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3.5 Encoding a set of corresponding pixels ${(p_{1}, p_{2}, p_{3})}_{j}^{1}$ ${(p_{1}, p_{2}, p_{3})}_{j}^{1}$ into $a_{j}^{1}$ $a_{j}^{1}$ ( $a_{j}^{2}$ $a_{j}^{2}$ and $a_{j}^{3}$ $a_{j}^{3}$ ) and $b_{j}^{1}$ $b_{j}^{1}$ in terms of $s_{A}^{1}$ $s_{A}^{1}$ ( $s_{A}^{3}, s_{A}^{1}$ $s_{A}^{3}, s_{A}^{1}$ , respectively) and s_B in the first chords of A and B, respectively for visual 3-secret sharing. 73

3.6 Encoding a set of corresponding pixels ${(p_{1}, p_{2}, p_{3})}_{j}^{2}$ ${(p_{1}, p_{2}, p_{3})}_{j}^{2}$ into $a_{j}^{2}$ $a_{j}^{2}$ ( $a_{j}^{3}$ $a_{j}^{3}$ and $a_{j}^{1}$ $a_{j}^{1}$ ) and $b_{j}^{2}$ $b_{j}^{2}$ in terms of $s_{A}^{2}$ $s_{A}^{2}$ ( $s_{A}^{2}, s_{A}^{1}$ $s_{A}^{2}, s_{A}^{1}$ , respectively) and s_B in the first chords of A and B, respectively for visual 3-secret sharing. 76

3.7 Encoding scheme equivalent to Table 6 for the second chords of A and B for visual 3-secret sharing 77

3.8 Encoding a set of corresponding pixels ${(p_{1}, p_{2}, p_{3})}_{j}^{3}$ ${(p_{1}, p_{2}, p_{3})}_{j}^{3}$ into $a_{j}^{3}$ $a_{j}^{3}$ ( $a_{j}^{1}$ $a_{j}^{1}$ and $a_{j}^{2}$ $a_{j}^{2}$ ) and $b_{j}^{3}$ $b_{j}^{3}$ in terms of $s_{A}^{3}$ $s_{A}^{3}$ ( $s_{A}^{1}, s_{A}^{2}$ $s_{A}^{1}, s_{A}^{2}$ , respectively) and s_B in the first chords of A and B, respectively for visual 3-secret sharing. 78

3.9 Possible set of encoded patterns for $s_{A}^{1}, s_{A}^{2}, s_{A}^{3}$ $s_{A}^{1}, s_{A}^{2}, s_{A}^{3}$ , and s_B; and their stacked results for x = 3 84

3.10 Comparison of visual multiple secrets sharing schemes 90

4.1 Comparison of visual cryptography schemes 108

4.2 Approaches to photograph visual cryptography 110

5.1 Values of β for scheme S′ for n = 3. The max over each row is in boldface 146

5.2 Values of β for n = 4. The max over each row is in boldface 146

5.3 Values of β for n = 5. The max over each row is in boldface and ∊₁ and ∊₂ denote very small values 147

6.1 Distribution phase of (2, 4)-VCS using Yang et al.’s [25] algorithm 170

6.2 Reconstruction of participant 1 and 2 using Yang et al.’s [25] algorithm 170

6.3 Reconstruction of participant 3 and 4 using Yang et al.’s [25] algorithm 170

6.4 Distribution phase of the visual cryptography with minimal access structure Γ₀ = {{1, 2}, {1, 3}, {2, 3}} using Hu and Tzeng [?] algorithm 173

6.5 Reconstruction of participant 1 and 2 using Hu and Tzeng [10] algorithm 174

6.6 Reconstruction of participant 1 and 3 using Hu and Tzeng [10] algorithm 174

6.7 Reconstruction of participant 2 and 3 using Hu and Tzeng [10] algorithm 174

7.1 Results of the superimposition of two random pixels 187

7.2 Encoding b into r₁ and r₂ and results of s = r₁ ⊗ r₂ by Algorithms 1, 2, and 3. 192

7.3 Light contrasts by Algorithms 1–3 in producing VCRG-2 196

7.4 All possible combinations of b, r₁, and r₂, the corresponding results of a₁ (= r₁), a₂ = f (r₂, a1), r₃ = f (b, a₂), and r₁ ⊗ r₂, r₁ ⊗ r₃, r₂ ⊗ r₃ as well as r₁ ⊗ r₂ ⊗ r₃ by Algorithm 4 for (3, 3)-VCRG 203

7.5 Light transmissions of the corresponding pixels in Table 7.3. 204

7.6 Light contrasts by Algorithms 4–6 for ( n, n)-VCRG. 206

7.7 Comparison of pixel expansions for conventional (n, n) visual cryptographic and (n, n)-VCRG schemes 219

9.1 Comparison between VCSs with reversing based on perfect black VCSs 276

10.1 Comparisons of different cheating prevention visual secret sharing schemes 295

11.1 The advantages and disadvantages of large and small subpixels 298

11.2 The probability of the four patterns appearing at the columns i and i + 1 in the collections C₀ and C₁ 307

11.3 The properties of the misaligned schemes 313

11.4 The relation of the ratios R_C_,c and R′_C, _c 317

11.5 The advantages and disadvantages of large and small subpixels 319

15.1 PSNR comparisons for past schemes and our scheme 419

15.2 PSNR measurements when embedding the two target secret images in Figure 15.5 (a) and (b) comparisons for past schemes and our scheme 421

15.3 PSNR comparisons with the Lin-Tsai scheme 421

15.4 PSNR comparisons with the Lin-Tsai scheme 423

16.1 The cover-to-secret image size ratios of the existing schemes and the proposed scheme 458

16.2 The detection ratios of different schemes 458

17.1 Pixel expansions, file sizes of shadows, and contrasts of the previewed images for some (k, n)-TiOISSSs 482

17.2 A comparison among three TiOISSSs 484

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