projection matrix

Decomposing Projection Using OpenCV and C++

Leave a Comment / C++, Computer Vision, Tutorials / admin

#include <opencv/cv.hpp>
#include <opencv2/core.hpp>
#include <opencv2/calib3d.hpp>
#include "transformation.hpp"

//https://www.cnblogs.com/shengguang/p/5932522.html
void HouseHolderQR(const cv::Mat &A, cv::Mat &Q, cv::Mat &R)
{
    assert ( A.channels() == 1 );
    assert ( A.rows >= A.cols );
    auto sign = [](double value) { return value >= 0 ? 1: -1; };
    const auto totalRows = A.rows;
    const auto totalCols = A.cols;
    R = A.clone();
    Q = cv::Mat::eye ( totalRows, totalRows, A.type() );
    for ( int col = 0; col < A.cols; ++ col )
    {
        cv::Mat matAROI = cv::Mat ( R, cv::Range ( col, totalRows ), cv::Range ( col, totalCols ) );
        cv::Mat y = matAROI.col ( 0 );
        auto yNorm = norm ( y );
        cv::Mat e1 = cv::Mat::eye ( y.rows, 1, A.type() );
        cv::Mat w = y + sign(y.at<double>(0,0)) *  yNorm * e1;
        cv::Mat v = w / norm( w );
        cv::Mat vT; cv::transpose(v, vT );
        cv::Mat I = cv::Mat::eye( matAROI.rows, matAROI.rows, A.type() );
        cv::Mat I_2VVT = I - 2 * v * vT;
        cv::Mat matH = cv::Mat::eye ( totalRows, totalRows, A.type() );
        cv::Mat matHROI = cv::Mat(matH, cv::Range ( col, totalRows ), cv::Range ( col, totalRows ) );
        I_2VVT.copyTo ( matHROI );
        R = matH * R;
        Q = Q * matH;
    }
}

int main()
{


/*
Thera are two notations for Projection Matrix

1)P=K[R|t]
                                                              ┌X┐
                                                      ┌R,t┐   |Y|
    X_Cam=R*X+t  ==> Homogeneous Coordinate ==> X_Cam=|0,1|   |Z|
                                                      └   ┘4x4└1┘4x1

                                         ┌X┐
                                ┌   ┐    |Y|
    image plane  <--x=K[I|0]3x4 |R,t|    |Z|
                                └0,1┘4x4 └1┘4x1

    P=K[R|t]

2)P=KR[I|-X0]
                                                              ┌X┐
                                                       ┌R -Rc┐|Y|
    X_Cam=R*[X_w-C]  ==> Homogeneous Coordinate ==>    |     ||Z|
                                                       └0  1 ┘└1┘
                                       ┌X┐
                                ┌R -Rc┐|Y|
    image plane  <--x=K[I|0]3x4 |0  1 ||Z|
                                └     ┘└1┘
    P=K[R|-RC]= KR[I|-C]


(1) and (2) ==> t=-RC

*/
    int numberOfPixelInHeight,numberOfPixelInWidth;
    double heightOfSensor, widthOfSensor;
    double focalLength=1.50;
    double mx, my, U0, V0;
    numberOfPixelInHeight=600;
    numberOfPixelInWidth=800;

    heightOfSensor=10;
    widthOfSensor=10;


    my=(numberOfPixelInHeight)/heightOfSensor ;
    U0=(numberOfPixelInHeight)/2 ;

    mx=(numberOfPixelInWidth)/widthOfSensor;
    V0=(numberOfPixelInWidth)/2;

    cv::Mat cameraMatrix = (cv::Mat_<double>(3,3) <<
    focalLength*mx, 0, V0,
    0,focalLength*my,U0,
    0,0,1);

    double tx,ty,tz,roll,pitch,yaw;

    tx=1.0;
    ty=2.1;
    tz=-1.4;

    cv::Mat translation = (cv::Mat_<double>(3,1) <<tx,ty,tz);

    cv::Vec3d theta;

    roll=+M_PI/4 ;
    pitch=+M_PI/10;
    yaw=-+M_PI/6;

