opencv仿射变换：平移，缩放和旋转

今天药忘吃喽~ 2022-12-29 11:20 133阅读 0赞

**目录**

仿射变换原理

opencv中仿射变换实现

使用getAffineTransform()函数求仿射变换矩阵

使用 getRotationMatrix2D()函数获取仿射矩阵

使用仿射矩阵对图像做仿射变换warpAffine

--------------------

# 仿射变换原理 #

仿射变换（Affine Transformation或 Affine Map），又称仿射映射，是指在几何中，一个向量空间进行一次线性变换并接上一个平移，变换为另一个向量空间的过程。它保持了二维图形的“平直性”（即：直线经过变换之后依然是直线）和“平行性”（即：二维图形之间的相对位置关系保持不变，平行线依然是平行线，且直线上点的位置顺序不变）。

一个任意的仿射变换都能表示为乘以一个矩阵(线性变换)接着再加上一个向量(平移)的形式。可以表示为线性变换和平移变换的叠加

我们能够用仿射变换来表示如下三种常见的变换形式：

*  旋转，rotation (线性变换)
 *  平移，translation(向量加)
 *  缩放，scale(线性变换)

如果进行更深层次的理解，仿射变换代表的是两幅图之间的一种映射关系。

仿射变换的数学表示是先**乘以一个线形变换**矩阵再**加上一个平移向量**，其中线性变换矩阵为2×2的矩阵，平移向量为2×1的向量

假设我们存在一个线性变换矩阵 A 和平移矩阵B ，两者与输入的M矩阵之间的关系如式

![20201222113135537.png][]

根据旋转矩阵 A和平移矩阵B 以及图像像素值![\\left \[ x\\, y \\right \]^\{T\}][left _ x_ y _right _T]，仿射变换的数学原理可以用式

![20201222113604109.png][]

仿射变换又称为三点变换，如果知道变换前后两张图像中三个像素点坐标的对应关系，就可以求得仿射变换中的变换矩阵 .

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg_size_16_color_FFFFFF_t_70][]

点1, 2 和 3 (在图一中形成一个三角形) 与图二中三个点一一映射, 仍然形成三角形, 但形状已经大大改变. 如果我们能通过这样两组三点求出仿射变换 (你能选择自己喜欢的点), 接下来我们就能把仿射变换应用到图像中所有的点.。

而我们通常使用2 x 3的矩阵来表示仿射变换。

![ea5dbb9aa7648380b3983987106944ae.png][]

# opencv中仿射变换实现 #

要求出图像的仿射变换（旋转缩放等），首先要求出**仿射变换矩阵。**

## 使用getAffineTransform()函数求仿射变换矩阵 ##

上文提到过，已知**变换前后两张图像中三个像素点坐标的对应关系，就可以求得仿射变换中的变换矩阵。**OpenCV 4提供了利用三个对应像素点来确定 矩阵的函数getAffineTransform()

函数原型：

CV_EXPORTS Mat getAffineTransform( const Point2f src[], const Point2f dst[] );

> Calculates an affine transform from three pairs of the corresponding points.  
> The function calculates the ![\\f$2 \\times 3\\f$][f_2 _times 3_f] matrix of an affine transform so that:
> 
> # ![\\large \\f\[\\begin\{bmatrix\} x'\_i \\\\ y'\_i \\end\{bmatrix\} = \\texttt\{map\_matrix\} \\cdot \\begin\{bmatrix\} x\_i \\\\ y\_i \\\\ 1 \\end\{bmatrix\}\\f\]][large _f_begin_bmatrix_ x_i _ y_i _end_bmatrix_ _ _texttt_map_matrix_ _cdot _begin_bmatrix_ x_i _ y_i _ 1 _end_bmatrix_f] #
> 
> where
> 
> # ![\\large \\f\[dst(i)=(x'\_i,y'\_i), src(i)=(x\_i, y\_i), i=0,1,2\\f\]][large _f_dst_i_x_i_y_i_ src_i_x_i_ y_i_ i_0_1_2_f] #
> 
> @param src Coordinates of triangle vertices in the source image.  
> @param dst Coordinates of the corresponding triangle vertices in the destination image.
> 
>  *  src\[\]：原图像中的三个像素坐标。
>  *  dst\[\]：目标图像中的三个像素坐标。

## 使用 getRotationMatrix2D()函数获取仿射矩阵 ##

函数原型：

CV_EXPORTS_W Mat getRotationMatrix2D(Point2f center, double angle, double scale);
    
    /** @sa getRotationMatrix2D */
    CV_EXPORTS Matx23d getRotationMatrix2D_(Point2f center, double angle, double scale);
    
    inline
    Mat getRotationMatrix2D(Point2f center, double angle, double scale)
    {
        return Mat(getRotationMatrix2D_(center, angle, scale), true);
    }

