前面已经利用分解本质矩阵E得到两幅图像的姿态R1,R2,t1,t2.
如何把这四个值组合成P1,得到正确的解,接下来就要分析。
1:利用三维重构求取在摄像机前面的重构点的所占比,得出最高占比的那个P1,就是要求得解。
P1 = Matx34d(R1(0,0), R1(0,1), R1(0,2), t1(0), R1(1,0), R1(1,1), R1(1,2), t1(1), R1(2,0), R1(2,1), R1(2,2), t1(2)); cout << "Testing P1 " << endl << Mat(P1) << endl; vector<CloudPoint> pcloud,pcloud1; vector<KeyPoint> corresp; double reproj_error1 = TriangulatePoints(imgpts1_good, imgpts2_good, K, Kinv, distcoeff, P, P1, pcloud, corresp); double reproj_error2 = TriangulatePoints(imgpts2_good, imgpts1_good, K, Kinv, distcoeff, P1, P, pcloud1, corresp); vector<uchar> tmp_status; //check if pointa are triangulated --in front-- of cameras for all 4 ambiguations if (!TestTriangulation(pcloud,P1,tmp_status) || !TestTriangulation(pcloud1,P,tmp_status) || reproj_error1 > 100.0 || reproj_error2 > 100.0) { P1 = Matx34d(R1(0,0), R1(0,1), R1(0,2), t2(0), R1(1,0), R1(1,1), R1(1,2), t2(1), R1(2,0), R1(2,1), R1(2,2), t2(2)); cout << "Testing P1 "<< endl << Mat(P1) << endl; pcloud.clear(); pcloud1.clear(); corresp.clear(); reproj_error1 = TriangulatePoints(imgpts1_good, imgpts2_good, K, Kinv, distcoeff, P, P1, pcloud, corresp); reproj_error2 = TriangulatePoints(imgpts2_good, imgpts1_good, K, Kinv, distcoeff, P1, P, pcloud1, corresp); if (!TestTriangulation(pcloud,P1,tmp_status) || !TestTriangulation(pcloud1,P,tmp_status) || reproj_error1 > 100.0 || reproj_error2 > 100.0) { if (!CheckCoherentRotation(R2)) { cout << "resulting rotation is not coherent\n"; P1 = 0; return false; } P1 = Matx34d(R2(0,0), R2(0,1), R2(0,2), t1(0), R2(1,0), R2(1,1), R2(1,2), t1(1), R2(2,0), R2(2,1), R2(2,2), t1(2)); cout << "Testing P1 "<< endl << Mat(P1) << endl; pcloud.clear(); pcloud1.clear(); corresp.clear(); reproj_error1 = TriangulatePoints(imgpts1_good, imgpts2_good, K, Kinv, distcoeff, P, P1, pcloud, corresp); reproj_error2 = TriangulatePoints(imgpts2_good, imgpts1_good, K, Kinv, distcoeff, P1, P, pcloud1, corresp); if (!TestTriangulation(pcloud,P1,tmp_status) || !TestTriangulation(pcloud1,P,tmp_status) || reproj_error1 > 100.0 || reproj_error2 > 100.0) { P1 = Matx34d(R2(0,0), R2(0,1), R2(0,2), t2(0), R2(1,0), R2(1,1), R2(1,2), t2(1), R2(2,0), R2(2,1), R2(2,2), t2(2)); cout << "Testing P1 "<< endl << Mat(P1) << endl; pcloud.clear(); pcloud1.clear(); corresp.clear(); reproj_error1 = TriangulatePoints(imgpts1_good, imgpts2_good, K, Kinv, distcoeff, P, P1, pcloud, corresp); reproj_error2 = TriangulatePoints(imgpts2_good, imgpts1_good, K, Kinv, distcoeff, P1, P, pcloud1, corresp); if (!TestTriangulation(pcloud,P1,tmp_status) || !TestTriangulation(pcloud1,P,tmp_status) || reproj_error1 > 100.0 || reproj_error2 > 100.0) { cout << "Shit." << endl; return false; } } } }这里主要用了两个函数TriangulatePoints和TestTriangulation。
四种组合依次进行判断,首先给定一种组合,我们有两个由2D点匹配和P矩阵产生的关键等式:x=PX和x’=P’X,x和x’是匹配的二维点,X是两个相机进行拍照的真实世界三维点。如果我们重写这个等式,我们可以公式化为一个线性方程系统,该系统可以解出X的值,X正是我们所期望寻找的值。假定X=(x,y,z,1)t(一个合理的点的假设,这些点离相机的中心不太近或者不太远)产生一个形式为AX=B的非齐次线性方程系统。
