Levenberg-Marquardt algorithm explained
Levenberg-Marquardt algorithm explained Read More »
Linear Algebra
Eigen unaryExpr (Function Pointer, Lambda Expression) Example
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double ramp(double x) { if (x > 0) return x; else return 0; } void unaryExprExample() { //unaryExpr is a const function so it will not give you the possibility to modify values in place Eigen::ArrayXd x = Eigen::ArrayXd::Random(5); std::cout<<x <<std::endl; x = x.unaryExpr([](double elem) // changed type of parameter { return elem < 0.0 ? 0.0 : 1.0; // return instead of assignment }); std::cout<<x <<std::endl; std::cout << x.unaryExpr(std::ptr_fun(ramp)) << std::endl; } |
Eigen unaryExpr (Function Pointer, Lambda Expression) Example Read More »
Matrix Decomposition with Eigen: QR, Cholesky Decomposition LU, UL
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void qRDecomposition(Eigen::MatrixXd &A,Eigen::MatrixXd &Q, Eigen::MatrixXd &R) { /* A=QR Q: is orthogonal matrix-> columns of Q are orthonormal R: is upper triangulate matrix this is possible when columns of A are linearly indipendent */ Eigen::MatrixXd thinQ(A.rows(),A.cols() ), q(A.rows(),A.rows()); Eigen::HouseholderQR<Eigen::MatrixXd> householderQR(A); q = householderQR.householderQ(); thinQ.setIdentity(); Q = householderQR.householderQ() * thinQ; R=Q.transpose()*A; } void qRExample() { Eigen::MatrixXd A; A.setRandom(3,4); std::cout<<"A" <<std::endl; std::cout<<A <<std::endl; Eigen::MatrixXd Q(A.rows(),A.rows()); Eigen::MatrixXd R(A.rows(),A.cols()); /////////////////////////////////HouseholderQR//////////////////////// Eigen::MatrixXd thinQ(A.rows(),A.cols() ), q(A.rows(),A.rows()); Eigen::HouseholderQR<Eigen::MatrixXd> householderQR(A); q = householderQR.householderQ(); thinQ.setIdentity(); Q = householderQR.householderQ() * thinQ; std::cout << "HouseholderQR" <<std::endl; std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; R = householderQR.matrixQR().template triangularView<Eigen::Upper>(); std::cout << R<<std::endl; std::cout << R.rows()<<std::endl; std::cout << R.cols()<<std::endl; R=Q.transpose()*A; // std::cout << "R" <<std::endl; // std::cout << R<<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; /////////////////////////////////ColPivHouseholderQR//////////////////////// Eigen::ColPivHouseholderQR<Eigen::MatrixXd> colPivHouseholderQR(A.rows(), A.cols()); colPivHouseholderQR.compute(A); //R = colPivHouseholderQR.matrixR().template triangularView<Upper>(); R = colPivHouseholderQR.matrixR(); Q = colPivHouseholderQR.matrixQ(); std::cout << "ColPivHouseholderQR" <<std::endl; std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; std::cout << "R" <<std::endl; std::cout << R <<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; /////////////////////////////////FullPivHouseholderQR//////////////////////// std::cout << "FullPivHouseholderQR" <<std::endl; Eigen::FullPivHouseholderQR<Eigen::MatrixXd> fullPivHouseholderQR(A.rows(), A.cols()); fullPivHouseholderQR.compute(A); Q=fullPivHouseholderQR.matrixQ(); R=fullPivHouseholderQR.matrixQR().template triangularView<Eigen::Upper>(); std::cout << "Q" <<std::endl; std::cout << Q <<std::endl; std::cout << "R" <<std::endl; std::cout << R <<std::endl; std::cout << "A-Q*R" <<std::endl; std::cout << A-Q*R <<std::endl; } void lDUDecomposition() { /* L: lower triangular matrix L U: upper triangular matrix U D: is a diagonal matrix A=LDU */ } void lUDecomposition() { /* L: lower triangular matrix L U: upper triangular matrix U A=LU */ } double exp(double x) // the functor we want to apply { std::setprecision(5); return std::trunc(x); } void gramSchmidtOrthogonalization(Eigen::MatrixXd &matrix,Eigen::MatrixXd &orthonormalMatrix) { /* In this method