秘密影像分享(SIS)是一種門檻式的影像分享機制，從眾多子影像中選取任意的子影像數目若高於或等於門檻則能解回秘密影像。SIS有兩大類型: 第一種是有「相疊即視」特性的視覺密碼(Visual cryptography scheme ; VCS)。使用VCS時無需額外設備可將子影像疊合即能用人眼解回秘密影像，但缺點是解回之影像視覺品質不佳。第二種是多項式秘密影像分享 (Polynomial based SIS; PSIS) 能使用拉格朗日插值多項式解回無失真秘密影像，但是插值多項式的計算比VCS的疊合運算複雜許多。所謂的雙重解碼功能秘密影像分享 (Two-in-One Secret Image Sharing; TiOSIS) 有兩種解碼方式: 第一個解碼階段使用VCS預覽圖像，第二解碼階段是使用PSIS解回原始的秘密影像。一般而言，用於TiOSIS機制第二階段PSIS解密的信息是嵌入於第一階段的VCS子像素排列。由於VCS本質上原就有某種程度的抗雜訊強韌性，子影像的黑、白點即使受雜訊干擾，解回的像素仍然有極高的正確率。因此，在傳輸或儲存子影像時，並不需要設抗雜訊的VCS。但用於TiOSIS時，若VCS子影像有錯就無法正確地攜帶PSIS 解密所需的信息。
植基於上述的觀察，開啟了具t個錯誤改正能力VCS(簡寫為VCS-tEC)的新研究方向，這個新型的VCS-tEC讓TiOSIS的應用更加落實。最近有些使用定量碼的VCS-tEC被提出，然而有些問題，例如信息攜帶量、與解碼複雜度、秘密的呈現方式都需要考量和改進。本論文“改進具糾錯能力之視覺密碼的信息攜帶量和解碼複雜度”提出了兩種方案: (i) 機率式(k, n)VCS-tEC (簡寫為PVCS-tEC)，和 (ii) 確定式(2, n)VCS-tEC (簡寫為DVCS-tEC)。我們以機率式VCS與BCH碼，實現方案(i)的(k, n)-PVCS-tEC。使用BCH線性區塊碼比非線性定量碼在解回信息的複雜度低、速度快，並且信息攜帶量也較大。方案(ii)中，我們以確定式VCS來改善解回影像的清晰度，並同時還保有高的信息攜帶量和簡單快速的解碼方法。若使用確定式VCS，只能完成(2, n)-VCS-tEC，且使用部分的BCH碼字。我們的方案(ii)成功使用了部分BCH碼字，符合VCS的安全與清晰度條件、又同時能以原有之線性BCH碼簡單解碼。

Secret Image Sharing (SIS) is a threshold image sharing scheme. If the number of involved participants in a privileged set greater than the threshold, they can recover secret from their shadow images (referred to as shadows). There are two major categories of SIS: one is Visual cryptography scheme (VCS) and the other is Polynomial SIS (PSIS). VCS has the benefit of stacking-to-see property. The secret image can be visually decoded by human visual system via stacking shadows. However, VCS has the vague recovered image. The other is PSIS, which can recover the original secret image via using Lagrange interpolation. The so-called two-in-one SIS (TiOSIS) is a new type of SIS with two decoding options: the secret image is previewed by VCS. In the second decoding phase, we then spend more computation via Lagrange interpolation to obtain the distortion-less image. In TiOSIS, the carrying information for PSIS is embedded in the combinations of subpixels in VCS shadow images. Because of the intrinsic noise robustness of VCS, even though black/white dots in shadow images suffer from interference by noise, the secret color may still retain the corresponding darkness with high probability. However, the subpixel still has error at this time. If the subpixel value is not correct, the incorrect carrying information for PSIS recovery yields an incorrect result.
The above observation starts a new research direction of VCS with t-error correcting capability (VCS-tEC) and makes the application of TiOSIS more practical. Recently, some VCS-tECs based constant weight code were proposed; however, their carrying-information capacity, decoding efficiency, and demonstration way of VCS secret could be further enhanced. In the thesis, we propose two schemes, (i) the probabilistic (k, n)-VCS-tEC (PVCS-tEC) and (ii) the deterministic (2, n)-VCS-tEC (DVCS-tEC) are proposed in the thesis. In Scheme (i), we adopt probabilistic VCS and BCH code to realize (k, n)-VCS-tEC. BCH code has low-complexity, fast decoding, and more carrying-information capacity than using constant weigh code. In Scheme (ii), we use deterministic VCS to enhance the contrast of VCS. However, we can only deal with (2, n)-VCS-tEC when using deterministic VCS. For the case, part of the BCH codewords are used, such that the decoding can be realized by BCH code to retain fast decoding and meantime has high carrying-information capacity. In addition, the security and contrast conditions of VCS are still satisfied.

