[1]李雄军,廖日军,李金龙,等.图像Arnold变换中的准对称性问题与半周期现象[J].深圳大学学报理工版,2015,32(6):551-562.[doi:10.3724/SP.J.1249.2015.06551]
　Li Xiongjun,Liao Rijun,Li Jinlong,et al.Quasi-symmetry and the half-cycle phenomenon in scrambling degrees for images with pixel locations scrambled by Arnold transformation[J].Journal of Shenzhen University Science and Engineering,2015,32(6):551-562.[doi:10.3724/SP.J.1249.2015.06551]
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图像Arnold变换中的准对称性问题与半周期现象()
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《深圳大学学报理工版》[ISSN:1000-2618/CN:44-1401/N]

卷:
第32卷
期数:
2015年第6期
页码:
551-562
栏目:
电子与信息科学
出版日期:
2015-11-23

文章信息/Info

Title:
Quasi-symmetry and the half-cycle phenomenon in scrambling degrees for images with pixel locations scrambled by Arnold transformation
文章编号:
201506001
作者:
李雄军; 廖日军; 李金龙; 冼建标; 徐健杰; 黄培; 何小雨
深圳大学物理科学与技术学院，深圳  518060
Author(s):
Li Xiongjun;  Liao Rijun;  Li Jinlong;  Xian Jianbiao;  Xu Jianjie;  Huang Pei;  and He Xiaoyu
College of Physics Science and Technology, Shenzhen University, Shenzhen 518060, P.R.China
关键词:
计算机应用; Arnold变换; 准对称性; 置乱变换; 置乱度; 半周期现象; Fibonacci变换; 图像加密
Keywords:
computer application;  Arnold transformation;  quasi-symmetry;  scrambling transform;  scrambling degree;  half-cycle phenomenon;  Fibonacci transformation;  image encryption
分类号:
U 491.1
DOI:
10.3724/SP.J.1249.2015.06551
文献标志码:
A
摘要:
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．
Abstract:
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.

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备注/Memo

备注/Memo:
Received：2015-10-22；Accepted：2015-10-31
Foundation：University Student’s Innovation and Entrepreneurship Training Program Foundation of Shenzhen University(201510590079); 2015 Foundation for College Students’ Science and Technology Innovation Training Program (“Climbing” Program) of Guangdong Province (201510590079)
 Corresponding author：Associate professor Li Xiongjun．E-mail：lixj@szu.edu.cn
Citation：Li Xiongjun, Liao Rijun, Li Jinlong, et al. Quasi-symmetry and the half-cycle phenomenon in scrambling degrees for images with pixel locations scrambled by Arnold transformation［J］. Journal of Shenzhen University Science and Engineering, 2015, 32(6): 551-562.（in Chinese）
基金项目：广东省“攀登计划”资助项目（201560020006）；深圳大学大学生创新创业训练计划资助项目（20151059 0079）
作者简介：李雄军（1966—），女（汉族），湖南省双峰县人，深圳大学副教授、博士．E-mail：lixj@szu.edu.cn
引文：李雄军，廖日军，李金龙，等．图像Arnold变换中的准对称性问题与半周期现象［J］. 深圳大学学报理工版，2015，32（6）：551-562.
更新日期/Last Update: 2015-11-06