基于准则匹配的图像对准

一、概述

  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$A=\left(a_{ij}\right)$,$B=\left(b_{ij}\right)$,$i=1,2,...,M$,$j=1,2,...,N$.则差向量$$\begin{array}{c}\begin{array}{rl}D& =A-B\\ & =\left(a_{ij}-b_{ij}\right)\end{array}\end{array}$$

(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$$\begin{array}{c}\begin{array}{r}SAD=\sum _{i=1}^M\sum _{j=1}^N\left|a_{ij}-b_{ij}\right|\end{array}\end{array}$$

(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$$\begin{array}{c}\begin{array}{r}MAD=\frac{1}{M\times N}\sum _{i=1}^M\sum _{j=1}^N\left|a_{ij}-b_{ij}\right|\end{array}\end{array}$$

(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$$\begin{array}{c}\begin{array}{r}SSD=\sum _{i=1}^M\sum _{j=1}^N{\left(a_{ij}-b_{ij}\right)}^2\end{array}\end{array}$$

(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$$\begin{array}{c}\begin{array}{r}MSD=\frac{1}{M\times N}\sum _{i=1}^M\sum _{j=1}^N{\left(a_{ij}-b_{ij}\right)}^2\end{array}\end{array}$$

(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$$\begin{array}{c}\begin{array}{r}NCC=\frac{{\sum }_{i=1}^M{\sum }_{j=1}^N\left(a_{ij}-E(A)\right)\left(b_{ij}-E(B)\right)}{\sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(a_{ij}-E(A)\right)}^2}\cdot \sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(b_{ij}-E(B)\right)}^2}}\end{array}\end{array}$$
  易知，ncc值的范围为 $[-1,1]$,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$$\begin{array}{c}\begin{array}{rl}Pearson& =\frac{Cov\left(X,Y\right)}{\sqrt{D\left(X\right)}\cdot \sqrt{D\left(Y\right)}}\\ & =\frac{E\left(X-EX\right)\left(Y-EY\right)}{\sqrt{D\left(X\right)}\cdot \sqrt{D\left(Y\right)}}\end{array}\end{array}$$  将两个矩阵看做两个随机变量代入，有$$\begin{array}{c}\begin{array}{rl}Pearson& =\frac{\frac{1}{M\times N}{\sum }_{i=1}^M{\sum }_{j=1}^N\left(a_{ij}-E(A)\right)\left(b_{ij}-E(B)\right)}{\sqrt{\frac{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(a_{ij}-E(A)\right)}^2}{M\times N}}\cdot \sqrt{\frac{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(b_{ij}-E(B)\right)}^2}{M\times N}}}\\ & =\frac{{\sum }_{i=1}^M{\sum }_{j=1}^N\left(a_{ij}-E(A)\right)\left(b_{ij}-E(B)\right)}{\sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(a_{ij}-E(A)\right)}^2}\sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(b_{ij}-E(B)\right)}^2}}\\ & =NCC\end{array}\end{array}$$

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$$\begin{array}{c}\begin{array}{r}cos\theta =\frac{<X,Y>}{\left|X\right|\cdot \left|Y\right|}\end{array}\end{array}$$  将两个矩阵向量去中心化后代入，有$$\begin{array}{c}\begin{array}{rl}cos\theta & =\frac{<A-E(A),B-E(B)>}{\left|A-E(A)\right|\cdot \left|B-E(B)\right|}\\ & =\frac{{\sum }_{i=1}^M{\sum }_{j=1}^N\left(a_{ij}-E(A)\right)\left(b_{ij}-E(B)\right)}{\sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(a_{ij}-E(A)\right)}^2}\sqrt{{\sum }_{i=1}^M{\sum }_{j=1}^N{\left(b_{ij}-E(B)\right)}^2}}\\ & =NCC\end{array}\end{array}$$

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');

 

  
 

代码和数据打包：
https://download.csdn.net/download/Albert201605/87356041?spm=1001.2014.3001.5503

本文pdf下载：
https://download.csdn.net/download/Albert201605/87362510?spm=1001.2014.3001.5503

End.

参考：
https://www.freesion.com/article/7622115606/

转载：https://blog.csdn.net/Albert201605/article/details/128518872