    theta[0]=roll;
    theta[1]=pitch;
    theta[2]=yaw;
    cv::Mat rotation=eulerAnglesToRotationMatrix(theta);
    //1)P=K[R|t]
    cv::Mat R_T;
    cv::hconcat(rotation, translation, R_T);
    cv::Mat projectionMatrix=cameraMatrix*R_T;



    cv::Mat calculatedCameraMatrix,calculatedRotation,calculatedTranslation;

    cv::decomposeProjectionMatrix(projectionMatrix,calculatedCameraMatrix,calculatedRotation,calculatedTranslation);


    std::cout<<"==============================Ground Truth==============================" <<std::endl;

    std::cout<<"Rotation Matrix (Ground Truth)" <<std::endl;
    std::cout<<rotation <<std::endl;

    std::cout<<"Translation Matrix (Ground Truth)" <<std::endl;
    std::cout<<translation <<std::endl;

    std::cout<<"Camera Matrix (Ground Truth)" <<std::endl;
    std::cout<<cameraMatrix <<std::endl;

    std::cout<<"==============================Decomposing Using OpenCV==============================" <<std::endl;


    std::cout<<"Computed Rotation Matrix (OpenCV)" <<std::endl;
    std::cout<<calculatedRotation <<std::endl;

    std::cout<<"Computed Translation Matrix (OpenCV)" <<std::endl;
    //std::cout<<calculatedTranslation/calculatedTranslation.at<double>(3,0) <<std::endl;
    cv::Mat tempT=(cv::Mat_<double>(3,1)<<
                   calculatedTranslation.at<double>(0,0)/calculatedTranslation.at<double>(3,0),
                   calculatedTranslation.at<double>(1,0)/calculatedTranslation.at<double>(3,0),
                   calculatedTranslation.at<double>(2,0)/calculatedTranslation.at<double>(3,0));
    std::cout<<-rotation*tempT<<std::endl;

    std::cout<<"Computed Camera Matrix (OpenCV)" <<std::endl;
    std::cout<<calculatedCameraMatrix <<std::endl;

///////////////////////////////////////decomposing the projection matrix///////////////////////////////////////////

/*

    P=KR[I|-X0]=[H_inf3x3|h3x1]
    KR=H_inf3x3
    1)X0
    -KRX0=h3x1 => X0=-(KR)^-1*h3x1 ==>X0=-(H_inf3x3)^-1*h3x1
    2)K,R

    KR=H_inf3x3 =>(KR)^-1= H_inf3x3^-1 =>R^-1*K^-1=H_inf3x3^-1 | R^-1*K^-1=Q*R => R=Q^-1, K=R^-1
                                         H_inf3x3^-1=Q*R       |

*/


    cv::Mat H_inf3x3=(cv::Mat_<double>(3,3)<< projectionMatrix.at<double>(0,0),projectionMatrix.at<double>(0,1),projectionMatrix.at<double>(0,2)
                      ,projectionMatrix.at<double>(1,0),projectionMatrix.at<double>(1,1),projectionMatrix.at<double>(1,2)
                      ,projectionMatrix.at<double>(2,0),projectionMatrix.at<double>(2,1),projectionMatrix.at<double>(2,2));

    cv::Mat h3x1=(cv::Mat_<double>(3,1)<<projectionMatrix.at<double>(0,3),projectionMatrix.at<double>(1,3),projectionMatrix.at<double>(2,3) );



    cv::Mat Q,R;
    cv::Mat H_inf3x3_inv=H_inf3x3.inv();
    //R=Q^-1, K=R^-1


    HouseHolderQR(H_inf3x3_inv, Q, R);
    cv::Mat K=R.inv();

    std::cout<<"==============================Decomposing Using My Code==============================" <<std::endl;
    //due to homogeneity we divide it by last element
    std::cout<<"Estimated Camera Matrix\n"<<K/K.at<double>(2,2) <<std::endl;

    cv::Mat rotationMatrix=Q.inv();
    std::cout<<"Estimated Camera Rotation\n"<<rotationMatrix*-1 <<std::endl;

    std::cout<<"Estimated Camera Translation" <<std::endl;

    //t=-R*C, Q.inv()=R
    std::cout<<-1*(-Q.inv()*(-H_inf3x3.inv()*h3x1 ))<<std::endl;

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#include <opencv/cv.hpp>

#include <opencv2/core.hpp>

#include <opencv2/calib3d.hpp>

#include "transformation.hpp"

