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

(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
\(\begin{equation} \begin{aligned}SAD=\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left| a_{ij}-b_{ij} \right|}}\end{aligned} \end{equation}\)

(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
\(\begin{equation} \begin{aligned}MAD=\frac{1}{M\times N}\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left| a_{ij}-b_{ij} \right|}}\end{aligned} \end{equation}\)

(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
\(\begin{equation} \begin{aligned}SSD=\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-b_{ij} \right)^{2}}}\end{aligned} \end{equation}\)

(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
\(\begin{equation} \begin{aligned}MSD=\frac{1}{M\times N}\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-b_{ij} \right)^{2}}}\end{aligned} \end{equation}\)

(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
\(\begin{equation} \begin{aligned}NCC=\frac{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)\left( b_{ij}-E(B) \right)}}}{\sqrt{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)^{2}}}}\cdot\sqrt{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( b_{ij}-E(B) \right)^{2}}}}}\end{aligned} \end{equation}\)
  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
\(\begin{equation} \begin{aligned} Pearson&=\frac{Cov\left( X,Y \right)}{\sqrt{D\left( X \right)}\cdot\sqrt{D\left( Y \right)}}\\&=\frac{E\left( X-EX \right)\left( Y-EY \right)}{\sqrt{D\left( X \right)}\cdot\sqrt{D\left( Y \right)}} \end{aligned} \end{equation}\)
  将两个矩阵看做两个随机变量代入,有
\(\begin{equation} \begin{aligned} Pearson&=\frac{\frac{1}{M\times N}\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)\left( b_{ij}-E(B) \right)}}}{\sqrt{\frac{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)^{2}}}}{M\times N}}\cdot\sqrt{\frac{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( b_{ij}-E(B) \right)^{2}}}}{M\times N}}}\\&=\frac{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)\left( b_{ij}-E(B) \right)}}}{\sqrt{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( a_{ij}-E(A) \right)^{2}}}} \sqrt{\sum_{i=1}^{M}{\sum_{j=1}^{N}{\left( b_{ij}-E(B) \right)^{2}}}}}\\&=NCC \end{aligned} \end{equation}\)

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

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.
posted @ 2023-01-02 12:20  归去_来兮  阅读(201)  评论(0编辑  收藏  举报