you make every column perpendicular to it's previous columns, here if a and b are representation vector of two columns, c=b-((b.a)/|a|).a ^ / b / / / ----------> a ^ /| b / | / | c / | ----------> a you just have to normilze every vector after make it perpendicular to previous columns so: q1=a.normalized(); q2=b-(b.q1).q1 q2=q2.normalized(); q3=c-(c.q1).q1 - (c.q2).q2 q3=q3.normalized(); Now we have Q, but we want A=QR so we just multiply both side by Q.transpose(), since Q is orthonormal, Q*Q.transpose() is I A=QR; Q.transpose()*A=R; */ Eigen::VectorXd col; for(int i=0;i<matrix.cols();i++) { col=matrix.col(i); col=col.normalized(); for(int j=0;j<i-1;j++) { //orthonormalMatrix.col(i) } orthonormalMatrix.col(i)=col; } Eigen::MatrixXd A(4,3); A<<1,2,3,-1,1,1,1,1,1,1,1,1; Eigen::Vector4d a=A.col(0); Eigen::Vector4d b=A.col(1); Eigen::Vector4d c=A.col(2); Eigen::Vector4d q1= a.normalized(); Eigen::Vector4d q2=b-(b.dot(q1))*q1; q2=q2.normalized(); Eigen::Vector4d q3=c-(c.dot(q1))*q1 - (c.dot(q2))*q2; q3=q3.normalized(); std::cout<< "q1:"<<std::endl; std::cout<< q1<<std::endl; std::cout<< "q2"<<std::endl; std::cout<< q2<<std::endl; std::cout<< "q3:"<<std::endl; std::cout<< q3<<std::endl; Eigen::MatrixXd Q(4,3); Q.col(0)=q1; Q.col(1)=q2; Q.col(2)=q3; Eigen::MatrixXd R(3,3); R=Q.transpose()*(A); std::cout<<"Q"<<std::endl; std::cout<< Q<<std::endl; std::cout<<"R"<<std::endl; std::cout<< R.unaryExpr(std::ptr_fun(exp))<<std::endl; //MatrixXd A(4,3), thinQ(4,3), Q(4,4); Eigen::MatrixXd thinQ(4,3), q(4,4); //A.setRandom(); Eigen::HouseholderQR<Eigen::MatrixXd> qr(A); q = qr.householderQ(); thinQ.setIdentity(); thinQ = qr.householderQ() * thinQ; std::cout << "Q computed by Eigen" << "\n\n" << thinQ << "\n\n"; std::cout << q << "\n\n" << thinQ << "\n\n"; } void gramSchmidtOrthogonalizationExample() { Eigen::MatrixXd matrix(3,4),orthonormalMatrix(3,4) ; matrix=Eigen::MatrixXd::Random(3,4);////A.setRandom(); gramSchmidtOrthogonalization(matrix,orthonormalMatrix); } void choleskyDecompositionExample() { /* Positive-definite matrix: Matrix Mnxn is said to be positive definite if the scalar zTMz is strictly positive for every non-zero column vector z n real numbers. zTMz>0 Hermitian matrix: Matrix Mnxn is said to be positive definite if the scalar z*Mz is strictly positive for every non-zero column vector z n real numbers. z*Mz>0 z* is the conjugate transpose of z. Positive semi-definite same as above except zTMz>=0 or z*Mz>=0 Example: ┌2 -1 0┐ M= |-1 2 -1| |0 - 1 2| └ ┘ ┌ a ┐ z=| b | | c | └ ┘ zTMz=a^2 +c^2+ (a-b)^2+ (b-c)^2 Cholesky decomposition: Cholesky decomposition of a Hermitian positive-definite matrix A is: A=LL* L is a lower triangular matrix with real and positive diagonal entries L* is the conjugate transpose of L */ Eigen::MatrixXd A(3,3); A << 6, 0, 0, 0, 4, 0, 0, 0, 7; Eigen::MatrixXd L( A.llt().matrixL() ); Eigen::MatrixXd L_T=L.adjoint();//conjugate transpose std::cout << "L" << std::endl; std::cout << L << std::endl; std::cout << "L_T" << std::endl; std::cout << L_T << std::endl; std::cout << "A" << std::endl; std::cout << A << std::endl; std::cout << "L*L_T" << std::endl; std::cout << L*L_T << std::endl; } |
Matrix Decomposition with Eigen: QR, Cholesky Decomposition LU, UL Read More »
Eigen Arrays, Matrices and Vectors: Definition, Initialization Resizing, Populating and Coefficient Wise Operations
