter," in Proceedings of the 5th international conference on Autonomic and Trusted Computing, Oslo, Norway, pp. 629-640, Jun 2008.
[15] X. Jiang and Q. Liu, "Semi-fragile watermarking algorithm for image tampers localization and recovery," Journal of Electronics, no. pp. 343-51, 2008.
[16] M. J. Tsai and C. C. Chien, "Authentication and recovery for wavelet-based semifragile watermarking," Optical Engineering, vol. 47, no. 6, p. 067005, 2008.
[17] M. J. Tsai and C. C. Chien, "A wavelet-based semi-fragile watermarking with recovery mechanism," in Proceedings of IEEE International Symposium on Circuits and Systems, Seattle, WA, vol. 1-10, pp. 3033-3036, May 2008.
[18] S. K. Chen and S. J. Lin, "Non-expansible Flip-flop Visual Cryptography with Perfect Security," in Fifth International Conference on Intelligent Information Hiding and Multimedia Signal Processing, Kyoto, pp. 949-952, Sep 2009.
118
[19] H. C. Wu and C. C. Chang, "Sharing visual multi-secrets using circle shares,"
Computer Standards & Interfaces, vol. 28, no. 1, pp. 123-135, 2005.
[20] W. P. Fang and J. C. Lin, "Visual cryptography with extra ability of hiding confidential data," Journal of Electronic Imaging, vol. 15, no. 2, p. 023020, 2006.
[21] S. J. Shyu, S. Y. Huang, Y. K. Lee, R. Z. Wang, and K. Chen, "Sharing multiple secrets in visual cryptography," Pattern Recognition, vol. 40, no. 12, pp. 3633-3651, 2007.
[22] A. De Bonis and A. De Santis, "Randomness in secret sharing and visual cryptography schemes," Theoretical Computer Science, vol. 314, no. 3, pp. 351-374, 2004.
[23] T. H. Chen and K. H. Tsao, "Visual secret sharing by random grids revisited," Pattern Recognition, vol. 42, no. 9, pp. 2203-2217, 2009.
[24] Y. F. Chen, Y. K. Chan, C. C. Huang, M. H. Tsai, and Y. P. Chu, "A multiple-level visual secret-sharing scheme without image size expansion," Information Sciences, vol.
177, no. 21, pp. 4696-4710, 2007.
[25] R. Ito, H. Kuwakado, and H. Tanaka, "Image size invariant visual cryptography,"
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, vol. E82a, no. 10, pp. 2172-2177, 1999.
[26] R. Z. Wang and S. J. Shyu, "Scalable secret image sharing," Signal Processing: Image Communication, vol. 22, no. 4, pp. 363-373, 2007.
[27] W. P. Fang, "Friendly progressive visual secret sharing," Pattern Recognition, vol. 41, no. 4, pp. 1410-1414, 2008.
[28] S. J. Lin and J. C. Lin, "VCPSS: A two-in-one two-decoding-options image sharing method combining visual cryptography (VC) and polynomial-style sharing (PSS) approaches," Pattern Recognition, vol. 40, no. 12, pp. 3652-3666, 2007.
[29] C. N. Yang and T. S. Chen, "Aspect ratio invariant visual secret sharing schemes with
[29] C. N. Yang and T. S. Chen, "Aspect ratio invariant visual secret sharing schemes with