[数字图像处理]简单的几何学图像变换与图像配准

1.图像的几何学变换

之前的博文里，我简单的介绍了图像的放大与缩小。放大与缩小也算是图像的几何学变换，本文介绍了其他的几何学变换，包括旋转、水平倾斜和垂直倾斜（当然，还有水平移动与垂直移动。这些变换很简单，不需要插值，所以这里就不着重介绍了）。

假设输入图像为G(u,v)，其变换后的图像为F(x,y)。其变化的方法，如下所示。

图像的几何学变换，主要有两种向前映射与向后映射。

1.1向前映射

所谓向前映射，就是从输入图像为g(u,v)的(0,0)点开始，将g(u,v)遍历一遍，依次计算g(u,v)变换后的坐标。当然，计算出来的坐标不会是整数（很大程度上不会是整数），就像下图所示那样。

借由上图，我们重新理解一些向前映射。假设图像的一部分

这5个点，经过变换，我们得到了右边

的5个点。其实，这个5个点经过变换之后，计算出来的坐标并不是整数。假设现在遍历到

这个点（图中以蓝色表示），通过计算得到的点是

。我们将

的坐标进行四舍五入，将

的值赋值给

（这种处理相当于选择最近的点，属于最初级的插值方法）。

使用上述的向前映射的思想，将图像旋转，我们能得到如下效果。

可以看出来，所得到的结果非常“斑驳”。其原因是，我们将g(u,v)进行变换，计算之后，所得到的坐标四舍五入之后，有的点没有被赋值，而有点被赋值了多次。这就产生了“空穴”，所以让变换后的图像非常斑驳。

我看了一些文章，基本到这里都会说，“由于向前映射会产生空穴，所以向前映射一般不使用”。诸如此类的话，其实这样理解不太对，产生空穴的根本原因是由于插值法的不恰当所产生的。使用向前映射，也可以不产生“空穴”，所使用的方法如下所示。

我们可以通过最接近

的四个点，去确定

的值。这样的话，就可以不产生空穴。但是，这个方法实现起来很困难，所以，向前映射才一般不被使用。

1.2 向后映射

向后映射，就是将输出图像f(x,y)遍历一遍，然后计算输出(x,y)的时候，所需要g(x,y)的坐标。当然，这个数不一定是整数。为了方便理解，还是看下图。

假设，我们遍历到

，通过计算，我们得到

这样一个点。也就意味着，如果需要确定

的灰度值，那么，我们需要

点处的灰度值。这样，几何变换的问题又归结到了插值问题。最方便的办法就是选择最近的点，入上图的例子，

的灰度值等于

的灰度值。当然，还有更加“高级”点的方法，精度也还算可以。

的灰度值的确定方法如下所示。

如上图，我们使用四个点（

），去确定

的值。首先

与

之间使用线性插值，确定出

的值。同样的方法，确定出

的值。

与

之间，进行线性插值，就可以得到

的值了。（图画的非常清楚了）

使用向后映射，将图像旋转，我们可以得到如下结果。

1.3 上述两部分的Matlab代码

 
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             close 
             all
             ;
            

             clear 
             all
             ;
            

             clc
             ;
            


            

             %% -----------Geometric_spatial_transform------------------
            

             f 
             = 
             imread
             (
             './letter_T.tif'
             );
            

             f 
             = 
             mat2gray
             (
             f
             ,[
             0 
             255
             ]);
            


            

             [
             M
             ,
             N
             ] 
             = 
             size
             (
             f
             );
            


            

             %-----by forward mapping-----%
            

             g_fm 
             = 
             zeros
             (
             M
             ,
             N
             );
            

             seta 
             = 
             -
             pi
             /
             8
             ;
            


