深圳大学学报理工版

从广义Arnold变换的周期性及标准Arnold变换与Fibonacci的关系出发,推导k步Arnold变换的一次性等效变换矩阵,特别是半周期处的一次性变换矩阵,并分析其特点,证明图像经广义Arnold变换位置置乱后在置乱周期内呈现图像置乱度的准对称性,讨论当置乱周期为偶数时的半周期现象和置乱周期为奇数时的各种不同情况.研究结果表明,无论置乱周期为奇数还是偶数,图像乱度存在前半周期和后半周期的准对称性; 对偶数周期情况,标准Arnold变换下,在置乱次数等于周期的一半时,一次性置乱变换矩阵为单位矩阵的整数倍; 半周期处置乱图像更易呈现与原图相似的结构或内容信息; 对于某些维数的图像,半周期处的一次性置乱变换为负的单位矩阵,此时图像为原图的水平加垂直镜像图像; 广义Arnold变换下,偶数置乱周期变换的半周期处的一次性变换矩阵可能是标准Arnold变换的结果,或在此基础上叠加了一个位移量为图像维数一半的水平或垂直平移,因而仍然存在较明显的半周期现象.对于奇数周期,半周期现象虽然存在但一般不如偶数周期情况明显,更不易出现镜像或提前恢复原图的情况. 该研究可用于指导图像加密预处理中置乱次数选择和置乱乱度计算方法的评价与比较.

By referring to the periodicity of the general Arnold transformation and the relationship between standard Arnold transformation and Fibonacci transformation, we deduce the equivalent one-step transformation matrix for k times of Arnold transformation with pixel position scrambled, especially the one at the half-cycle of the scrambling period. We analyze their characteristics and provide a proof of the quasi-symmetry in scrambling degrees for images in one cycle. We discuss the half-cycle effect in scrambling degrees in scrambled images with even and odd scrambling periods respectively. Results show that there exists a quasi-symmetry in scrambling performance between the two half cycles regardless of the period being even or odd. In a standard Arnold transformation with a commonly even period, the one-step transform is equivalent to a simple scaling matrix transform which leads to the scrambled image at the half period with an obvious lower scrambling degree, where being the minus unitary matrix as a special case results in the scrambled image being the horizontal mirror image with an overlying vertical mirror image of the original image. For any general Arnold transformation with an even scrambling period, the one-step transformation at half cycle may be the same as the one-step transform for standard Arnold transform or with a translation of half of the image dimension superimposed, thus leading to a little less salient half-cycle phenomenon. For an Arnold transformation with an odd scrambling period, no such situation happens in general unless for images with very special contents and structure. The results can be applied in choice of scrambling time for the pre-processing in image encryption and the evaluation and comparison of image scrambling degree criteria.

引言
1 准对称性问题
2 半周期现象讨论
3 实验与讨论
4 结语

图1 图像经Arnold变换置乱情况<br/><br/>Fig.1 Images scrambled by Arnold transformation and their scrambling degree curves during a scrambling cycle

图1 图像经Arnold变换置乱情况

Fig.1 Images scrambled by Arnold transformation and their scrambling degree curves during a scrambling cycle

图2 准对称性解释示意图<br/><br/>Fig.2 Schematic sketch for explanation of quasi-symmetry

图2 准对称性解释示意图

Fig.2 Schematic sketch for explanation of quasi-symmetry

表1 典型Arnold变换参数与特性<br/>Table 1 Some typical Arnold transforms for scrambling and their characteristics

表1 典型Arnold变换参数与特性
Table 1 Some typical Arnold transforms for scrambling and their characteristics

图3 图像在半周期处恢复原图的特例(b=c=1, N=128, mN=96)<br/>Fig.3 Image recovery at the half scrambling cycle for the original image with specific symmetry(b=c=1, N=128, mN=96)

图3 图像在半周期处恢复原图的特例(b=c=1, N=128, mN=96)
Fig.3 Image recovery at the half scrambling cycle for the original image with specific symmetry(b=c=1, N=128, mN=96)

图4 实例2的部分置乱图像(b=2, c=3,图像大小为122×122, mN =12)<br/>Fig.4 Some scrambled images by a general Arnold transformation with b=2 and c=3 for Test 2(N=122, mN=12)

图4 实例2的部分置乱图像(b=2, c=3,图像大小为122×122, mN =12)
Fig.4 Some scrambled images by a general Arnold transformation with b=2 and c=3 for Test 2(N=122, mN=12)

图5 实例2的置乱度曲线<br/>Fig.5 The scrambling degree curve for Test 2

图5 实例2的置乱度曲线
Fig.5 The scrambling degree curve for Test 2

图6 实例3的部分置乱结果和置乱曲线(b=c=1, N=124, mN =15)<br/>Fig.6 Some scrambled images for Test 3(b=c=1, N=124, mN =15)

