在图像处理相关的问题中，图像对准是一类典型的问题，也就是要将两幅图严丝合缝地对应起来。通常来讲，两幅图大小不一，一个是模板，一个是母图，也就是要在母图中搜寻定位到与模板图最为接近的区域。
  实现的方式有很多，惯常使用的是基于准则匹配的方法和基于特征匹配的方法。基于准则匹配，就是直接地对图的灰度值矩阵进行计算操作，以特定的准则遍历整个母图，找到与目标图（模板图）最相近的子区域；基于特征匹配，就是先提取出图像特征，再基于特征进行操作。这里对基于准则匹配的图像对准基本方法做简单介绍。

二、匹配准则

  常见的匹配准则有SAD、MAD、SSD、MSD、NCC。前四种是基于两个矩阵的向量差做运算，NCC是计算两个矩阵的相关系数。事实上，矩阵是一个高阶向量（二阶张量），对两个矩阵向量作差，就得到差向量，对差向量做分析运算，便可在一定程度上获得两个矩阵间的差异性信息。
设A=(aij)A=(aij),B=(bij)B=(bij),i=1,2,...,Mi=1,2,...,M,j=1,2,...,Nj=1,2,...,N.则差向量
D=A−B=(aij−bij)D=A−B=(aij−bij)

(1) SAD

  SAD，绝对误差算法（Sum of Absolute Differences），它是差向量D中各元素的绝对值之和，也就是L1范数，是两个向量间的曼哈顿距离。表达式为
SAD=∑i=1M∑j=1N|aij−bij|SAD=∑i=1M∑j=1N|aij−bij|

(2) MAD

  MAD，平均绝对误差算法（Mean Absolute Differences），它是在SAD基础上进一步求平均值。表达式为
MAD=1M×N∑i=1M∑j=1N|aij−bij|MAD=1M×N∑i=1M∑j=1N|aij−bij|

(3) SSD

  SSD，误差平方和算法（Sum of Squared Differences）,它是差向量D中各元素的平方和。表达式为
SSD=∑i=1M∑j=1N(aij−bij)2SSD=∑i=1M∑j=1N(aij−bij)2

(4) MSD

  MSD，平均误差平方和算法（Mean Square Differences），它是在SSD的基础上进一步求平均值。表达式为
MSD=1M×N∑i=1M∑j=1N(aij−bij)2MSD=1M×N∑i=1M∑j=1N(aij−bij)2

(5) NCC

  NCC，归一化互相关算法（Normalized Cross Correlation）。若将两个矩阵看做两个随机变量，那么NCC就是两个变量之间的皮尔逊相关系数。同时，它也是两个矩阵向量在各自中心化之后彼此间空间夹角的余弦值。它的表达式为
NCC=∑Mi=1∑Nj=1(aij−E(A))(bij−E(B))∑Mi=1∑Nj=1(aij−E(A))2−−−−−−−−−−−−−−−−−−−−√⋅∑Mi=1∑Nj=1(bij−E(B))2−−−−−−−−−−−−−−−−−−−−√NCC=∑i=1M∑j=1N(aij−E(A))(bij−E(B))∑i=1M∑j=1N(aij−E(A))2⋅∑i=1M∑j=1N(bij−E(B))2
  易知，ncc值的范围为 [−1,1][−1,1],越接近1，两个矩阵越相关；越接近-1，两个矩阵越不相关。


等同于皮尔逊相关系数
  皮尔逊相关系数，用以衡量两个变量间的线性相关性。它的表达式为
Pearson=Cov(X,Y)D(X)−−−−−√⋅D(Y)−−−−−√=E(X−EX)(Y−EY)D(X)−−−−−√⋅D(Y)−−−−−√Pearson=Cov(X,Y)D(X)⋅D(Y)=E(X−EX)(Y−EY)D(X)⋅D(Y)
  将两个矩阵看做两个随机变量代入，有
Pearson=1M×N∑Mi=1∑Nj=1(aij−E(A))(bij−E(B))∑Mi=1∑Nj=1(aij−E(A))2M×N−−−−−−−−−−−−−−−√⋅∑Mi=1∑Nj=1(bij−E(B))2M×N−−−−−−−−−−−−−−−√=∑Mi=1∑Nj=1(aij−E(A))(bij−E(B))∑Mi=1∑Nj=1(aij−E(A))2−−−−−−−−−−−−−−−−−−−−√∑Mi=1∑Nj=1(bij−E(B))2−−−−−−−−−−−−−−−−−−−−√=NCCPearson=1M×N∑i=1M∑j=1N(aij−E(A))(bij−E(B))∑i=1M∑j=1N(aij−E(A))2M×N⋅∑i=1M∑j=1N(bij−E(B))2M×N=∑i=1M∑j=1N(aij−E(A))(bij−E(B))∑i=1M∑j=1N(aij−E(A))2∑i=1M∑j=1N(bij−E(B))2=NCC