//https://www.cnblogs.com/shengguang/p/5932522.html

void HouseHolderQR(const cv::Mat &A, cv::Mat &Q, cv::Mat &R)

{

assert ( A.channels() == 1 );

assert ( A.rows >= A.cols );

auto sign = [](double value) { return value >= 0 ? 1: -1; };

const auto totalRows = A.rows;

const auto totalCols = A.cols;

R = A.clone();

Q = cv::Mat::eye ( totalRows, totalRows, A.type() );

for ( int col = 0; col < A.cols; ++ col )

{

cv::Mat matAROI = cv::Mat ( R, cv::Range ( col, totalRows ), cv::Range ( col, totalCols ) );

cv::Mat y = matAROI.col ( 0 );

auto yNorm = norm ( y );

cv::Mat e1 = cv::Mat::eye ( y.rows, 1, A.type() );

cv::Mat w = y + sign(y.at<double>(0,0)) * yNorm * e1;

cv::Mat v = w / norm( w );

cv::Mat vT; cv::transpose(v, vT );

cv::Mat I = cv::Mat::eye( matAROI.rows, matAROI.rows, A.type() );

cv::Mat I_2VVT = I - 2 * v * vT;

cv::Mat matH = cv::Mat::eye ( totalRows, totalRows, A.type() );

cv::Mat matHROI = cv::Mat(matH, cv::Range ( col, totalRows ), cv::Range ( col, totalRows ) );

I_2VVT.copyTo ( matHROI );

R = matH * R;

Q = Q * matH;

}

int main()

{

Thera are two notations for Projection Matrix

1)P=K[R|t]

┌X┐

┌R,t┐ |Y|

X_Cam=R*X+t ==> Homogeneous Coordinate ==> X_Cam=|0,1| |Z|

└ ┘4x4└1┘4x1

┌X┐

┌ ┐ |Y|

image plane <--x=K[I|0]3x4 |R,t| |Z|

└0,1┘4x4 └1┘4x1

P=K[R|t]

2)P=KR[I|-X0]

┌X┐

┌R -Rc┐|Y|

X_Cam=R*[X_w-C] ==> Homogeneous Coordinate ==> | ||Z|

└0 1 ┘└1┘

┌X┐

┌R -Rc┐|Y|

image plane <--x=K[I|0]3x4 |0 1 ||Z|

└ ┘└1┘

P=K[R|-RC]= KR[I|-C]

(1) and (2) ==> t=-RC

int numberOfPixelInHeight,numberOfPixelInWidth;

double heightOfSensor, widthOfSensor;

double focalLength=1.50;

double mx, my, U0, V0;

numberOfPixelInHeight=600;

numberOfPixelInWidth=800;

heightOfSensor=10;

widthOfSensor=10;

my=(numberOfPixelInHeight)/heightOfSensor ;

U0=(numberOfPixelInHeight)/2 ;

mx=(numberOfPixelInWidth)/widthOfSensor;

V0=(numberOfPixelInWidth)/2;

cv::Mat cameraMatrix = (cv::Mat_<double>(3,3) <<

focalLength*mx, 0, V0,

0,focalLength*my,U0,

0,0,1);

double tx,ty,tz,roll,pitch,yaw;

tx=1.0;

ty=2.1;

tz=-1.4;

cv::Mat translation = (cv::Mat_<double>(3,1) <<tx,ty,tz);

cv::Vec3d theta;

roll=+M_PI/4 ;

pitch=+M_PI/10;

yaw=-+M_PI/6;

theta[0]=roll;

theta[1]=pitch;

theta[2]=yaw;

cv::Mat rotation=eulerAnglesToRotationMatrix(theta);

