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An image fusion algorithm based on lifting wavelet transform and IHS transform[J]. Journal of Image and Graphics, 2009, 14(2): 340-345. [16] 赵福才, 胡以华, 张立. 利用小波变换进行MC-MPSK信号载频盲估计[J]. 北京邮电大学学报, 2008, 31(3): 133-136. Zhao Fucai, Hu Yihua, Zhang Li. Carrier frequency blind estimation of MC-MPSK signal using wavelet transform[J]. Journal of Beijing University of Posts and Telecommunications, 2008, 31(3): 133-136.附录定理1 对于式(1),假设m=n,且源信号矩阵S各行Si(i=1,2,…,n)均为t×t的图像按行排列而成. 则混合图像一层小波变换后的高频部分仍可表示为源图像高频部分线性混合的方式,且A保持不变. 定义5个函数操作的概念. IM(v):R1×t→Rt×t,表示将行一维行向量信号v转化为二维矩阵,即 IM(v)=v1[]v2[]…[]vt vt+1[]vt+2[]…[]vt+t [][][] vT-t+1[]vT-t+2[]…[]vT 其中v=(v1,v2,…,vT)是一个T=t×t的行向量. IM-1(M):Rt×t→R1×t,表示将二维矩阵M按行排列成的一维行向量信号,即 IM-1(M)=(M11,M12,…,M1t,M21,…,Mtt) 其中 M=M11,[]M12,[]…,[]M1t M21,[]M22,[]…,[]M2t [][][] Mt1,[]Mt2,[]…,[]Mtt∈Rt×t v(i,j)表示获取矩阵v第i行第j列的元素值. JQ(M):Rt×t→R×,表示截取图像矩阵M∈Rt×t右下角M∈R×的子矩阵,即JQ(M)=M(t/2+1)(t/2+1),[]M(t/2+1)(t/2+2),[]…,[]M(t/2+1)t M(t/2+2)(t/2+1),[]M(t/2+2)(t/2+2),[]…,[]M(t/2+2)t [][][] Mt(t/2+1),[]Mt(t/2+2),[]…,[]Mtt∈R(t/2)×(t/2) 其中M=M11,[]M12,[]…,[]M1t M21,[]M22,[]…,[]M2t [][][] Mt1,[]Mt2,[]…,[]Mtt∈Rt×t WT(X):Rm×T→Rm×(T/4) WT(X)=IM-1(JQ(WT(IM(X1)))) IM-1(JQ(WT(IM(X2)))) IM-1(JQ(WT(IM(Xm)))) 其中X∈Rm×T,Xi表示取X的第i行. 证明 由X=AS可知 Xi,j=∑n[]t=1Ai,tSt,j, i=1,2,…,m,j=1,2,…,T(a1) 假设R=WT(X)=IM-1(JQ(WT(IM(X1)))) IM-1(JQ(WT(IM(X2)))) IM-1(JQ(WT(IM(Xm)))),则i=1,2,…,m,有Ri=IM-1(JQ(WT(IM(Xi))))=IM-1(JQ(WT(Xi1,[]Xi2,[]…,[]Xit Xi(t+1),[]Xi(t+2),[]…,[]Xi(t+t) [][][] Xi(T-t+1),[]Xi(T-t+2),[]…,[]XiT)))= IM-1JQ∫t</sup>x1=1∫t</sup>x2=1Xi(x1,x2)φ,dx1dx2= IM-1JQ∫t</sup>x1=1∫t</sup>x2=1Xi(x1,x2)φ,dx1dx2= IM-1JQ∫t</sup>x1=1∫t</sup>x2=1Xi,(x1-1)t+x2φ,dx1dx2(a2) 将式(a1)代入式(a2)得 Ri=IM-1JQ∫t</sup>x1=1∫t</sup>x2=1∑n[]t=1Ai,tSt,(x1-1)t+x2φ,dx1dx2= IM-1JQ∑n[]t=1Ai,t∫t</sup>x1=1∫t</sup>x2=1St,(x1-1)t+x2φ,dx1dx2= IM-1JQ∑n[]t=1Ai,t∫t</sup>x1=1∫t</sup>x2=1St(x1,x2)φ,dx1dx2))= IM-1JQ∑n[]t=1Ai,tWT(IM(St))=∑n[]t=1Ai,tIM-1(JQ(WT(IM(St))))=∑n[]t=1Ai,tWT(St) 于是WT(X)=AWT(S). 证毕. 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