> @brief Calculates an affine matrix of 2D rotation.
> 
> The function calculates the following matrix:以此得出仿射矩阵：
> 
> # ![\\f\[\\begin\{bmatrix\} \\alpha & \\beta & (1- \\alpha ) \\cdot \\texttt\{center.x\} - \\beta \\cdot \\texttt\{center.y\} \\\\ - \\beta & \\alpha & \\beta \\cdot \\texttt\{center.x\} + (1- \\alpha ) \\cdot \\texttt\{center.y\} \\end\{bmatrix\}\\f\]][f_begin_bmatrix_ _alpha _ _beta _ _1- _alpha _ _cdot _texttt_center.x_ - _beta _cdot _texttt_center.y_ _ - _beta _ _alpha _ _beta _cdot _texttt_center.x_ _ _1- _alpha _ _cdot _texttt_center.y_ _end_bmatrix_f] #
> 
> where
> 
> ![\\begin\{array\}\{l\} \\alpha = \\texttt\{scale\} \\cdot \\cos \\texttt\{angle\} , \\\\ \\beta = \\texttt\{scale\} \\cdot \\sin \\texttt\{angle\} \\end\{array\}][begin_array_l_ _alpha _ _texttt_scale_ _cdot _cos _texttt_angle_ _ _ _beta _ _texttt_scale_ _cdot _sin _texttt_angle_ _end_array]
> 
> The transformation maps the rotation center to itself. If this is not the target, adjust the shift.
> 
> @param center Center of the rotation in the source image.  
> @param angle Rotation angle in degrees. Positive values mean counter-clockwise rotation (the  
> coordinate origin is assumed to be the top-left corner).  
> @param scale Isotropic scale factor.
> 
>  *  第一个参数，Point2f类型的center，表示源图像的旋转中心。
>  *  第二个参数，double类型的angle，旋转角度。角度为正值表示向逆时针旋转（坐标原点是左上角）。
>  *  第三个参数，double类型的scale，缩放系数。

getRotationMatrix2D函数源代码：

cv::Mat cv::getRotationMatrix2D( Point2f center,double angle, double scale )
    {
       angle *= CV_PI/180;
       double alpha = cos(angle)*scale;
       double beta = sin(angle)*scale;
     
       Mat M(2, 3, CV_64F);//生成一个2*3的矩阵
       double* m = (double*)M.data;
     
        //按照上文公式填充内容
       m[0] = alpha;
       m[1] = beta;
       m[2] = (1-alpha)*center.x - beta*center.y;
       m[3] = -beta;
       m[4] = alpha;
       m[5] = beta*center.x + (1-alpha)*center.y;
     
       return M;
    }

## 使用仿射矩阵对图像做仿射变换warpAffine ##

warpAffine函数原型：

CV_EXPORTS_W void warpAffine( InputArray src, OutputArray dst,
                                  InputArray M, Size dsize,
                                  int flags = INTER_LINEAR,
                                  int borderMode = BORDER_CONSTANT,
                                  const Scalar& borderValue = Scalar());