将获得由两个2维点产生的3D点的一个近似。还有一个要注意的事情是,2D点用齐次坐标表示,意味着x和y的值后面追加一个1。我们必须确保这些点在标准化的坐标系中,这意味着它们必须乘以先前的标定矩阵K.我们可能注意到我们简单地利用KP矩阵来替代k矩阵和每一个点相乘(每一点乘以KP)。
通过计算重构所有当前两幅图像的点,计算在摄像机前面的重构点的占比,判断当前设置的P1是否正确。
//Triagulate points double TriangulatePoints(const vector<KeyPoint>& pt_set1, const vector<KeyPoint>& pt_set2, const Mat& K, const Mat& Kinv, const Mat& distcoeff, const Matx34d& P, const Matx34d& P1, vector<CloudPoint>& pointcloud, vector<KeyPoint>& correspImg1Pt) { #ifdef __SFM__DEBUG__ vector<double> depths; #endif // pointcloud.clear(); correspImg1Pt.clear(); Matx44d P1_(P1(0,0),P1(0,1),P1(0,2),P1(0,3), P1(1,0),P1(1,1),P1(1,2),P1(1,3), P1(2,0),P1(2,1),P1(2,2),P1(2,3), 0, 0, 0, 1); Matx44d P1inv(P1_.inv()); cout << "Triangulating..."; double t = getTickCount(); vector<double> reproj_error; unsigned int pts_size = pt_set1.size(); #if 0 //Using OpenCV's triangulation //convert to Point2f vector<Point2f> _pt_set1_pt,_pt_set2_pt; KeyPointsToPoints(pt_set1,_pt_set1_pt); KeyPointsToPoints(pt_set2,_pt_set2_pt); //undistort Mat pt_set1_pt,pt_set2_pt; undistortPoints(_pt_set1_pt, pt_set1_pt, K, distcoeff); undistortPoints(_pt_set2_pt, pt_set2_pt, K, distcoeff); //triangulate Mat pt_set1_pt_2r = pt_set1_pt.reshape(1, 2);//通道数变为1,行数为2,为什么这么干? Mat pt_set2_pt_2r = pt_set2_pt.reshape(1, 2); Mat pt_3d_h(1,pts_size,CV_32FC4); cv::triangulatePoints(P,P1,pt_set1_pt_2r,pt_set2_pt_2r,pt_3d_h); //calculate reprojection vector<Point3f> pt_3d; convertPointsHomogeneous(pt_3d_h.reshape(4, 1), pt_3d); cv::Mat_<double> R = (cv::Mat_<double>(3,3) << P(0,0),P(0,1),P(0,2), P(1,0),P(1,1),P(1,2), P(2,0),P(2,1),P(2,2)); Vec3d rvec; Rodrigues(R ,rvec); Vec3d tvec(P(0,3),P(1,3),P(2,3)); vector<Point2f> reprojected_pt_set1; projectPoints(pt_3d,rvec,tvec,K,distcoeff,reprojected_pt_set1); for (unsigned int i=0; i<pts_size; i++) { CloudPoint cp; cp.pt = pt_3d[i]; pointcloud.push_back(cp); reproj_error.push_back(norm(_pt_set1_pt[i]-reprojected_pt_set1[i])); } #else Mat_<double> KP1 = K * Mat(P1); #pragma omp parallel for num_threads(1) for (int i=0; i<pts_size; i++) {//对每一个关键点,先转化成齐次坐标模式,然后在左边乘以Kinv,得到带标定参数的点。 Point2f kp = pt_set1[i].pt; Point3d u(kp.x,kp.y,1.0);//齐次坐标 Mat_<double> um = Kinv * Mat_<double>(u); u.x = um(0); u.y = um(1); u.z = um(2); Point2f kp1 = pt_set2[i].pt; Point3d u1(kp1.x,kp1.y,1.0); Mat_<double> um1 = Kinv * Mat_<double>(u1); u1.x = um1(0); u1.y = um1(1); u1.z = um1(2); Mat_<double> X = IterativeLinearLSTriangulation(u,P,u1,P1);//进行迭代线性重构 // cout << "3D Point: " << X << endl; // Mat_<double> x = Mat(P1) * X; // cout << "P1 * Point: " << x << endl; // Mat_<double> xPt = (Mat_<double>(3,1) << x(0),x(1),x(2)); // cout << "Point: " << xPt << endl; Mat_<double> xPt_img = KP1 * X; //reproject // cout << "Point * K: " << xPt_img << endl; Point2f xPt_img_(xPt_img(0)/xPt_img(2),xPt_img(1)/xPt_img(2)); #pragma omp critical { double reprj_err = norm(xPt_img_-kp1); reproj_error.push_back(reprj_err); CloudPoint cp; cp.pt = Point3d(X(0),X(1),X(2)); cp.reprojection_error = reprj_err; pointcloud.push_back(cp); correspImg1Pt.push_back(pt_set1[i]); #ifdef __SFM__DEBUG__ depths.push_back(X(2)); #endif } } #endif Scalar mse = mean(reproj_error); t = ((double)getTickCount() - t)/getTickFrequency(); cout << "Done. ("<<pointcloud.size()<<"points, " << t <<"s, mean reproj err = " << mse[0] << ")"<< endl; //show "range image" #ifdef __SFM__DEBUG__ { double minVal,maxVal; minMaxLoc(depths, &minVal, &maxVal); Mat tmp(240,320,CV_8UC3,Scalar(0,0,0)); //cvtColor(img_1_orig, tmp, CV_BGR2HSV); for (unsigned int i=0; i<pointcloud.size(); i++) { double _d = MAX(MIN((pointcloud[i].z-minVal)/(maxVal-minVal),1.0),0.0); circle(tmp, correspImg1Pt[i].pt, 1, Scalar(255 * (1.0-(_d)),255,255), CV_FILLED); } cvtColor(tmp, tmp, CV_HSV2BGR); imshow("Depth Map", tmp); waitKey(0); destroyWindow("Depth Map"); } #endif return mse[0]; }其中IterativeLinearLSTriangulation(u,P,u1,P1);//进行迭代线性重构的函数形式:
Mat_<double> IterativeLinearLSTriangulation(Point3d u, //homogenous image point (u,v,1) Matx34d P, //camera 1 matrix Point3d u1, //homogenous image point in 2nd camera Matx34d P1 //camera 2 matrix ) { double wi = 1, wi1 = 1; Mat_<double> X(4,1); Mat_<double> X_ = LinearLSTriangulation(u,P,u1,P1); X(0) = X_(0); X(1) = X_(1); X(2) = X_(2); X(3) = 1.0; for (int i=0; i<10; i++) { //Hartley suggests 10 iterations at most //recalculate weights double p2x = Mat_<double>(Mat_<double>(P).row(2)*X)(0); double p2x1 = Mat_<double>(Mat_<double>(P1).row(2)*X)(0); //breaking point if(fabsf(wi - p2x) <= EPSILON && fabsf(wi1 - p2x1) <= EPSILON) break; wi = p2x; wi1 = p2x1; //reweight equations and solve Matx43d A((u.x*P(2,0)-P(0,0))/wi, (u.x*P(2,1)-P(0,1))/wi, (u.x*P(2,2)-P(0,2))/wi, (u.y*P(2,0)-P(1,0))/wi, (u.y*P(2,1)-P(1,1))/wi, (u.y*P(2,2)-P(1,2))/wi, (u1.x*P1(2,0)-P1(0,0))/wi1, (u1.x*P1(2,1)-P1(0,1))/wi1, (u1.x*P1(2,2)-P1(0,2))/wi1, (u1.y*P1(2,0)-P1(1,0))/wi1, (u1.y*P1(2,1)-P1(1,1))/wi1, (u1.y*P1(2,2)-P1(1,2))/wi1 ); Mat_<double> B = (Mat_<double>(4,1) << -(u.x*P(2,3) -P(0,3))/wi, -(u.y*P(2,3) -P(1,3))/wi, -(u1.x*P1(2,3) -P1(0,3))/wi1, -(u1.y*P1(2,3) -P1(1,3))/wi1 ); solve(A,B,X_,DECOMP_SVD); X(0) = X_(0); X(1) = X_(1); X(2) = X_(2); X(3) = 1.0; } return X; }在这个函数中,首先通过Mat_ X_ = LinearLSTriangulation(u,P,u1,P1);计算出初始的X点。
Mat_<double> LinearLSTriangulation(Point3d u, //homogenous image point (u,v,1) Matx34d P, //camera 1 matrix Point3d u1, //homogenous image point in 2nd camera Matx34d P1 //camera 2 matrix ) { //build matrix A for homogenous equation system Ax = 0 //assume X = (x,y,z,1), for Linear-LS method //which turns it into a AX = B system, where A is 4x3, X is 3x1 and B is 4x1 // cout << "u " << u <<", u1 " << u1 << endl; // Matx<double,6,4> A; //this is for the AX=0 case, and with linear dependence.. // A(0) = u.x*P(2)-P(0); // A(1) = u.y*P(2)-P(1); // A(2) = u.x*P(1)-u.y*P(0); // A(3) = u1.x*P1(2)-P1(0); // A(4) = u1.y*P1(2)-P1(1); // A(5) = u1.x*P(1)-u1.y*P1(0); // Matx43d A; //not working for some