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std::cout <<"/////////////////Definition////////////////"<<std::endl; /* Definition of vectors and matrices in Eigen comes in the following form: Eigen::MatrixSizeType Eigen::VectorSizeType Eigen::ArraySizeType Size can be 2,3,4 for fixed size square matrices or X for dynamic size Type can be: i for integer, f for float, d for double, c for complex, cf for complex float, cd for complex double. */ // Vector3f is a fixed column vector of 3 floats: Eigen::Vector3f objVector3f; // RowVector2i is a fixed row vector of 3 integer: Eigen::RowVector2i objRowVector2i; // VectorXf is a column vector of size 10 floats: Eigen::VectorXf objv(10); Eigen::Matrix4d m; // 4x4 double Eigen::Matrix4cd objMatrix4cd; // 4x4 double complex //a is a 3x3 matrix, with a static float[9] array of uninitialized coefficients, Eigen::Matrix3f a; //b is a dynamic-size matrix whose size is currently 0x0, and whose array of coefficients hasn't yet been allocated at all. Eigen::MatrixXf b; //A is a 10x15 dynamic-size matrix, with allocated but currently uninitialized coefficients. Eigen::MatrixXf A(10, 15); //V is a dynamic-size vector of size 30, with allocated but currently uninitialized coefficients. Eigen::VectorXf V(30); //Template style definition // Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> &matrix Eigen::Matrix<double, 2, 3> my_matrix; my_matrix << 1, 2, 3, 4, 5, 6; // ArrayXf Eigen::Array<float, Eigen::Dynamic, 1> a1; // Array3f Eigen::Array<float, 3, 1> a2; // ArrayXXd Eigen::Array<double, Eigen::Dynamic, Eigen::Dynamic> a3; // Array33d Eigen::Array<double, 3, 3> a4; Eigen::Matrix3d matrix_from_array = a4.matrix(); std::cout <<"///////////////////Initialization//////////////////"<< std::endl; Eigen::Matrix2d a_2d; a_2d.setRandom(); a_2d.setConstant(4.3); Eigen::MatrixXd identity=Eigen::MatrixXd::Identity(6,6); Eigen::MatrixXd zeros=Eigen::MatrixXd::Zero(3, 3); Eigen::ArrayXXf table(10, 4); table.col(0) = Eigen::ArrayXf::LinSpaced(10, 0, 90); std::cout <<"/////////////Matrix Coefficient Wise Operations///////////////////"<< std::endl; int i,j; std::cout << my_matrix << std::endl; std::cout << my_matrix.transpose()<< std::endl; std::cout<<my_matrix.minCoeff(&i, &j)<<std::endl; std::cout<<my_matrix.maxCoeff(&i, &j)<<std::endl; std::cout<<my_matrix.prod()<<std::endl; std::cout<<my_matrix.sum()<<std::endl; std::cout<<my_matrix.mean()<<std::endl; std::cout<<my_matrix.trace()<<std::endl; std::cout<<my_matrix.colwise().mean()<<std::endl; std::cout<<my_matrix.rowwise().maxCoeff()<<std::endl; std::cout<<my_matrix.lpNorm<2>()<<std::endl; std::cout<<my_matrix.lpNorm<Eigen::Infinity>()<<std::endl; std::cout<<(my_matrix.array()>0).all()<<std::endl;// if all elemnts are positive std::cout<<(my_matrix.array()>2).any()<<std::endl;//if any element is greater than 2 std::cout<<(my_matrix.array()>1).count()<<std::endl;// count the number of elements greater than 1 std::cout << my_matrix.array() - 2 << std::endl; std::cout << my_matrix.array().abs() << std::endl; std::cout << my_matrix.array().square() << std::endl; std::cout << my_matrix.array() * my_matrix.array() << std::endl; std::cout << my_matrix.array().exp() << std::endl; std::cout << my_matrix.array().log() << std::endl; std::cout << my_matrix.array().sqrt() << std::endl; std::cout <<"//////////////////Block Elements Access////////////////////"<< std::endl; //Block of size (p,q), starting at (i,j) matrix.block(i,j,p,q) Eigen::MatrixXf mat(4, 4); mat << 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16; std::cout << "Block in the middle" << std::endl; std::cout << mat.block<2, 2>(1, 1) << std::endl; for (int i = 1; i <= 3; ++i) { std::cout << "Block of size " << i << "x" << i << std::endl; std::cout << mat.block(0, 0, i, i) << std::endl; } /////////////build matrix from vector, resizing matrix, dynamic size //////////////// Eigen::MatrixXd dynamicMatrix; int rows, cols; rows=3; cols=2; dynamicMatrix.resize(rows,cols); dynamicMatrix<<-1,7,3,4,5,1; //If you want a conservative variant of resize() which does not change the coefficients, use conservativeResize() dynamicMatrix.conservativeResize(dynamicMatrix.rows(), dynamicMatrix.cols()+1); dynamicMatrix.col(dynamicMatrix.cols()-1) = Eigen::Vector3d(1, 4, 0); dynamicMatrix.conservativeResize(dynamicMatrix.rows(), dynamicMatrix.cols()+1); dynamicMatrix.col(dynamicMatrix.cols()-1) = Eigen::Vector3d(5, -8, 6); /* you should expect this: 1 7 1 5 3 4 4 -8 5 1 0 6 */ std::cout<< dynamicMatrix<<std::endl; |
