基于矩阵变换的图像置乱逆问题求解  

Solution for the Inverse Problem of Matrix Transform Based Image Scrambling

作　　者：邵利平[1] 覃征[1] 衡星辰[1] 高洪江[1] 

机构地区：[1]西安交通大学电子商务研究所,陕西西安710049

出　　处：《电子学报》2008年第7期1355-1363,共9页Acta Electronica Sinica

基　　金：国家973重点基础研究发展规划(No.2004CB719401);国防“十一五”预研基金(No.402040202,No.102010302)

摘　　要：基于矩阵变换的图像置乱用周期性对图像恢复,代价高昂;而通过逆变换恢复,已有方法未解决ZN上逆阵求解问题.针对此问题,通过构造映射规则,将伴随矩阵求逆方法推广到ZN,解决了n维矩阵变换的逆问题.为减小该方法计算代价,将杜里特尔分解和克劳特分解求逆推广到ZN,解决了变换阵顺阶主子式模N互质,矩阵变换的逆问题.为弱化扩展杜里特尔分解和克劳特分解求逆条件,将高斯-约当消去法推广到ZN,给出了任意变换阵在ZN上求逆算法和简化求逆算法.所提方法可用于得到任意变换阵在ZN上的逆变换阵,从而可直接对图像恢复,而不必计算可恢复周期.实验表明所提方法的可行性和有效性.In matrix transform based image scrambling, the cost to restore image by periodicity is expensive. To restore image by inverse mapping,existing methods do not solve the problem computing inverse matrix in ZN. To address it, this paper expanded the method getting inverse matrix by adjoint matrix to ZN through constructing mapping regulations. To decrease the expensive cost of it, this paper generalized Doolittle decomposition and Crout decomposition into ZN, which solves the inverse problem of any n dimensional matrix scrambling transform where every order rank principal minor of transform matrix is prime with N. To weak the expanded Doolittle decomposition and Crout decomposition condition, this paper expanded Gauss-Jordan elimination into ZN and gave the inverse matrix generation algorithm and simplified algorithm. The proposed methods can get the inverse matrix in ZN directly to restore image without computing the periodicity. The experiments show the proposed methods feasibility and validity.

关 键 词：图像置乱变换 杜里特尔分解 克劳特分解 高斯-约当消去 逆变换 乘法逆元 

分 类 号：TP309.7[自动化与计算机技术—计算机系统结构]