reason... // A(0) = u.x*P(2)-P(0); // A(1) = u.y*P(2)-P(1); // A(2) = u1.x*P1(2)-P1(0); // A(3) = u1.y*P1(2)-P1(1); Matx43d A(u.x*P(2,0)-P(0,0), u.x*P(2,1)-P(0,1), u.x*P(2,2)-P(0,2), u.y*P(2,0)-P(1,0), u.y*P(2,1)-P(1,1), u.y*P(2,2)-P(1,2), u1.x*P1(2,0)-P1(0,0), u1.x*P1(2,1)-P1(0,1), u1.x*P1(2,2)-P1(0,2), u1.y*P1(2,0)-P1(1,0), u1.y*P1(2,1)-P1(1,1), u1.y*P1(2,2)-P1(1,2) ); Matx41d B(-(u.x*P(2,3) -P(0,3)), -(u.y*P(2,3) -P(1,3)), -(u1.x*P1(2,3) -P1(0,3)), -(u1.y*P1(2,3) -P1(1,3))); Mat_<double> X; solve(A,B,X,DECOMP_SVD); return X; }计算出初始的重构三维点后,用P0,P1矩阵的第二行与重构点坐标进行相乘,取行向量的第零个值作为权值。再次计算加权值的三维点重构。
其中solve (Ax=B)用来求X。这样就得到了三维重构点X的三维坐标。
然后再投射在图像上,使用
Mat_<double> xPt_img = KP1 * X;
计算重投影误差:
double reprj_err = norm(xPt_img_-kp1);计算出所有点的重投影误差后,求平均,作为该P1组合下的误差。若是大于100.0,则该P1是无效的。
double reproj_error1 = TriangulatePoints(imgpts1_good, imgpts2_good, K, Kinv, distcoeff, P, P1, pcloud, corresp); reproj_error1 > 100.0下面看判断P1是否有效的另一个条件:
!TestTriangulation(pcloud,P1,tmp_status)
bool TestTriangulation(const vector<CloudPoint>& pcloud, const Matx34d& P, vector<uchar>& status) { vector<Point3d> pcloud_pt3d = CloudPointsToPoints(pcloud);//把CloudPoint转化成Point3d vector<Point3d> pcloud_pt3d_projected(pcloud_pt3d.size()); Matx44d P4x4 = Matx44d::eye(); for(int i=0;i<12;i++) P4x4.val[i] = P.val[i];//复制 perspectiveTransform(pcloud_pt3d, pcloud_pt3d_projected, P4x4);//这里是应用P对三维点做透视变换 status.resize(pcloud.size(),0); for (int i=0; i<pcloud.size(); i++) { status[i] = (pcloud_pt3d_projected[i].z > 0) ? 1 : 0;//判断z坐标是否在摄像机前面 } int count = countNonZero(status);//计数,得到在摄像机前面点的个数 double percentage = ((double)count / (double)pcloud.size());//摄像机前面的点在总点数中的占比,少于75%就代表P1无效 cout << count << "/" << pcloud.size() << " = " << percentage*100.0 << "% are in front of camera" << endl; if(percentage < 0.75) return false; //less than 75% of the points are in front of the camera //check for coplanarity of points检查共平面的点 if(false) //not这里没有执行 { cv::Mat_<double> cldm(pcloud.size(),3); for(unsigned int i=0;i<pcloud.size();i++) { cldm.row(i)(0) = pcloud[i].pt.x; cldm.row(i)(1) = pcloud[i].pt.y; cldm.row(i)(2) = pcloud[i].pt.z; } cv::Mat_<double> mean; cv::PCA pca(cldm,mean,CV_PCA_DATA_AS_ROW); int num_inliers = 0; cv::Vec3d nrm = pca.eigenvectors.row(2); nrm = nrm / norm(nrm); cv::Vec3d x0 = pca.mean; double p_to_plane_thresh = sqrt(pca.eigenvalues.at<double>(2)); for (int i=0; i<pcloud.size(); i++) { Vec3d w = Vec3d(pcloud[i].pt) - x0; double D = fabs(nrm.dot(w)); if(D < p_to_plane_thresh) num_inliers++; } cout << num_inliers << "/" << pcloud.size() << " are coplanar" << endl; if((double)num_inliers / (double)(pcloud.size()) > 0.85) return false; } return true; }
这个函数的主要目的是对三维点用于P1进行透视变换,经过透视变换后的点还在摄像机前面的数量占总点数的比率大于75%表示P1有效。
以上就是判断P1是哪种组合的分析,可能不是很清楚,只是略记一下。想起来再补充。
有个疑问是第一个条件已经基于KP1计算重投影误差了,为什么第二个条件还要进行基于P1的透视变换,判断是否在摄像机前面。这样是不是重复,难道相差一个K标定参数,就有必要了。
我感觉是K很重要,直接决定了重投影投射在图像上的坐标,而没有K的P1透视变换只是把三维点从世界坐标系变换到相机坐标系,这样就可判断是不是在相机前面了,这样才会有必要。
先分析到这吧,以后再分析,光束平差法BA进行多视图重建,以及PCL进行三维显示的内容。话说结果图的三维显示还是很赞。