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图像算法中会经常用到摄像机的畸变校正，有必要总结分析OpenCV中畸变校正方法，其中包括普通针孔相机模型和鱼眼相机模型fisheye两种畸变校正方法。

普通相机模型畸变校正函数针对OpenCV中的cv::initUndistortRectifyMap()，鱼眼相机模型畸变校正函数对应OpenCV中的cv::fisheye::initUndistortRectifyMap()。两种方法算出映射Mapx和Mapy后，统一用cv::Remap()函数进行插值得到校正后的图像。

1. FishEye模型的畸变校正。

方便起见，直接贴出OpenCV源码，我在里面加了注释说明。建议参考OpenCV官方文档看畸变模型原理会更清楚：

简要流程就是：

1. 求内参矩阵的逆，由于摄像机坐标系的三维点到二维图像平面，需要乘以旋转矩阵R和内参矩阵K。那么反向投影回去则是二维图像坐标乘以  K*R的逆矩阵。

2. 将目标图像中的每一个像素点坐标(j,i)，乘以1中求出的逆矩阵iR，转换到摄像机坐标系（_x,_y,_w）,并归一化得到z=1平面下的三维坐标(x,y,1)；

3.求出平面模型下像素点对应鱼眼半球模型下的极坐标(r, theta)。

4.利用鱼眼畸变模型求出拥有畸变时像素点对应的theta_d。

5.利用求出的theta_d值将三维坐标点重投影到二维图像平面得到(u,v)，(u,v)即为目标图像对应的畸变图像中像素点坐标

6.使用cv::Remap（）函数，根据mapx,mapy取出对应坐标位置的像素值赋值给目标图像，一般采用双线性插值法，得到畸变校正后的目标图像。

#include 
   
    void cv::fisheye::initUndistortRectifyMap( InputArray K, InputArray D, InputArray R, InputArray P,    const cv::Size& size, int m1type, OutputArray map1, OutputArray map2 ){    CV_Assert( m1type == CV_16SC2 || m1type == CV_32F || m1type <=0 );    map1.create( size, m1type <= 0 ? CV_16SC2 : m1type );    map2.create( size, map1.type() == CV_16SC2 ? CV_16UC1 : CV_32F );    CV_Assert((K.depth() == CV_32F || K.depth() == CV_64F) && (D.depth() == CV_32F || D.depth() == CV_64F));    CV_Assert((P.empty() || P.depth() == CV_32F || P.depth() == CV_64F) && (R.empty() || R.depth() == CV_32F || R.depth() == CV_64F));    CV_Assert(K.size() == Size(3, 3) && (D.empty() || D.total() == 4));    CV_Assert(R.empty() || R.size() == Size(3, 3) || R.total() * R.channels() == 3);    CV_Assert(P.empty() || P.size() == Size(3, 3) || P.size() == Size(4, 3));    //从内参矩阵K中取出归一化焦距fx,fy; cx,cy    cv::Vec2d f, c;    if (K.depth() == CV_32F)    {        Matx33f camMat = K.getMat();        f = Vec2f(camMat(0, 0), camMat(1, 1));        c = Vec2f(camMat(0, 2), camMat(1, 2));    }    else    {        Matx33d camMat = K.getMat();        f = Vec2d(camMat(0, 0), camMat(1, 1));        c = Vec2d(camMat(0, 2), camMat(1, 2));    }    //从畸变系数矩阵D中取出畸变系数k1,k2,k3,k4    Vec4d k = Vec4d::all(0);    if (!D.empty())        k = D.depth() == CV_32F ? (Vec4d)*D.getMat().ptr
    
     (): *D.getMat().ptr
     
      ();    //旋转矩阵RR转换数据类型为CV_64F，如果不需要旋转，则RR为单位阵    cv::Matx33d RR  = cv::Matx33d::eye();    if (!R.empty() && R.total() * R.channels() == 3)    {        cv::Vec3d rvec;        R.getMat().convertTo(rvec, CV_64F);        RR = Affine3d(rvec).rotation();    }    else if (!R.empty() && R.size() == Size(3, 3))        R.getMat().convertTo(RR, CV_64F);        //新的内参矩阵PP转换数据类型为CV_64F    cv::Matx33d PP = cv::Matx33d::eye();    if (!P.empty())        P.getMat().colRange(0, 3).convertTo(PP, CV_64F);    //关键一步：新的内参矩阵*旋转矩阵，然后利用SVD分解求出逆矩阵iR，后面用到    cv::Matx33d iR = (PP * RR).inv(cv::DECOMP_SVD);    //反向映射，遍历目标图像所有像素位置，找到畸变图像中对应位置坐标(u,v)，并分别保存坐标(u,v)到mapx和mapy中    for( int i = 0; i < size.height; ++i)    {        float* m1f = map1.getMat().ptr
      