> @brief Applies an affine transformation to an image.
> 
> The function warpAffine transforms the source image using the specified matrix:依据如下式子，对图像做仿射变换
> 
> ![\\f\[\\texttt\{dst\} (x,y) = \\texttt\{src\} ( \\texttt\{M\} \_\{11\} x + \\texttt\{M\} \_\{12\} y + \\texttt\{M\} \_\{13\}, \\texttt\{M\} \_\{21\} x + \\texttt\{M\} \_\{22\} y + \\texttt\{M\} \_\{23\})\\f\]][f_texttt_dst_ _x_y_ _ _texttt_src_ _ _texttt_M_ _11_ x _ _texttt_M_ _12_ y _ _texttt_M_ _13_ _texttt_M_ _21_ x _ _texttt_M_ _22_ y _ _texttt_M_ _23_f]
> 
> when the flag \#WARP\_INVERSE\_MAP is set. Otherwise, the transformation is first inverted  
> with \#invertAffineTransform and then put in the formula above instead of M. The function cannot  
> operate in-place.
> 
> @param src input image.  
> @param dst output image that has the size dsize and the same type as src .  
> @param M ![\\f$2\\times 3\\f$][f_2_times 3_f] transformation matrix.  
> @param dsize size of the output image.  
> @param flags combination of interpolation methods (see \#InterpolationFlags) and the optional  
> flag \#WARP\_INVERSE\_MAP that means that M is the inverse transformation (  
> ![\\f$\\texttt\{dst\}\\rightarrow\\texttt\{src\}\\f$][f_texttt_dst_rightarrow_texttt_src_f]).  
> @param borderMode pixel extrapolation method (see \#BorderTypes); when  
> borderMode=\#BORDER\_TRANSPARENT, it means that the pixels in the destination image corresponding to  
> the "outliers" in the source image are not modified by the function.  
> @param borderValue value used in case of a constant border; by default, it is 0.
> 
>  *  第一个参数，InputArray类型的src，输入图像，即源图像，填Mat类的对象即可。
>  *  第二个参数，OutputArray类型的dst，函数调用后的运算结果存在这里，需和源图片有一样的尺寸和类型。
>  *  第三个参数，InputArray类型的M，2×3的变换矩阵。
>  *  第四个参数，Size类型的dsize，表示输出图像的尺寸。
>  *  第五个参数，int类型的flags，插值方法的标识符。此参数有默认值INTER\_LINEAR(线性插值)，可选的插值方式如下：
> 
>  *   *  INTER\_NEAREST - 最近邻插值
>      *  INTER\_LINEAR - 线性插值（默认值）
>      *  INTER\_AREA - 区域插值
>      *  INTER\_CUBIC –三次样条插值
>      *  INTER\_LANCZOS4 -Lanczos插值
>      *  CV\_WARP\_FILL\_OUTLIERS - 填充所有输出图像的象素。如果部分象素落在输入图像的边界外，那么它们的值设定为 fillval.
>      *  CV\_WARP\_INVERSE\_MAP –表示M为输出图像到输入图像的反变换，即 。因此可以直接用来做象素插值。否则, warpAffine函数从M矩阵得到反变换。
> 
>  *  第六个参数，int类型的borderMode，边界像素模式，默认值为BORDER\_CONSTANT。
>  *  第七个参数，const Scalar&类型的borderValue，在恒定的边界情况下取的值，默认值为Scalar()，即0。

使用两种方式进行仿射变换：

#include <opencv2\opencv.hpp>
    #include <iostream>
    #include <vector>
    
    using namespace std;
    using namespace cv;
    
    int main()
    {
    	Mat img = imread("5.jpg");
    	if (img.empty())
    	{
    		cout << "请确认图像文件名称是否正确" << endl;
    		return -1;
    	}
    
    	Mat rotation0, rotation1, img_warp0, img_warp1;
    	double angle = 30;  //设置图像旋转的角度
    	Size dst_size(img.rows, img.cols);  //设置输出图像的尺寸
    	Point2f center(img.rows / 2.0, img.cols / 2.0);  //设置图像的旋转中心
    	rotation0 = getRotationMatrix2D(center, angle, 1);  //计算放射变换矩阵
    	warpAffine(img, img_warp0, rotation0, dst_size);  //进行仿射变换
    	imshow("img_warp0", img_warp0);
    	//根据定义的三个点进行仿射变换
    	Point2f src_points[3];
    	Point2f dst_points[3];
    	src_points[0] = Point2f(0, 0);  //原始图像中的三个点
    	src_points[1] = Point2f(0, (float)(img.cols - 1));
    	src_points[2] = Point2f((float)(img.rows - 1), (float)(img.cols - 1));
    	//放射变换后图像中的三个点
    	dst_points[0] = Point2f((float)(img.rows)*0.11, (float)(img.cols)*0.20);
    	dst_points[1] = Point2f((float)(img.rows)*0.15, (float)(img.cols)*0.70);
    	dst_points[2] = Point2f((float)(img.rows)*0.81, (float)(img.cols)*0.85);
    	rotation1 = getAffineTransform(src_points, dst_points);  //根据对应点求取仿射变换矩阵
    	warpAffine(img, img_warp1, rotation1, dst_size);  //进行仿射变换
    	imshow("img_warp1", img_warp1);
    	waitKey(0);
    	return 0;
    }

实验效果：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg_size_16_color_FFFFFF_t_70 1][]

参考文章：

[【从零学习OpenCV 4】图像仿射变换][OpenCV 4]

[【OpenCV入门教程之十八】OpenCV仿射变换 & SURF特征点描述合辑][OpenCV_OpenCV_ _ SURF]