Creating a Simulated Stereo Vision Cameras With OpenCV and C++
n this tutorial, I created an ellipsoid in 3D space and create two cameras in right and left and captured the image:
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cv::Mat leftCameraRotation,rightCameraRotation; double rollLeft, pitchLeft, yawLeft,rollRight, pitchRight, yawRight ,txLeft,tyLeft,tzLeft,txRight,tyRight,tzRight; cv::Vec3d thetaLeft,thetaRight; rollLeft=0 ; pitchLeft=+M_PI/10; yawLeft=0; thetaLeft[0]=rollLeft; thetaLeft[1]=pitchLeft; thetaLeft[2]=yawLeft; rollRight=0; pitchRight= -M_PI/10; yawRight= 0; thetaRight[0]=rollRight; thetaRight[1]=pitchRight; thetaRight[2]=yawRight; txLeft=-1; tyLeft=0.0; tzLeft=-4.0; txRight=1.0; tyRight=0.0; tzRight=-4.0; leftCameraRotation =eulerAnglesToRotationMatrix(thetaLeft); rightCameraRotation =eulerAnglesToRotationMatrix(thetaRight); cv::Mat leftCameraTranslation = (cv::Mat_<double>(3,1) <<txLeft,tyLeft,tzLeft); cv::Mat rightCameraTranslation = (cv::Mat_<double>(3,1) <<txRight,tyRight,tzRight); std::vector<cv::Point3d> objectPointsInWorldCoordinate; double X,Y,Z,radius; ////////////////////////////////////////creating ellipsoid in the world coordinate/////////////////////// double phiStepSize,thetaStepSize; phiStepSize=0.7; thetaStepSize=0.6; double a,b,c; a=2; b=3; c=1.6; for(double phi=-M_PI;phi<M_PI;phi=phi+phiStepSize) { for(double theta=-M_PI/2;theta<M_PI/2;theta=theta+thetaStepSize) { X=a*cos(theta)*cos(phi); Y=b*cos(theta)*sin(phi); Z=c*sin(theta); objectPointsInWorldCoordinate.push_back(cv::Point3d(X, Y, Z)); } } int numberOfPixelInHeight,numberOfPixelInWidth; double heightOfSensor, widthOfSensor; double focalLength=2.0; double mx, my, U0, V0; numberOfPixelInHeight=600; numberOfPixelInWidth=600; heightOfSensor=10; widthOfSensor=10; my=(numberOfPixelInHeight)/heightOfSensor ; U0=(numberOfPixelInHeight)/2 ; mx=(numberOfPixelInWidth)/widthOfSensor; V0=(numberOfPixelInWidth)/2; cv::Mat K = (cv::Mat_<double>(3,3) << focalLength*mx, 0, V0, 0,focalLength*my,U0, 0,0,1); std::vector<cv::Point2d> imagePointsLeftCamera,imagePointsRightCamera; cv::projectPoints(objectPointsInWorldCoordinate, leftCameraRotation, leftCameraTranslation, K, cv::noArray(), imagePointsLeftCamera); cv::projectPoints(objectPointsInWorldCoordinate, rightCameraRotation, rightCameraTranslation, K, cv::noArray(), imagePointsRightCamera); ////////////////////////////////////////////////storing images from right and left camera////////////////////////////////////////////// std::string fileName; cv::Mat rightImage,leftImage; int U,V; leftImage=cv::Mat::zeros(numberOfPixelInHeight,numberOfPixelInWidth,CV_8UC1); for(std::size_t i=0;i<imagePointsLeftCamera.size();i++) { V=int(imagePointsLeftCamera.at(i).x); U=int(imagePointsLeftCamera.at(i).y); leftImage.at<char>(U,V)=255; } fileName=std::string("imagePointsLeftCamera")+std::to_string(focalLength)+ std::string("_.jpg"); cv::imwrite(fileName,leftImage); rightImage=cv::Mat::zeros(numberOfPixelInHeight,numberOfPixelInWidth,CV_8UC1); for(std::size_t i=0;i<imagePointsRightCamera.size();i++) { V=int(imagePointsRightCamera.at(i).x); U=int(imagePointsRightCamera.at(i).y); rightImage.at<char>(U,V)=255; } fileName=std::string("imagePointsRightCamera")+std::to_string(focalLength)+ std::string("_.jpg"); cv::imwrite(fileName,rightImage); |
Creating a Simulated Stereo Vision Cameras With OpenCV and C++ Read More »