图6 实例3的部分置乱结果和置乱曲线(b=c=1, N=124, mN =15)
Fig.6 Some scrambled images for Test 3(b=c=1, N=124, mN =15)

图7 实例3的置乱度曲线<br/>Fig.7 The scrambling degree curve for Test 3

图7 实例3的置乱度曲线
Fig.7 The scrambling degree curve for Test 3

表2 乱度对照表与半周期显著性<br/>Table 2 Contrast of scrambling degrees and salience test of the half-cycle phenomenon for Fig.1(a)and Fig.5

表2 乱度对照表与半周期显著性
Table 2 Contrast of scrambling degrees and salience test of the half-cycle phenomenon for Fig.1(a)and Fig.5

[1] Arnold V J, Avea A. Ergodic problems of classica mechanics, mathematical Physics monograph series[M]. New York(USA):W A Benjamin Inc, 1968.
[2] Li Xiongjun. A generalized matrix-based scrambling transformation and its properties[C]// Proceedings of the 9th International Conference for Young Computer Scientists. Huangshan(China): IEEE, 2008: 1429-1434.
[3] Zou Jiancheng, Tie Xiaoyun. Arnold transformation of digital image with two dimensions and its periodicity[J]. Journal of North China University of Technology, 2000, 12(1): 10-14.(in Chinese)
[4] Mao Leibo. Research on Arnold transformation algorithm and anti-Arnold transformation algorithm[J]. Journal of Chongqing Technology and Business University Natural Science Edition, 2012, 29(3): 16-21.(in Chinese)
[5] Wu Faen, Zou Jiancheng. Some necessary conditions for the periodicity of Arnold transformation of digital image with two dimensions[J]. Journal of North China University of Technology, 2001, 125(6): 66- 69.(in Chinese)
[6] Wu Chengmao. An improved discrete arnold transform and its application in image scrambling and encryption[J]. Acta Physica Sinica, 2014, 63(9): 90504-1- 090504-20.(in Chinese)
[7] Radharani S, Valarmathi M L. Content based image watermarking scheme using block SVD and Arnold transform [C]// International Conference on Electronics and Communication Systems(ICECS). Coimbatore(India): IEEE, 2014: 1- 4.
[8] Mehta R, Vishwakarma V P, Rajpal N. Lagrangian support vector regression based image watermarking in wavelet domain [C]// The 2nd International Conference on Signal Processing and Integrated Networks(SPIN). Noida(India): IEEE, 2015: 854- 859.
[9] Li Xiongjun.A new measure of image scrambling degree based on grey level difference and information entropy[C]// Proceedings of International Conference on Computational Intelligence and Security. Washington D C:IEEE, 2008,1: 350 -354.
[10] Wang Xinxin,Bu Ting.An evaluation algorithm of image scrambling degree based on the image area[J]. Journal of Anhui University Natural Science Edition, 2011, 35(4): 48-52.(in Chinese)
[11] Xu Jiangfeng,Yang You.Analysis of scrambling performance of encrypted image[J]. Computer Science,2006, 33(3): 110-113.(in Chinese)
[12] Tan Yongjie, Hu Haizhi. A survey of assessment of image scrambling degree[J]. Computer and Digital Engineering, 2012, 40(4): 93-95.(in Chinese)
[13] Huang Liangyong, Xiao Degui. The best image scrambling degree of binary image based on Arnold transform[J]. Journal of Computer Applications, 2009, 29(2): 474- 476.(in Chinese)
[14] Feng Xingang, Zhou Quan. A novel scrambling degree rule of digital image based on center of mass[J]. Journal of Electronics & Information Technology, 2008, 30(11): 2684-2687.(in Chinese)
[15] Huang Xing, Zhang Minrui. Study on the image scrambling extent[J]. Geomatics and Information Science of Wuhan University, 2008, 33(5): 465- 468.(in Chinese)
[16] Guo Linqin. On the image scrambling and scrambling degree[J]. Journal of Xi'an University of Arts and Science Natural Science Edition, 2013, 16(3): 49-52.(in Chinese)
[17] Yang Xiyang, Li Zhiwei. Image scrambling measurement based on chi-square statistic[J]. Journal of Xiamen University Natural Science, 2010, 49(6): 778-781.(in Chinese)
[18] Liao Rijun, Li Xiongjun, Xu Jianjie, et al. Discussions on applications of Arnold transformation in binary image scrambling[J].Journal of Shenzhen University Science and Engineering, 2015, 32(4): 428- 433.(in Chinese)

备注

引言

1 准对称性问题

2 半周期现象讨论

3 实验与讨论

4 结语

期刊信息

备注

引言

1 准对称性问题

2 半周期现象讨论

3 实验与讨论

4 结 语

期刊信息

4 结语