小波变换与小波包变换
人脸图像f(x,y) 的一层小波变换如下图所示：
图中L 和H 分别表示低通滤波器和高通滤波器，l(n) 和h(n) 分别表示它们相应的脉冲响应，2↓1表示降2采样fLL和fHH分别表示双向低频和高频小波子空间图像，而fLH和fHL分别表示一个方向低频、另一个方向高频的小波子空间图像。
若继续对fLL 做小波分解，则称为二层小波变换；若对4个子图都继续做小波分解，则称为二层小波包变换。对只含有低频成分的子图做小波分解，则称为不完全小波包变换。
图像小波变换的Matlab代码
clc;clear;close all;Xa =[40,10,56,46]; % 56*2=112,46*2-92(图片分辨率为112*92)Xh =[40,10,56,46];Xv =[40,10,56,46];Xd =[40,10,56,46];for i = 1:40 % 40个人脸 for j = 1:10 % 1人10个表情 I = imread(strcat('D:\Faces\ORL\s',num2str(i),'\',num2str(j),'.bmp')); [c1,s1]=wavedec2(double(I),1,'db1'); % 分别做4个方向的小波分解 ca = appcoef2(c1,s1,'db1',1); ch = detcoef2('h',c1,s1,1); cv = detcoef2('v',c1,s1,1); cd = detcoef2('d',c1,s1,1); % A = uint8([wcodemat(ca,255),wcodemat(ch,255);wcodemat(cv,255),wcodemat(cd,255)]); % subplot(2,5,j),imshow(A,[],'InitialMagnification',100); for k = 1:56 for l = 1:46 Xa(i,j,k,l) = ca(k,l); Xh(i,j,k,l) = ch(k,l); Xv(i,j,k,l) = cv(k,l); Xd(i,j,k,l) = cd(k,l); end end end endfor i = 1:1 % 查看第1个人的10个表情图的小波分解效果 for j = 1:10 xa = [56,46]; xh = [56,46]; xv = [56,46]; xd = [56,46]; for k = 1:56 for l = 1:46 xa(k,l) = Xa(i,j,k,l); xh(k,l) = Xh(i,j,k,l); xv(k,l) = Xv(i,j,k,l); xd(k,l) = Xd(i,j,k,l); end end A = uint8([wcodemat(xa,255),wcodemat(xh,255);wcodemat(xv,255),wcodemat(xd,255)]); subplot(2,5,j),imshow(A,[],'InitialMagnification',100); endend 效果图
后记
本例仅给出图像1层小波变换，可以根据实际项目需求做n层图像小波分解，用于提取人脸图像特征。例如：
[c,s] = wavedec2(double(ca),1,'db1');a = appcoef2(c,s,'db1',1);% 构造基本分类器D1for m = 1:28 for n = 1:23 D1(i,j,m,n) = a(m,n); endend h = detcoef2('h',c,s,1);v = detcoef2('v',c,s,1);d = detcoef2('d',c,s,1); 具体可以参考论文《基于模糊积分的不完全小波包子空间集成人脸识别 翟俊海 王熙照 张素芳》。
Reference
帖中代码：
clc;clear;close allI = imread('lena.bmp');%subplot(221);subplot(221),imshow(I,[],'InitialMagnification',100);title('原始载体图像')%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%图像的一级小波分解 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[c1,s1]=wavedec2(double(I),1,'db1');ca88 = appcoef2(c1,s1,'db1',1);ch88 = detcoef2('h',c1,s1,1);cv88 = detcoef2('v',c1,s1,1);cd88 = detcoef2('d',c1,s1,1);A88 = uint8([wcodemat(ca88,255),wcodemat(ch88,255);wcodemat(cv88,255),wcodemat(cd88,255)]);%subplot(222); subplot(222),imshow(A88,[],'InitialMagnification',100);title('图像的一级小波分解');%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%图像的二级小波分解 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[c,s]=wavedec2(double(I),2,'db1');ca2 = appcoef2(c,s,'db1',2);ch2 = detcoef2('h',c,s,2);cv2 = detcoef2('v',c,s,2);cd2 = detcoef2('d',c,s,2);%ca1 = appcoef2(c,s,'db2',1);ch1 = detcoef2('h',c,s,1);cv1 = detcoef2('v',c,s,1);cd1 = detcoef2('d',c,s,1);A1 = [ca2*4,ch2*4;cv2*4,cd2*4];k=s(2,1)*2-s(3,1);ch1 = padarray(ch1,[k k],1,'post');cv1 = padarray(cv1,[k k],1,'post');cd1 = padarray(cd1,[k k],1,'post');%A = [A1,ch1*4; cv1*4,cd1*4];A = uint8([wcodemat(A1,255),wcodemat(ch1*10,255); wcodemat(cv1*10,255),wcodemat(cd1*10,255)]);% min = min(A(:));% max = max(A(:));%subplot(223); subplot(223),imshow(A,[],'InitialMagnification',100);title('图像的二级小波分解');%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%图像的三级小波分解 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%[c,s]=wavedec2(double(I),3,'db1');ca3 = appcoef2(c,s,'db1',3);ch3 = detcoef2('h',c,s,3);cv3 = detcoef2('v',c,s,3);cd3 = detcoef2('d',c,s,3);ch2 = detcoef2('h',c,s,2);cv2 = detcoef2('v',c,s,2);cd2 = detcoef2('d',c,s,2);ch1 = detcoef2('h',c,s,1);cv1 = detcoef2('v',c,s,1);cd1 = detcoef2('d',c,s,1);A2 = [ca3*4,ch3*4;cv3*4,cd3*4];k=s(2,1)*2-s(3,1);ch2 = padarray(ch2,[k k],1,'post');cv2 = padarray(cv2,[k k],1,'post');cd2 = padarray(cd2,[k k],1,'post');A1= [A2,ch2*4; cv2*4,cd2*4];k=s(2,1)*4-s(4,1);ch1 = padarray(ch1,[k k],1,'post');cv1 = padarray(cv1,[k k],1,'post');cd1 = padarray(cd1,[k k],1,'post');AA = uint8([wcodemat(A1,255),wcodemat(ch1*4,255); wcodemat(cv1*4,255),wcodemat(cd1*4,255)]);% min = min(A(:));% max = max(A(:));%subplot(224);subplot(224),imshow(AA,[],'InitialMagnification',100);title('图像的三级小波分解'); 效果图：