Chapter 1 Introduction 1
1.1 Background 1
1.2 Contribution of The Thesis 3
1.3 Organization of The Thesis 4
Chapter 2 Previous Works 5
2.1 VCS 5
2.2 PVCS 7
Chapter 3 Observations on Visual Cryptography with Error Correcting Capability 9
3.1 Pros and Cons of Various VCS-tEC 9
3.2 Concerns in Designing VCS-tEC 10
Chapter 4 The Proposed Probabilistic (k, n)-VCS-tEC 13
4.1 Motivation and Design Concept 13
4.2 Proposed Probabilistic (k, n)-VCS-tEC Using BCH Code 14
4.3 Numerical Examples 23
Chapter 5 The Proposed Deterministic (2, n)-VCS-tEC 39
5.1 Deterministic (2, n)-VCS-tEC 42
5.2 Numerical Examples 46
Chapter 6 Experimental Results and Discussions 55
6.1 Visual Example 55
6.2 More Results for Natural Images 60
6.3 Discussion and Comparison 67
Chapter 7 Conclusion and Future Work 77
References 79

[BOSE82] B. Bose and T. R. Rao, “Theory of unidirectional error correcting/detecting codes,” IEEE Transactions on Computers, no. 6, pp. 521– 530, 1982.
[BLUN03] C. Blundo, P. D’Arco, A. De Santis, and D. R. Stinson, “Contrast optimal threshold visual cryptography schemes,” SIAM Journal on Discrete Mathematics, vol. 16, no. 2, pp. 224–261, 2003.
[CIMA05] S. Cimato, A. De Santis, A. L. Ferrara, and B. Masucci, “Ideal contrast visual cryptography schemes with reversing,” Information Processing Letters, vol. 93, no. 4, pp. 199–206, 2005.[CIMA06] S. Cimato, R. De Prisco, and A. De Santis, “Probabilistic visual cryptography schemes,” The Computer Journal, vol. 49, no. 1, pp. 97– 107, 2006.
[CHEN09] T. Chen and K. Tsao, “Visual secret sharing by random grids revisited,” Pattern Recognition, vol. 42, no. 9, pp. 2203–2217, 2009.
[EISE02] P. A. Eisen and D. R. Stinson, “Threshold visual cryptography schemes with specified whiteness levels of reconstructed pixels,” Designs, Codes and Cryptography, vol. 25, no. 1, pp. 15–61, 2002.
[HOU03] Y. Hou, “Visual cryptography for color images,” Pattern Recognition, vol. 36, no. 7, pp. 1619–1629, 2003.
[HU16] H. Hu, G. Shen, Z. Fu, B. Yu, and J. Wang, “General construction for xor-based visual cryptography and its extended capability,” Multimedia Tools and Applications, vol. 75, no. 21, pp. 13 883–13 911, 2016.
[JIN05] D. Jin, W.Q. Yan, and M.S. Kankanhalli, “Progressive color visual cryptography,” Journal of Electronic Imaging, vol. 14, no. 3, p. 033019, 2005.
[JIA18] X. Jia, D. Wang, D. Nie, and C. Zhang, “Collaborative visual cryptography schemes,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 28, no. 5, pp. 1056–1070, 2018.
[JIA19] X. Jia, D. Wang, Q. Chu, and Z. Chen, “An efficient xor-based verifiable visual cryptographic scheme,” Multimedia Tools and Applications, vol. 78, no. 7, pp. 8207–8223, 2019.
[KUND90] S. Kundu and S.M. Reddy, “On symmetric error correcting and all unidirectional error detecting codes,” IEEE transactions on computers, vol. 39, no. 6, pp. 752–761, 1990.
[LAIH95] C.S. Laih and C.N. Yang, “On the analysis and design of group theoretical t-syec/aued codes,” IEEE transactions on computers, vol. 45, no. 1, pp. 103–108, 1996.
[LI12] P. Li, P.J. Ma, X.H. Su, and C.N. Yang, “Improvements of a two-inone image secret sharing scheme based on gray mixing model,” Journal of Visual Communication and Image Representation, vol. 23, no. 3, pp. 441–453, 2012.
[LI18] P. Li, Z. Liu, and C.N. Yang, “A construction method of (t, k, n)-essential secret image sharing scheme,” Signal Processing: Image Communication, vol. 65, pp. 210–220, 2018.
