圖像算法中會經常用到攝像機的畸變校正，有必要總結分析OpenCV中畸變校正方法，其中包括普通針孔相機模型和魚眼相機模型fisheye兩種畸變校正方法。
普通相機模型畸變校正函數針對OpenCV中的cv::initUndistortRectifyMap()，魚眼相機模型畸變校正函數對應OpenCV中的cv::fisheye::initUndistortRectifyMap()。兩種方法算出映射Mapx和Mapy后，統一用cv::Remap()函數進行插值得到校正后的圖像。
1. FishEye模型的畸變校正。
方便起見，直接貼出OpenCV源碼，我在里面加了注釋說明。建議參考OpenCV官方文檔看畸變模型原理會更清楚：https://docs.opencv.org/3.0-beta/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html#fisheye-initundistortrectifymap
簡要流程就是：
1. 求內參矩陣的逆，由於攝像機坐標系的三維點到二維圖像平面，需要乘以旋轉矩陣R和內參矩陣K。那么反向投影回去則是二維圖像坐標乘以 K*R的逆矩陣。
2. 將目標圖像中的每一個像素點坐標(j,i)，乘以1中求出的逆矩陣iR，轉換到攝像機坐標系（_x,_y,_w）,並歸一化得到z=1平面下的三維坐標(x,y,1)；
3.求出平面模型下像素點對應魚眼半球模型下的極坐標(r, theta)。
4.利用魚眼畸變模型求出擁有畸變時像素點對應的theta_d。
5.利用求出的theta_d值將三維坐標點重投影到二維圖像平面得到(u,v)，(u,v)即為目標圖像對應的畸變圖像中像素點坐標
6.使用cv::Remap（）函數，根據mapx,mapy取出對應坐標位置的像素值賦值給目標圖像，一般采用雙線性插值法，得到畸變校正后的目標圖像。
#include <opencv2\opencv.hpp> 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)); //從內參矩陣K中取出歸一化焦距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)); } //從畸變系數矩陣D中取出畸變系數k1,k2,k3,k4 Vec4d k = Vec4d::all(0); if (!D.empty()) k = D.depth() == CV_32F ? (Vec4d)*D.getMat().ptr<Vec4f>(): *D.getMat().ptr<Vec4d>(); //旋轉矩陣RR轉換數據類型為CV_64F，如果不需要旋轉，則RR為單位陣 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); //新的內參矩陣PP轉換數據類型為CV_64F cv::Matx33d PP = cv::Matx33d::eye(); if (!P.empty()) P.getMat().colRange(0, 3).convertTo(PP, CV_64F); //關鍵一步：新的內參矩陣*旋轉矩陣，然后利用SVD分解求出逆矩陣iR，后面用到 cv::Matx33d iR = (PP * RR).inv(cv::DECOMP_SVD); //反向映射，遍歷目標圖像所有像素位置，找到畸變圖像中對應位置坐標(u,v)，並分別保存坐標(u,v)到mapx和mapy中 for( int i = 0; i < size.height; ++i) { float* m1f = map1.getMat().ptr<float>(i); float* m2f = map2.getMat().ptr<float>(i); short* m1 = (short*)m1f; ushort* m2 = (ushort*)m2f; //二維圖像平面坐標系->攝像機坐標系 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) { //歸一化攝像機坐標系，相當於假定在Z=1平面上 double x = _x/_w, y = _y/_w; //求魚眼半球體截面半徑r double r = sqrt(x*x + y*y); //求魚眼半球面上一點與光心的連線和光軸的夾角Theta double theta = atan(r); //畸變模型求出theta_d，相當於有畸變的角度值 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); //利用有畸變的Theta值，將攝像機坐標系下的歸一化三維坐標，重投影到二維圖像平面，得到(j,i)對應畸變圖像中的(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)坐標到mapx,mapy if( m1type == CV_16SC2 ) { int iu = cv::saturate_cast<int>(u*cv::INTER_TAB_SIZE); int iv = cv::saturate_cast<int>(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; } //這三條語句是上面 ”//二維圖像平面坐標系->攝像機坐標系“的一部分，是矩陣iR的第一列，這樣寫能夠簡化計算 _x += iR(0, 0); _y += iR(1, 0); _w += iR(2, 0); } } }
2.普通相機模型的畸變校正
同樣建議參考OpenCV官方文檔閱讀代碼 https://docs.opencv.org/3.0-beta/modules/calib3d/doc/camera_calibration_and_3d_reconstruction.html# 。
主要流程和上面Fisheye模型差不多，只有第4部分的畸變模型不一樣，普通相機的畸變模型如下：
同樣把源代碼貼上，並加上注解：
#include <opencv2\opencv.hpp> 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_<double> R = Mat_<double>::eye(3, 3); Mat_<double> A = Mat_<double>(cameraMatrix), Ar; if( !newCameraMatrix.empty() ) Ar = Mat_<double>(newCameraMatrix); else Ar = getDefaultNewCameraMatrix( A, size, true ); if( !matR.empty() ) R = Mat_<double>(matR); if( !distCoeffs.empty() ) distCoeffs = Mat_<double>(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分解求新的內參矩陣Ar與旋轉矩陣R乘積的逆矩陣iR Mat_<double> iR = (Ar.colRange(0,3)*R).inv(DECOMP_LU); const double* ir = &iR(0,0); //從舊的內參矩陣中取出光心位置u0,v0,和歸一化焦距fx,fy double u0 = A(0, 2), v0 = A(1, 2); double fx = A(0, 0), fy = A(1, 1); //尼瑪14個畸變系數，不過大多用到的只有(k1,k2,p1,p2)，最多加一個k3，用不到的置為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>(); 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這個是什么梯形畸變，用不到的話matTilt為單位陣 // 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<float>(i); float* m2f = map2.empty() ? 0 : map2.ptr<float>(i); short* m1 = (short*)m1f; ushort* m2 = (ushort*)m2f; //利用逆矩陣iR將二維圖像坐標(j,i)轉換到攝像機坐標系(_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] ) { //攝像機坐標系歸一化，令Z=1平面 double w = 1./_w, x = _x*w, y = _y*w; //這一部分請看OpenCV官方文檔，畸變模型部分 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); //根據求取的xd,yd將三維坐標重投影到二維畸變圖像坐標(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<int>(u*INTER_TAB_SIZE); int iv = saturate_cast<int>(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; } } } }
如有錯誤，望不吝賜教！
另附上CUDA實現兩種畸變校正方法的代碼，放在我的碼雲上：https://gitee.com/rxdj/camera-calibration.git。見cudaUndistort中的兩個.cu文件