       (i);        float* m2f = map2.getMat().ptr
       
        (i);        short*  m1 = (short*)m1f;        ushort* m2 = (ushort*)m2f;        //二维图像平面坐标系->摄像机坐标系        double _x = i*iR(0, 1) + iR(0, 2),               _y = i*iR(1, 1) + iR(1, 2),               _w = i*iR(2, 1) + iR(2, 2);        for( int j = 0; j < size.width; ++j)        {            //归一化摄像机坐标系，相当于假定在Z=1平面上            double x = _x/_w, y = _y/_w;            //求鱼眼半球体截面半径r            double r = sqrt(x*x + y*y);            //求鱼眼半球面上一点与光心的连线和光轴的夹角Theta            double theta = atan(r);            //畸变模型求出theta_d，相当于有畸变的角度值            double theta2 = theta*theta, theta4 = theta2*theta2, theta6 = theta4*theta2, theta8 = theta4*theta4;            double theta_d = theta * (1 + k[0]*theta2 + k[1]*theta4 + k[2]*theta6 + k[3]*theta8);            //利用有畸变的Theta值，将摄像机坐标系下的归一化三维坐标，重投影到二维图像平面，得到(j,i)对应畸变图像中的(u,v)            double scale = (r == 0) ? 1.0 : theta_d / r;            double u = f[0]*x*scale + c[0];            double v = f[1]*y*scale + c[1];            //保存(u,v)坐标到mapx,mapy            if( m1type == CV_16SC2 )            {                int iu = cv::saturate_cast
        
         (u*cv::INTER_TAB_SIZE); int iv = cv::saturate_cast
         
          (v*cv::INTER_TAB_SIZE); m1[j*2+0] = (short)(iu >> cv::INTER_BITS); m1[j*2+1] = (short)(iv >> cv::INTER_BITS); m2[j] = (ushort)((iv & (cv::INTER_TAB_SIZE-1))*cv::INTER_TAB_SIZE + (iu & (cv::INTER_TAB_SIZE-1))); } else if( m1type == CV_32FC1 ) { m1f[j] = (float)u; m2f[j] = (float)v; } //这三条语句是上面 ”//二维图像平面坐标系->摄像机坐标系“的一部分，是矩阵iR的第一列，这样写能够简化计算 _x += iR(0, 0); _y += iR(1, 0); _w += iR(2, 0); } }}
         
        
       
      
     
    
   

2.普通相机模型的畸变校正

 同样建议参考OpenCV官方文档阅读代码 。

主要流程和上面Fisheye模型差不多，只有第4部分的畸变模型不一样，普通相机的畸变模型如下：

 同样把源代码贴上，并加上注解：

#include 
   
    void cv::initUndistortRectifyMap( InputArray _cameraMatrix, InputArray _distCoeffs,                              InputArray _matR, InputArray _newCameraMatrix,                              Size size, int m1type, OutputArray _map1, OutputArray _map2 ){    Mat cameraMatrix = _cameraMatrix.getMat(), distCoeffs = _distCoeffs.getMat();    Mat matR = _matR.getMat(), newCameraMatrix = _newCameraMatrix.getMat();    if( m1type <= 0 )        m1type = CV_16SC2;    CV_Assert( m1type == CV_16SC2 || m1type == CV_32FC1 || m1type == CV_32FC2 );    _map1.create( size, m1type );    Mat map1 = _map1.getMat(), map2;    if( m1type != CV_32FC2 )    {        _map2.create( size, m1type == CV_16SC2 ? CV_16UC1 : CV_32FC1 );        map2 = _map2.getMat();    }    else        _map2.release();    Mat_
    
      R = Mat_
     
      ::eye(3, 3);    Mat_
      
        A = Mat_
       
        (cameraMatrix), Ar;    if( !newCameraMatrix.empty() )        Ar = Mat_
        
         (newCameraMatrix); else Ar = getDefaultNewCameraMatrix( A, size, true ); if( !matR.empty() ) R = Mat_
         
          (matR); if( !distCoeffs.empty() ) distCoeffs = Mat_
          
           (distCoeffs); else { distCoeffs.create(14, 1, CV_64F); distCoeffs = 0.; } CV_Assert( A.size() == Size(3,3) && A.size() == R.size() ); CV_Assert( Ar.size() == Size(3,3) || Ar.size() == Size(4, 3)); //LU分解求新的内参矩阵Ar与旋转矩阵R乘积的逆矩阵iR Mat_
           
             iR = (Ar.colRange(0,3)*R).inv(DECOMP_LU); const double* ir = &iR(0,0); //从旧的内参矩阵中取出光心位置u0,v0,和归一化焦距fx,fy double u0 = A(0, 2), v0 = A(1, 2); double fx = A(0, 0), fy = A(1, 1); //尼玛14个畸变系数，不过大多用到的只有(k1,k2,p1,p2)，最多加一个k3，用不到的置为0 CV_Assert( distCoeffs.size() == Size(1, 4) || distCoeffs.size() == Size(4, 1) || distCoeffs.size() == Size(1, 5) || distCoeffs.size() == Size(5, 1) || distCoeffs.size() == Size(1, 8) || distCoeffs.size() == Size(8, 1) || distCoeffs.size() == Size(1, 12) || distCoeffs.size() == Size(12, 1) || distCoeffs.size() == Size(1, 14) || distCoeffs.size() == Size(14, 1)); if( distCoeffs.rows != 1 && !distCoeffs.isContinuous() ) distCoeffs = distCoeffs.t(); const double* const distPtr = distCoeffs.ptr
            