等同于余弦距离
  余弦距离即空间向量夹角的余弦值，通常用以衡量两个向量间的差异度。它的表达式为
cosθ=<X,Y>|X|⋅|Y|cosθ=<X,Y>|X|⋅|Y|
  将两个矩阵向量去中心化后代入，有
cosθ=<A−E(A),B−E(B)>|A−E(A)|⋅|B−E(B)|=∑Mi=1∑Nj=1(aij−E(A))(bij−E(B))∑Mi=1∑Nj=1(aij−E(A))2−−−−−−−−−−−−−−−−−−−−√∑Mi=1∑Nj=1(bij−E(B))2−−−−−−−−−−−−−−−−−−−−√=NCCcosθ=<A−E(A),B−E(B)>|A−E(A)|⋅|B−E(B)|=∑i=1M∑j=1N(aij−E(A))(bij−E(B))∑i=1M∑j=1N(aij−E(A))2∑i=1M∑j=1N(bij−E(B))2=NCC

三、matlab实现

(1) SAD

clear all;
close all; clc;

%1.读取图片
img_A_dir = '.\data\lena.bmp';  %待寻母图
img_A_raw = imread(img_A_dir);
[r1,c1,d1] = size(img_A_raw);
if d1==3 %灰度化
    img_A = rgb2gray(img_A_raw);
else
    img_A = img_A_raw;
end

img_B_dir = '.\data\refer.bmp';  %模板图
img_B_raw = imread(img_B_dir);
[r2,c2,d2] = size(img_B_raw);
if d2==3
    img_B = rgb2gray(img_B_raw);
else
    img_B = img_B_raw;
end

%2.计算SAD矩阵
msad = zeros(r1-r2,c1-c2);

for i = 1:r1-r2
    for j = 1:c1-c2
        temp = img_A(i:i+r2-1,j:j+c2-1);        
        msad(i,j) = msad(i,j) + sum(sum(abs(temp - img_B)));    
    end
end

%3.定位匹配位置
min_sad = min(min(msad));
[x,y] = find(msad == min_sad);
x = x(1); %定位到的第一个位置
y = y(1);

%4.保存结果图
getImg = img_A_raw(x:x+r2-1,y:y+c2-1,1:3);
imwrite(getImg,'.\output\SAD_match.bmp');

fprintf('\n Done. \n');
在这里插入代码片

(2) MAD

clear all;
close all; clc;

%1.读取图片
img_A_dir = '.\data\lena.bmp';  %待寻母图
img_A_raw = imread(img_A_dir);
[r1,c1,d1] = size(img_A_raw);
if d1==3 %灰度化
    img_A = rgb2gray(img_A_raw);
else
    img_A = img_A_raw;
end

img_B_dir = '.\data\refer.bmp';  %模板图
img_B_raw = imread(img_B_dir);
[r2,c2,d2] = size(img_B_raw);
if d2==3
    img_B = rgb2gray(img_B_raw);
else
    img_B = img_B_raw;
end

%2.计算MAD矩阵
mmad = zeros(r1-r2,c1-c2);

for i = 1:r1-r2
    for j = 1:c1-c2
        temp = img_A(i:i+r2-1,j:j+c2-1);        
        mmad(i,j) = mmad(i,j) + sum(sum(abs(temp - img_B)))/(r2*c2);    
    end
end

%3.定位匹配位置
min_mad = min(min(mmad));
[x,y] = find(mmad == min_mad);
x = x(1); %定位到的第一个位置
y = y(1);

%4.保存结果图
getImg = img_A_raw(x:x+r2-1,y:y+c2-1,1:3);
imwrite(getImg,'.\output\MAD_match.bmp');

fprintf('\n Done. \n');

(3) SSD

clear all;
close all; clc;