            

             for 
             v 
             = 
             (
             -
             M
             /
             2
             ):
             1
             :(
             M
             /
             2
             )
             -
             1
            
    
             for 
             w 
             = 
             (
             -
             N
             /
             2
             ):
             1
             :(
             N
             /
             2
             )
             -
             1
            
        
             x 
             = 
             round
             (
             v
             *
             cos
             (
             seta
             ) 
             - 
             w
             *
             sin
             (
             seta
             ));
            
        
             y 
             = 
             round
             (
             v
             *
             sin
             (
             seta
             ) 
             + 
             w
             *
             cos
             (
             seta
             ));
            
        
             if 
             (((
             y
             >
             =(
             -
             N
             /
             2
             ))
             &&
             (
             y
             <
             =(
             N
             /
             2
             )))
             &&
             ((
             x
             >
             =(
             -
             M
             /
             2
             ))
             &&
             (
             x
             <
             =(
             M
             /
             2
             ))))
            
            
             g_fm
             (
             x
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             y
             +
             (
             N
             /
             2
             )
             +
             1
             ) 
             = 
             f
             (
             v
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             w
             +
             (
             N
             /
             2
             )
             +
             1
             );
            
        
             end
            
    
             end
            

             end
            


            

             figure
             ();
            

             subplot
             (
             1
             ,
             2
             ,
             1
             );
            

             imshow
             (
             f
             ,[
             0 
             1
             ]);
            

             xlabel
             (
             'a).Original Image'
             );
            


            

             subplot
             (
             1
             ,
             2
             ,
             2
             );
            

             imshow
             (
             g_fm
             ,[
             0 
             1
             ]);
            

             xlabel
             (
             'b).Ruselt of Geometric spatial transform'
             );
            


            

             %-----by inverse mapping----
            

             g_im 
             = 
             zeros
             (
             M
             ,
             N
             );
            

             seta 
             = 
             -
             pi
             /
             8
             ;
            


            

             for 
             x 
             = 
             (
             -
             M
             /
             2
             ):
             1
             :(
             M
             /
             2
             )
             -
             1
            
    
             for 
             y 
             = 
             (
             -
             N
             /
             2
             ):
             1
             :(
             N
             /
             2
             )
             -
             1
            
        
             v 
             = 
             x
             *
             cos
             (
             -
             seta
             ) 
             - 
             y
             *
             sin
             (
             -
             seta
             );
            
        
             w 
             = 
             x
             *
             sin
             (
             -
             seta
             ) 
             + 
             y
             *
             cos
             (
             -
             seta
             );
            
        
             if 
             (((
             w
             >
             =(
             -
             N
             /
             2
             )
             +
             1
             )
             &&
             (
             w
             <
             =(
             N
             /
             2
             )
             -
             1
             ))
             &&
             ((
             v
             >
             =(
             -
             M
             /
             2
             )
             +
             1
             )
             &&
             (
             v
             <
             =(
             M
             /
             2
             )
             -
             1
             )))
            
            
             %g_im(x+(M/2)+1,y+(N/2)+1) = 1;%f(round(v+(M/2)+1),round(w+(N/2)+1));
            
            
             Q_11 
             = 
             f
             (
             floor
             (
             v
             )
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             floor
             (
             w
             )
             +
             (
             N
             /
             2
             )
             +
             1
             );
            
            
             Q_21 
             =  
             f
             (
             floor
             (
             v
             )
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             ceil
             (
             w
             )
             +
             (
             N
             /
             2
             )
             +
             1
             );
            
            
             Q_12 
             =  
             f
             (
             ceil
             (
             v
             )
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             floor
             (
             w
             )
             +
             (
             N
             /
             2
             )
             +
             1
             );
            
            
             Q_22 
             =   
             f
             (
             ceil
             (
             v
             )
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             ceil
             (
             w
             )
             +
             (
             N
             /
             2
             )
             +
             1
             );
            
            
            
            
             R1 
             = 
             (
             Q_21 
             - 
             Q_11
             )
             *
             (
             w
             -
             floor
             (
             w
             )) 
             + 
             Q_11
             ;
            
            
             R2 
             = 
             (
             Q_22 
             - 
             Q_12
             )
             *
             (
             w
             -
             floor
             (
             w
             )) 
             + 
             Q_12
             ;
            