//1)P=K[R|t]

cv::Mat R_T;

cv::hconcat(rotation, translation, R_T);

cv::Mat projectionMatrix=cameraMatrix*R_T;

cv::Mat calculatedCameraMatrix,calculatedRotation,calculatedTranslation;

cv::decomposeProjectionMatrix(projectionMatrix,calculatedCameraMatrix,calculatedRotation,calculatedTranslation);

std::cout<<"==============================Ground Truth==============================" <<std::endl;

std::cout<<"Rotation Matrix (Ground Truth)" <<std::endl;

std::cout<<rotation <<std::endl;

std::cout<<"Translation Matrix (Ground Truth)" <<std::endl;

std::cout<<translation <<std::endl;

std::cout<<"Camera Matrix (Ground Truth)" <<std::endl;

std::cout<<cameraMatrix <<std::endl;

std::cout<<"==============================Decomposing Using OpenCV==============================" <<std::endl;

std::cout<<"Computed Rotation Matrix (OpenCV)" <<std::endl;

std::cout<<calculatedRotation <<std::endl;

std::cout<<"Computed Translation Matrix (OpenCV)" <<std::endl;

//std::cout<<calculatedTranslation/calculatedTranslation.at<double>(3,0) <<std::endl;

cv::Mat tempT=(cv::Mat_<double>(3,1)<<

calculatedTranslation.at<double>(0,0)/calculatedTranslation.at<double>(3,0),

calculatedTranslation.at<double>(1,0)/calculatedTranslation.at<double>(3,0),

calculatedTranslation.at<double>(2,0)/calculatedTranslation.at<double>(3,0));

std::cout<<-rotation*tempT<<std::endl;

std::cout<<"Computed Camera Matrix (OpenCV)" <<std::endl;

std::cout<<calculatedCameraMatrix <<std::endl;

///////////////////////////////////////decomposing the projection matrix///////////////////////////////////////////

P=KR[I|-X0]=[H_inf3x3|h3x1]

KR=H_inf3x3

1)X0

-KRX0=h3x1 => X0=-(KR)^-1*h3x1 ==>X0=-(H_inf3x3)^-1*h3x1

2)K,R

KR=H_inf3x3 =>(KR)^-1= H_inf3x3^-1 =>R^-1*K^-1=H_inf3x3^-1 | R^-1*K^-1=Q*R => R=Q^-1, K=R^-1

H_inf3x3^-1=Q*R |

cv::Mat H_inf3x3=(cv::Mat_<double>(3,3)<< projectionMatrix.at<double>(0,0),projectionMatrix.at<double>(0,1),projectionMatrix.at<double>(0,2)

,projectionMatrix.at<double>(1,0),projectionMatrix.at<double>(1,1),projectionMatrix.at<double>(1,2)

,projectionMatrix.at<double>(2,0),projectionMatrix.at<double>(2,1),projectionMatrix.at<double>(2,2));

cv::Mat h3x1=(cv::Mat_<double>(3,1)<<projectionMatrix.at<double>(0,3),projectionMatrix.at<double>(1,3),projectionMatrix.at<double>(2,3) );

cv::Mat Q,R;

cv::Mat H_inf3x3_inv=H_inf3x3.inv();

//R=Q^-1, K=R^-1

HouseHolderQR(H_inf3x3_inv, Q, R);

cv::Mat K=R.inv();

std::cout<<"==============================Decomposing Using My Code==============================" <<std::endl;

//due to homogeneity we divide it by last element

std::cout<<"Estimated Camera Matrix\n"<<K/K.at<double>(2,2) <<std::endl;

cv::Mat rotationMatrix=Q.inv();

std::cout<<"Estimated Camera Rotation\n"<<rotationMatrix*-1 <<std::endl;

std::cout<<"Estimated Camera Translation" <<std::endl;

//t=-R*C, Q.inv()=R

std::cout<<-1*(-Q.inv()*(-H_inf3x3.inv()*h3x1 ))<<std::endl;

And the output is:

==============================Ground Truth==============================
Rotation Matrix (Ground Truth)
[0.823639103546332, 0.5427868801100539, -0.1643199010764935;
 -0.4755282581475767, 0.5031184295835893, -0.7216264418079997;
 -0.3090169943749474, 0.6724985119639573, 0.6724985119639574]
Translation Matrix (Ground Truth)
[1;
 2.1;
 -1.4]
Camera Matrix (Ground Truth)
[120, 0, 400;
 0, 90, 300;
 0, 0, 1]
==============================Decomposing Using OpenCV==============================
Computed Rotation Matrix (OpenCV)
[0.823639103546332, 0.5427868801100539, -0.1643199010764936;
 -0.4755282581475769, 0.5031184295835893, -0.7216264418079997;
 -0.3090169943749474, 0.6724985119639573, 0.6724985119639574]
Computed Translation Matrix (OpenCV)
[0.9999999999999988;
 2.1;
 -1.4]
Computed Camera Matrix (OpenCV)
[120, -1.4210854715202e-14, 400;
 0, 90.00000000000001, 300;
 0, 0, 1]
==============================Decomposing Using My Code==============================
Estimated Camera Matrix
[120, -5.776629175002766e-14, 399.9999999999999;
 -5.608036291626306e-15, 89.99999999999994, 299.9999999999999;
 -1.49192016182457e-17, -1.561251128379126e-16, 1]
Estimated Camera Rotation
[0.8236391035463322, 0.5427868801100542, -0.1643199010764936;
 -0.4755282581475767, 0.5031184295835892, -0.7216264418079998;
 -0.3090169943749475, 0.6724985119639573, 0.6724985119639575]
Estimated Camera Translation
[1;
 2.099999999999999;
 -1.4]

==============================Ground Truth==============================

Rotation Matrix (Ground Truth)

[0.823639103546332, 0.5427868801100539, -0.1643199010764935;

-0.4755282581475767, 0.5031184295835893, -0.7216264418079997;

-0.3090169943749474, 0.6724985119639573, 0.6724985119639574]

Translation Matrix (Ground Truth)

[1;

2.1;

-1.4]

Camera Matrix (Ground Truth)

[120, 0, 400;

0, 90, 300;

0, 0, 1]

==============================Decomposing Using OpenCV==============================

Computed Rotation Matrix (OpenCV)

[0.823639103546332, 0.5427868801100539, -0.1643199010764936;

-0.4755282581475769, 0.5031184295835893, -0.7216264418079997;

-0.3090169943749474, 0.6724985119639573, 0.6724985119639574]

Computed Translation Matrix (OpenCV)

[0.9999999999999988;

2.1;

-1.4]

Computed Camera Matrix (OpenCV)

[120, -1.4210854715202e-14, 400;

0, 90.00000000000001, 300;

0, 0, 1]

==============================Decomposing Using My Code==============================

Estimated Camera Matrix

[120, -5.776629175002766e-14, 399.9999999999999;

-5.608036291626306e-15, 89.99999999999994, 299.9999999999999;

-1.49192016182457e-17, -1.561251128379126e-16, 1]

Estimated Camera Rotation

[0.8236391035463322, 0.5427868801100542, -0.1643199010764936;

-0.4755282581475767, 0.5031184295835892, -0.7216264418079998;

-0.3090169943749475, 0.6724985119639573, 0.6724985119639575]

Estimated Camera Translation

[1;

2.099999999999999;

-1.4]

Decomposing Projection Using OpenCV and C++ Read More »

Camera Projection Matrix with Eigen

Leave a Comment / C++, Computer Vision, Tutorials / admin

void projectFromCameraCoordinateToCameraPlane( Eigen::Matrix3Xd &pointsInCameraCoordinate, Eigen::Matrix3d &cameraIntrinsicMatrix,Eigen::Matrix2Xd &pointsInCameraPlane)
{
/*
Intrinsic camera Matrix is something like this:

focalLength  ===> unit is mm
mx=  ==>unit is Pixel/mm
U0=  ===> unit is Pixel

my=-(numberOfPixelInHeight)/heightOfSensor ;
U0=(numberOfPixelInHeight)/2 ;

mx=(numberOfPixelInWidth)/heightOfSensor; 
V0=(numberOfPixelInWidth)/2;

Gamma=0

X,Y,Z,W are homogeneous position of point in camera coordinate

┌ ┐	 ┌                         	 ┐┌ ┐
v`	 |f*mx		Gamma		V0	0||X|	
u`	=|0			f*my		U0	0||Y|
w`	 |0			0			 1	0||Z|
└ ┘	 └                         	 ┘└W┘

									 ▲y
									 |	
			----►V					 |	
			|	 ____________________|_____________________
			|	|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
		U	▼	|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|____________► x
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|
				|__|__|__|__|__|__|__|__|__|__|__|__|__|__|



To project points from camera coordinate to camera plane:

V`=X*f*mx + V0*Z 
U`=Y*f*my + U0*Z
W`=Z

U=U`/W` ,V=V`/W`


V=f*mx*X/Z + V0 
U=f*my*Y/Z + U0

(U,V) is the index of corresponding point in the image
*/
	
	Eigen::Matrix3Xd pointsInCameraPlaneHomogeneous(pointsInCameraCoordinate.rows(),pointsInCameraCoordinate.cols());	

	pointsInCameraPlaneHomogeneous=cameraIntrinsicMatrix*pointsInCameraCoordinate;
	std::cout<<"pointsInCameraPlaneHomogeneous"<<std::endl;	
 	std::cout<<pointsInCameraPlaneHomogeneous<<std::endl;	
	for(int i=0;i<pointsInCameraPlaneHomogeneous.cols();i++  )
	{
		pointsInCameraPlaneHomogeneous(0,i)=pointsInCameraPlaneHomogeneous(0,i)/pointsInCameraPlaneHomogeneous(2,i);
		pointsInCameraPlaneHomogeneous(1,i)=pointsInCameraPlaneHomogeneous(1,i)/pointsInCameraPlaneHomogeneous(2,i);
	}
	pointsInCameraPlane=pointsInCameraPlaneHomogeneous.block(0,0,pointsInCameraPlaneHomogeneous.rows()-1,pointsInCameraPlaneHomogeneous.cols());
}

void transformPoints(Eigen::Matrix3Xd &points, Eigen::Matrix3d &rotationMatrix, Eigen::Vector3d &translationVector,Eigen::Matrix3Xd &transformedPoints)
{
	transformedPoints=(rotationMatrix*points).colwise() + translationVector ;
}

void projectFromCameraCoordinateToCameraPlane_Example()
{
	
/*
 OpenCV coordinate system
                  ▲
                 /
                /
              Z/
              /
             /
------------------------------► X
            |
            |
            |
            |
          Y |
            ▼
*/	
	
	int numberOfPixelInHeight,numberOfPixelInWidth;
	double heightOfSensor, widthOfSensor;
	double focalLength=1.5;
	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;
	
	double L = 0.2;
	Eigen::Matrix3Xd controlPointsInWorldCoordinate(3,6);
	Eigen::Matrix3Xd controlPointsInCameraCoordinate(3,6);

	
	Eigen::Matrix3d cameraIntrinsicMatrix;
	cameraIntrinsicMatrix<<focalLength*mx, 0, V0,
						   0,focalLength*my,U0,
						   0,0,1;

	controlPointsInWorldCoordinate.col(0)= Eigen::Vector3d(-L, -L, 0);
	controlPointsInWorldCoordinate.col(1)= Eigen::Vector3d(2 * L, -L, 0.2);
	controlPointsInWorldCoordinate.col(2)= Eigen::Vector3d(L, L, 0.2);
	controlPointsInWorldCoordinate.col(3)= Eigen::Vector3d(-L, L, 0);
	controlPointsInWorldCoordinate.col(4)= Eigen::Vector3d(-2 * L, L, 0);
	controlPointsInWorldCoordinate.col(5)= Eigen::Vector3d(0, 0, 0.5);
	
	
	Eigen::Matrix3d rotationMatrix;
	rotationMatrix<< 1,0,0
					,0,1,0
					,0,0,1;

	Eigen::Vector3d translationVector(3,1);
	translationVector<<-0.1, 0.1, 1.2;
	
	transformPoints(controlPointsInWorldCoordinate, rotationMatrix, translationVector, controlPointsInCameraCoordinate );

	Eigen::Matrix2Xd pointsInCameraPlane(2,controlPointsInCameraCoordinate.cols());
	projectFromCameraCoordinateToCameraPlane(controlPointsInCameraCoordinate,cameraIntrinsicMatrix,pointsInCameraPlane);
    
    
    std::vector<cv::Point3d> cvPointsInCameraCoordinate;
    std::vector<cv::Point2d> cvpointsInCameraPlane;
    cvPointsInCameraCoordinate.push_back(cv::Point3d(-L, -L, 0));
    cvPointsInCameraCoordinate.push_back(cv::Point3d(2 * L, -L, 0.2));
    cvPointsInCameraCoordinate.push_back(cv::Point3d(L, L, 0.2));
    cvPointsInCameraCoordinate.push_back(cv::Point3d(-L, L, 0));
    cvPointsInCameraCoordinate.push_back(cv::Point3d(-2 * L, L, 0));
    cvPointsInCameraCoordinate.push_back(cv::Point3d(0, 0, 0.5));