             (); double k1 = distPtr[0]; double k2 = distPtr[1]; double p1 = distPtr[2]; double p2 = distPtr[3]; double k3 = distCoeffs.cols + distCoeffs.rows - 1 >= 5 ? distPtr[4] : 0.; double k4 = distCoeffs.cols + distCoeffs.rows - 1 >= 8 ? distPtr[5] : 0.; double k5 = distCoeffs.cols + distCoeffs.rows - 1 >= 8 ? distPtr[6] : 0.; double k6 = distCoeffs.cols + distCoeffs.rows - 1 >= 8 ? distPtr[7] : 0.; double s1 = distCoeffs.cols + distCoeffs.rows - 1 >= 12 ? distPtr[8] : 0.; double s2 = distCoeffs.cols + distCoeffs.rows - 1 >= 12 ? distPtr[9] : 0.; double s3 = distCoeffs.cols + distCoeffs.rows - 1 >= 12 ? distPtr[10] : 0.; double s4 = distCoeffs.cols + distCoeffs.rows - 1 >= 12 ? distPtr[11] : 0.; double tauX = distCoeffs.cols + distCoeffs.rows - 1 >= 14 ? distPtr[12] : 0.; double tauY = distCoeffs.cols + distCoeffs.rows - 1 >= 14 ? distPtr[13] : 0.; //tauX,tauY这个是什么梯形畸变，用不到的话matTilt为单位阵 // Matrix for trapezoidal distortion of tilted image sensor cv::Matx33d matTilt = cv::Matx33d::eye(); cv::detail::computeTiltProjectionMatrix(tauX, tauY, &matTilt); for( int i = 0; i < size.height; i++ ) { float* m1f = map1.ptr
             
              (i); float* m2f = map2.empty() ? 0 : map2.ptr
              
               (i); short* m1 = (short*)m1f; ushort* m2 = (ushort*)m2f; //利用逆矩阵iR将二维图像坐标(j,i)转换到摄像机坐标系(_x,_y,_w) double _x = i*ir[1] + ir[2], _y = i*ir[4] + ir[5], _w = i*ir[7] + ir[8]; for( int j = 0; j < size.width; j++, _x += ir[0], _y += ir[3], _w += ir[6] ) { //摄像机坐标系归一化，令Z=1平面 double w = 1./_w, x = _x*w, y = _y*w; //这一部分请看OpenCV官方文档，畸变模型部分 double x2 = x*x, y2 = y*y; double r2 = x2 + y2, _2xy = 2*x*y; double kr = (1 + ((k3*r2 + k2)*r2 + k1)*r2)/(1 + ((k6*r2 + k5)*r2 + k4)*r2); double xd = (x*kr + p1*_2xy + p2*(r2 + 2*x2) + s1*r2+s2*r2*r2); double yd = (y*kr + p1*(r2 + 2*y2) + p2*_2xy + s3*r2+s4*r2*r2); //根据求取的xd,yd将三维坐标重投影到二维畸变图像坐标(u,v) cv::Vec3d vecTilt = matTilt*cv::Vec3d(xd, yd, 1); double invProj = vecTilt(2) ? 1./vecTilt(2) : 1; double u = fx*invProj*vecTilt(0) + u0; double v = fy*invProj*vecTilt(1) + v0; //保存u,v的值到Mapx,Mapy中 if( m1type == CV_16SC2 ) { int iu = saturate_cast
               
                (u*INTER_TAB_SIZE); int iv = saturate_cast
                
                 (v*INTER_TAB_SIZE); m1[j*2] = (short)(iu >> INTER_BITS); m1[j*2+1] = (short)(iv >> INTER_BITS); m2[j] = (ushort)((iv & (INTER_TAB_SIZE-1))*INTER_TAB_SIZE + (iu & (INTER_TAB_SIZE-1))); } else if( m1type == CV_32FC1 ) { m1f[j] = (float)u; m2f[j] = (float)v; } else { m1f[j*2] = (float)u; m1f[j*2+1] = (float)v; } } }}
                
               
              
             
            
           
          
         
        
       
      
     
    
   

 如有错误，望不吝赐教！

另附上CUDA实现两种畸变校正方法的代码，放在我的码云上：。见cudaUndistort中的两个.cu文件

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