            
            
             g_im
             (
             x
             +
             (
             M
             /
             2
             )
             +
             1
             ,
             y
             +
             (
             N
             /
             2
             )
             +
             1
             ) 
             = 
             (
             R2
             -
             R1
             )
             *
             (
             v
             -
             floor
             (
             v
             )) 
             + 
             R1
             ;
            
        
             end
            
    
             end
            

             end
            


            

             figure
             ();
            

             subplot
             (
             1
             ,
             2
             ,
             1
             );
            

             imshow
             (
             f
             ,[
             0 
             1
             ]);
            

             xlabel
             (
             'a).Original Image'
             );
            


            

             subplot
             (
             1
             ,
             2
             ,
             2
             );
            

             imshow
             (
             g_im
             ,[
             0 
             1
             ]);
            

             xlabel
             (
             'b).Ruselt of Geometric spatial transform'
             );
            
 
          
 
     

来自CODE的代码片
Geometric_spatial_transform.m

2.图像的配准

图像的配准，常常用于超解析领域。当然了，作为基础学习，博主没学那么深。这次的图像配准，试图去还原被倾斜变换的图像。首先，先将图像进行两个方向的倾斜。

我们这次的目的是，将图像b).还原为a).。这次变换相对简单，我们使用如下模型去拟合变换关系。

我们在原图与变换后的图像上，选择出四个标准点，然后带入方程，并求出系数

。使用这个方程，将图像b).还原为a)。这个过程比较简单，下面是结果。另外，所选择的标准点，我已经在上图中用红圈标定出来了。

可以看到，还原的效果非常的好（额，其实这也是应为畸变比较简单的缘故）。为了看出于原图的区别，我们做出了差分图像。可以看出来，还原不是完美的。

至此，上个代码作为本文的结尾。额，由于特征点的选择是手工的，这个代码可能没有多少意义。（不要吐槽最后这一句话。一般的，在使用特征点做图象配准的时候，一般选择在某个实际的物体上放入某个实际的标志，然后通过检测这个实际的标志，去进行变换。【这里可以参考《Digital Image Processing》 Rafael C. Gonzalez / Richard E. Woods的第二章的相关内容，这里有叙述的！！】）

 
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             close all;
            
             clear all;
            
             clc;
            
             %% --------------image registration---------------------------
            
             f_Original = imread('./characters_test_pattern.tif');
            
             f_Original = mat2gray(f_Original,[0 255]);
            
             [M,N] = size(f_Original);
            
             f = zeros(M+4,N+4);
            
             for x = 1:M
            
                 f(x+2,:) = [0 0 f_Original(x,:) 0 0];
            
             end
            
             [M,N] = size(f);
            
             s_v = 0.3;
            
             s_w = 0.02;
            
             P = round(M+s_v*N);
            
             Q = round(N+s_w*M);
            
             g_1 = zeros(P,N);
            
             for x = (-P/2):1:(P/2)-1
            
                 for y = (-N/2):1:(N/2)-1
            
                     v = x - s_v * y;
            
                     w = y;
            
                     if (((w>=(-N/2))&&(w<=(N/2)-1))&&((v>=(-M/2))&&(v<=(M/2)-1)))
            
                         Q_11 = f(floor(v)+(M/2)+1,floor(w)+(N/2)+1);
            
                         Q_21 =  f(floor(v)+(M/2)+1,ceil(w)+(N/2)+1);
            
                         Q_12 =  f(ceil(v)+(M/2)+1,floor(w)+(N/2)+1);
            
                         Q_22 =   f(ceil(v)+(M/2)+1,ceil(w)+(N/2)+1);
            
                         R1 = (Q_21 - Q_11)*(w-floor(w)) + Q_11;
            
                         R2 = (Q_22 - Q_12)*(w-floor(w)) + Q_12;
            
                         g_1(x+(P/2)+1,y+(N/2)+1) = (R2-R1)*(v-floor(v)) + R1;
            
                     end
            
                 end
            
             end
            
             g = zeros(P,Q);
            
             for x = (-P/2):1:(P/2)-1
            
                 for y = (-Q/2):1:(Q/2)-1
            
                     v = x;
            
                     w = y - s_w * x;
            
                     if (((w>=(-N/2))&&(w<=(N/2)-1))&&((v>=(-P/2))&&(v<=(P/2)-1)))
            