[20201222113135537.png]: /images/20221120/a2502bf0358345a49b860cd84af784a0.png
[left _ x_ y _right _T]: https://private.codecogs.com/gif.latex?%5Cleft%20%5B%20x%5C%2C%20y%20%5Cright%20%5D%5E%7BT%7D
[20201222113604109.png]: https://img-blog.csdnimg.cn/20201222113604109.png
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg_size_16_color_FFFFFF_t_70]: https://img-blog.csdnimg.cn/20201222142901170.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg=,size_16,color_FFFFFF,t_70
[ea5dbb9aa7648380b3983987106944ae.png]: https://img-blog.csdnimg.cn/img_convert/ea5dbb9aa7648380b3983987106944ae.png
[f_2 _times 3_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cf%242%20%5Ctimes%203%5Cf%24
[large _f_begin_bmatrix_ x_i _ y_i _end_bmatrix_ _ _texttt_map_matrix_ _cdot _begin_bmatrix_ x_i _ y_i _ 1 _end_bmatrix_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Clarge%20%5Cf%5B%5Cbegin%7Bbmatrix%7D%20x%27_i%20%5C%5C%20y%27_i%20%5Cend%7Bbmatrix%7D%20%3D%20%5Ctexttt%7Bmap_matrix%7D%20%5Ccdot%20%5Cbegin%7Bbmatrix%7D%20x_i%20%5C%5C%20y_i%20%5C%5C%201%20%5Cend%7Bbmatrix%7D%5Cf%5D
[large _f_dst_i_x_i_y_i_ src_i_x_i_ y_i_ i_0_1_2_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Clarge%20%5Cf%5Bdst%28i%29%3D%28x%27_i%2Cy%27_i%29%2C%20src%28i%29%3D%28x_i%2C%20y_i%29%2C%20i%3D0%2C1%2C2%5Cf%5D
[f_begin_bmatrix_ _alpha _ _beta _ _1- _alpha _ _cdot _texttt_center.x_ - _beta _cdot _texttt_center.y_ _ - _beta _ _alpha _ _beta _cdot _texttt_center.x_ _ _1- _alpha _ _cdot _texttt_center.y_ _end_bmatrix_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cf%5B%5Cbegin%7Bbmatrix%7D%20%5Calpha%20%26%20%5Cbeta%20%26%20%281-%20%5Calpha%20%29%20%5Ccdot%20%5Ctexttt%7Bcenter.x%7D%20-%20%5Cbeta%20%5Ccdot%20%5Ctexttt%7Bcenter.y%7D%20%5C%5C%20-%20%5Cbeta%20%26%20%5Calpha%20%26%20%5Cbeta%20%5Ccdot%20%5Ctexttt%7Bcenter.x%7D%20&plus;%20%281-%20%5Calpha%20%29%20%5Ccdot%20%5Ctexttt%7Bcenter.y%7D%20%5Cend%7Bbmatrix%7D%5Cf%5D
[begin_array_l_ _alpha _ _texttt_scale_ _cdot _cos _texttt_angle_ _ _ _beta _ _texttt_scale_ _cdot _sin _texttt_angle_ _end_array]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cbegin%7Barray%7D%7Bl%7D%20%5Calpha%20%3D%20%5Ctexttt%7Bscale%7D%20%5Ccdot%20%5Ccos%20%5Ctexttt%7Bangle%7D%20%2C%20%5C%5C%20%5Cbeta%20%3D%20%5Ctexttt%7Bscale%7D%20%5Ccdot%20%5Csin%20%5Ctexttt%7Bangle%7D%20%5Cend%7Barray%7D
[f_texttt_dst_ _x_y_ _ _texttt_src_ _ _texttt_M_ _11_ x _ _texttt_M_ _12_ y _ _texttt_M_ _13_ _texttt_M_ _21_ x _ _texttt_M_ _22_ y _ _texttt_M_ _23_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cf%5B%5Ctexttt%7Bdst%7D%20%28x%2Cy%29%20%3D%20%5Ctexttt%7Bsrc%7D%20%28%20%5Ctexttt%7BM%7D%20_%7B11%7D%20x%20&plus;%20%5Ctexttt%7BM%7D%20_%7B12%7D%20y%20&plus;%20%5Ctexttt%7BM%7D%20_%7B13%7D%2C%20%5Ctexttt%7BM%7D%20_%7B21%7D%20x%20&plus;%20%5Ctexttt%7BM%7D%20_%7B22%7D%20y%20&plus;%20%5Ctexttt%7BM%7D%20_%7B23%7D%29%5Cf%5D
[f_2_times 3_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cf%242%5Ctimes%203%5Cf%24
[f_texttt_dst_rightarrow_texttt_src_f]: https://private.codecogs.com/gif.latex?%5Cdpi%7B200%7D%20%5Cf%24%5Ctexttt%7Bdst%7D%5Crightarrow%5Ctexttt%7Bsrc%7D%5Cf%24
[watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg_size_16_color_FFFFFF_t_70 1]: https://img-blog.csdnimg.cn/20201222145053336.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L2x1b2xhaWh1YTIwMTg=,size_16,color_FFFFFF,t_70
[OpenCV 4]: https://zhuanlan.zhihu.com/p/101292646
[OpenCV_OpenCV_ _ SURF]: https://blog.csdn.net/poem_qianmo/article/details/33320997