[LIN07] S.J. Lin and J.C. Lin, “Vcpss: A two-in-one two-decoding-options image sharing method combining visual cryptography (vc) and polynomialstyle sharing (pss) approaches,” Pattern Recognition, vol. 40, no. 12, pp. 3652–3666, 2007.
[LIU10] F. Liu, C. Wu, and X. Lin, “Step construction of visual cryptography schemes,” Information Forensics and Security, IEEE Transactions on, vol. 5, no. 1, pp. 27–38, 2010.
[LIU17] Y. Liu and C. Yang, “Scalable secret image sharing scheme with essential shadows,” Signal Processing: Image Communication, vol. 58, pp. 49–55, 2017.
[LIU18] Y.N. Liu, Q. Zhong, M. Xie, and Z.B. Chen, “A novel multiplelevel secret image sharing scheme,” Multimedia Tools and Applications, vol. 77, no. 5, pp. 6017–6031, 2018.
[NAOR95] M. Naor and A. Shamir, “Visual cryptography,” Lecture Notes in Computer Science, vol. 950, no. 1, pp. 1–12, 1995.
[SHEN17] G. Shen, F. Liu, Z. Fu, and B. Yu, “Perfect contrast xor-based visual cryptography schemes via linear algebra,” Designs, Codes and Cryptography, vol. 85, no. 1, pp. 15–37, 2017.
[SHYU11] S. J. Shyu and M. C. Chen, “Optimum pixel expansions for threshold visual secret sharing schemes,” IEEE Transactions on Information Forensics and Security, vol. 6, no. 3, pp. 960–969, 2011.[THIE02] C.C. Thien and J.C. Lin, “Secret image sharing,” Computers & Graphics, vol. 26, no. 5, pp. 765–770, 2002.
[TUYL05] P. Tuyls, H. Hollmann, J. Lint, and L. Tolhuizen, “Xor-based visual cryptography schemes,” Designs, Codes and Cryptography, vol. 37, no. 1, pp. 169–186, 2005.
[VOLD95] E. Volden, G. Giraudon, and M. Berthod, “Image redundancy and classification,” in International Conference on Computer Analysis of Images and Patterns. Springer, pp. 206–213, 1995.
[VERH97] E.R. Verheul and H.C. Van Tilborg, “Constructions and properties of k out of n visual secret sharing schemes,” Designs, Codes and Cryptography, vol. 11, no. 2, pp. 179–196, 1997.
[WU12] X. Wu and W. Sun, “Visual secret sharing for general access structures by random grids,” IET information security, vol. 6, no. 4, pp. 299–309, 2012.
[WANG13] D. Wang, T. Song, L. Dong, and C.N. Yang, “Optimal contrast grayscale visual cryptography schemes with reversing.” IEEE Trans. Information Forensics and Security, vol. 8, no. 12, pp. 2059–2072, 2013.
[WU13] X. Wu and W. Sun, “Improving the visual quality of random grid-based visual secret sharing,” Signal Processing, vol. 93, no. 5, pp. 977–995, 2013.
[WU19] X. Wu and C.N. Yang, “A combination of color-black-and-white visual cryptography and polynomial based secret image sharing,” Journal of Visual Communication and Image Representation, vol. 61, pp. 74–84, 2019.
[WU20] X. Wu and C.N. Yang, “Probabilistic color visual cryptography schemes for black and white secret images,” Journal of Visual Communication and Image Representation, p. 102793, 2020.
[YAN17] X. Yan and Y. Lu, “Contrast-improved visual secret sharing based on random grid for general access structure,” Digital Signal Processing, vol. 71, pp. 36–45, 2017.
[YAN20] X. Yan, Y. Lu, L. Liu, and X. Song, “Reversible image secret sharing,” IEEE Transactions on Information Forensics and Security, vol. 15, pp. 3848–3858, 2020.
[YANG04] C. Yang, “New visual secret sharing schemes using probabilistic method,” Pattern Recognition Letters, vol. 25, no. 4, pp. 486–494, 2004.
[YANG14a] C.N. Yang and D.S. Wang, “Property analysis of xor-based visual cryptography,” IEEE transactions on circuits and systems for video technology, vol. 24, no. 2, pp. 189–197, 2014.
[YANG14b] C.N. Yang, C.C. Wu, and D.S. Wang, “A discussion on the relationship between probabilistic visual cryptography and random grid,” Information Sciences, vol. 278, pp. 141–173, 2014.
[YANG20] C.N. Yang and Y.Y. Yang, “On the analysis and design of visual cryptography with error correcting capability,” IEEE Transactions on Circuits and Systems for Video Technology, 2020.