%1.读取图片
img_A_dir = '.\data\lena.bmp';  %待寻母图
img_A_raw = imread(img_A_dir);
[r1,c1,d1] = size(img_A_raw);
if d1==3 %灰度化
    img_A = rgb2gray(img_A_raw);
else
    img_A = img_A_raw;
end

img_B_dir = '.\data\refer.bmp';  %模板图
img_B_raw = imread(img_B_dir);
[r2,c2,d2] = size(img_B_raw);
if d2==3
    img_B = rgb2gray(img_B_raw);
else
    img_B = img_B_raw;
end

%2.计算SSD矩阵
mssd = zeros(r1-r2,c1-c2);

for i = 1:r1-r2
    for j = 1:c1-c2
        temp = img_A(i:i+r2-1,j:j+c2-1);        
        mssd(i,j) = mssd(i,j) + sum(sum((temp - img_B).^2));    
    end
end

%3.定位匹配位置
min_ssd = min(min(mssd));
[x,y] = find(mssd == min_ssd);
x = x(1); %定位到的第一个位置
y = y(1);

%4.保存结果图
getImg = img_A_raw(x:x+r2-1,y:y+c2-1,1:3);
imwrite(getImg,'.\output\SSD_match.bmp');

fprintf('\n Done. \n');

(4) MSD

clear all;
close all; clc;

%1.读取图片
img_A_dir = '.\data\lena.bmp';  %待寻母图
img_A_raw = imread(img_A_dir);
[r1,c1,d1] = size(img_A_raw);
if d1==3 %灰度化
    img_A = rgb2gray(img_A_raw);
else
    img_A = img_A_raw;
end

img_B_dir = '.\data\refer.bmp';  %模板图
img_B_raw = imread(img_B_dir);
[r2,c2,d2] = size(img_B_raw);
if d2==3
    img_B = rgb2gray(img_B_raw);
else
    img_B = img_B_raw;
end

%2.计算MSD矩阵
mmsd = zeros(r1-r2,c1-c2);

for i = 1:r1-r2
    for j = 1:c1-c2
        temp = img_A(i:i+r2-1,j:j+c2-1);        
        mmsd(i,j) = mmsd(i,j) + sum(sum((temp - img_B).^2))/(r2*c2);    
    end
end

%3.定位匹配位置
min_msd = min(min(mmsd));
[x,y] = find(mmsd == min_msd);
x = x(1); %定位到的第一个位置
y = y(1);

%4.保存结果图
getImg = img_A_raw(x:x+r2-1,y:y+c2-1,1:3);
imwrite(getImg,'.\output\MSD_match.bmp');

fprintf('\n Done. \n');

(5) NCC

clear all;
close all; clc;

%1.读取图片
img_A_dir = '.\data\lena.bmp';  %待寻母图
img_A_raw = imread(img_A_dir);
[r1,c1,d1] = size(img_A_raw);
if d1==3 %灰度化
    img_A = rgb2gray(img_A_raw);
else
    img_A = img_A_raw;
end

img_B_dir = '.\data\refer.bmp';  %模板图
img_B_raw = imread(img_B_dir);
[r2,c2,d2] = size(img_B_raw);
if d2==3
    img_B = rgb2gray(img_B_raw);
else
    img_B = img_B_raw;
end

%2.计算NCC矩阵
mNCC = zeros(r1-r2,c1-c2);

for i = 1:r1-r2
    for j = 1:c1-c2
        
        temp = img_A(i:i+r2-1,j:j+c2-1);   
        
        mean_temp = mean(temp(:)); %temp均值
        mean_B = mean(img_B(:));  %img_B均值      
        
        inp = sum(sum((temp - mean_temp).*(img_B - mean_B))); %两向量内积        
        mod1 = sqrt(sum(sum((temp - mean_temp).^2))); %模长1
        mod2 = sqrt(sum(sum((img_B - mean_B).^2))); %模长2        
        ncc = inp / (mod1*mod2);       
        
        mNCC(i,j) = mNCC(i,j) + ncc;                             
    end
end


%3.定位匹配位置
max_ncc = max(max(mNCC)); %最大ncc值
[x,y] = find(mNCC == max_ncc);
x = x(1); %定位到的第一个位置
y = y(1);

%4.保存结果图
getImg = img_A_raw(x:x+r2-1,y:y+c2-1,1:3);
imwrite(getImg,'.\output\NCC_match.bmp');

fprintf('\n Done. \n');

End.