                         Q_11 = g_1(floor(v)+(P/2)+1,floor(w)+(N/2)+1);
            
                         Q_21 =  g_1(floor(v)+(P/2)+1,ceil(w)+(N/2)+1);
            
                         Q_12 =  g_1(ceil(v)+(P/2)+1,floor(w)+(N/2)+1);
            
                         Q_22 =   g_1(ceil(v)+(P/2)+1,ceil(w)+(N/2)+1);
            
                         R1 = (Q_21 - Q_11)*(w-floor(w)) + Q_11;
            
                         R2 = (Q_22 - Q_12)*(w-floor(w)) + Q_12;
            
                         g(x+(P/2)+1,y+(Q/2)+1) = (R2-R1)*(v-floor(v)) + R1;
            
                     end
            
                 end
            
             end
            
             figure();
            
             subplot(1,2,1);
            
             imshow(f,[0 1]);
            
             xlabel('a).Original Image');
            
             hold on;
            
             plot(621, 78,'ro','MarkerSize',7);
            
             plot(116,113,'ro','MarkerSize',7);
            
             plot( 85,649,'ro','MarkerSize',7);
            
             plot(624,641,'ro','MarkerSize',7);
            
             subplot(1,2,2);
            
             imshow(g,[0 1]);
            
             xlabel('b).Ruselt of Geometric spatial transform');
            
             hold on;
            
             plot(625,264,'ro','MarkerSize',7);
            
             plot(116,147,'ro','MarkerSize',7);
            
             plot( 96,674,'ro','MarkerSize',7);
            
             plot(638,828,'ro','MarkerSize',7);
            
             %% 
            
             f_reg = zeros(M,N);
            
             Original = [621-(N/2)  78-(M/2)  (621-(N/2))*(78-(M/2)) 1;
            
                         116-(N/2) 113-(M/2) (116-(N/2))*(113-(M/2)) 1;
            
                          85-(N/2) 649-(M/2)  (85-(N/2))*(649-(M/2)) 1;
            
                         624-(N/2) 641-(M/2) (624-(N/2))*(641-(M/2)) 1];
            
             Ruselt = [625-(Q/2);116-(Q/2);96-(Q/2);638-(Q/2)];
            
             c = (Original)\Ruselt;  %C1~C4
            
             Ruselt = [264-(P/2);147-(P/2);674-(P/2);828-(P/2)];
            
             c = [c (Original)\Ruselt];  %C5~C8 wwwwww
            
             for y = (-M/2):1:(M/2)-1
            
                 for x = (-N/2):1:(N/2)-1
            
                     w = c(1,1)*y + c(2,1)*x + c(3,1)*x*y + c(4,1);
            
                     v = c(1,2)*y + c(2,2)*x + c(3,2)*x*y + c(4,2);
            
                     if (((w>=(-Q/2))&&(w<=(Q/2)-1))&&((v>=(-P/2))&&(v<=(P/2)-1)))
            
                         Q_11 = g(floor(v)+(P/2)+1,floor(w)+(Q/2)+1);
            
                         Q_21 =  g(floor(v)+(P/2)+1,ceil(w)+(Q/2)+1);
            
                         Q_12 =  g(ceil(v)+(P/2)+1,floor(w)+(Q/2)+1);
            
                         Q_22 =   g(ceil(v)+(P/2)+1,ceil(w)+(Q/2)+1);
            
                         R1 = (Q_21 - Q_11)*(w-floor(w)) + Q_11;
            
                         R2 = (Q_22 - Q_12)*(w-floor(w)) + Q_12;
            
                         f_reg(x+(N/2)+1,y+(M/2)+1) = (R2-R1)*(v-floor(v)) + R1;
            
                         %g(round(v+(P/2)+1),round(w+(Q/2)+1)) = 0.5;
            
                     end
            
                 end
            
             end
            
             g_diff = abs(f - f_reg);
            
             figure();
            
             subplot(1,2,1);
            
             imshow(f_reg,[0 1]);
            
             xlabel('c).Ruselt of image registration');
            
             subplot(1,2,2);
            
             imshow(g_diff,[0 1]);
            
             xlabel('d).Difference image');