    cv::Mat cameraMatrix= (cv::Mat_<double>(3,3) <<
    focalLength*mx, 0, V0,
    0,focalLength*my,U0,
    0,0,1);


    cv::Mat rvec= cv::Mat::eye(3,3, CV_64F);
    cv::Mat tvec=(cv::Mat_<double>(3,1)<<-0.1, 0.1, 1.2);
    cv::projectPoints(cvPointsInCameraCoordinate,rvec,tvec,cameraMatrix,cv::noArray(),cvpointsInCameraPlane);
	
    std::cout<<"cameraIntrinsicMatrix" <<std::endl;
    std::cout<<cameraIntrinsicMatrix <<std::endl;

    std::cout<<cameraMatrix <<std::endl;

	
	
    Eigen::MatrixXd image(numberOfPixelInHeight,numberOfPixelInWidth);
    image=MatrixXd::Zero(numberOfPixelInHeight,numberOfPixelInWidth);
    int U,V;
    for(int i=0;i<pointsInCameraPlane.cols();i++)
    {
        U=int(pointsInCameraPlane(0,i));
        V=int(pointsInCameraPlane(1,i));
        std::cout<<U<<","<<V <<std::endl;
        image(U,V)=255;
        std::cout<<cvpointsInCameraPlane.at(i)<<std::endl;

    }
    Mat dst;
    eigen2cv(image, dst);
    std::string fileName=std::string("eigen_camera_projection_result_focal_")+std::to_string(focalLength)+ std::string("_.jpg");
    cv::imwrite(fileName,dst);
}

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void projectFromCameraCoordinateToCameraPlane( Eigen::Matrix3Xd &pointsInCameraCoordinate, Eigen::Matrix3d &cameraIntrinsicMatrix,Eigen::Matrix2Xd &pointsInCameraPlane)

{

Intrinsic camera Matrix is something like this:

focalLength ===> unit is mm

mx= ==>unit is Pixel/mm

U0= ===> unit is Pixel

my=-(numberOfPixelInHeight)/heightOfSensor ;

U0=(numberOfPixelInHeight)/2 ;

mx=(numberOfPixelInWidth)/heightOfSensor;

V0=(numberOfPixelInWidth)/2;

Gamma=0

X,Y,Z,W are homogeneous position of point in camera coordinate

┌ ┐ ┌ ┐┌ ┐

v` |f*mx Gamma V0 0||X|

u` =|0 f*my U0 0||Y|

w` |0 0 1 0||Z|

└ ┘ └ ┘└W┘

▲y

----►V |

| ____________________|_____________________

| |__|__|__|__|__|__|__|__|__|__|__|__|__|__|

U ▼ |__|__|__|__|__|__|__|__|__|__|__|__|__|__|

|__|__|__|__|__|__|__|__|__|__|__|__|__|__|

|__|__|__|__|__|__|__|__|__|__|__|__|__|__|____________► x

|__|__|__|__|__|__|__|__|__|__|__|__|__|__|

To project points from camera coordinate to camera plane:

V`=X*f*mx + V0*Z

U`=Y*f*my + U0*Z

W`=Z

U=U`/W` ,V=V`/W`

V=f*mx*X/Z + V0

U=f*my*Y/Z + U0

(U,V) is the index of corresponding point in the image

Eigen::Matrix3Xd pointsInCameraPlaneHomogeneous(pointsInCameraCoordinate.rows(),pointsInCameraCoordinate.cols());

pointsInCameraPlaneHomogeneous=cameraIntrinsicMatrix*pointsInCameraCoordinate;

std::cout<<"pointsInCameraPlaneHomogeneous"<<std::endl;

std::cout<<pointsInCameraPlaneHomogeneous<<std::endl;

for(int i=0;i<pointsInCameraPlaneHomogeneous.cols();i++ )

{

pointsInCameraPlaneHomogeneous(0,i)=pointsInCameraPlaneHomogeneous(0,i)/pointsInCameraPlaneHomogeneous(2,i);

pointsInCameraPlaneHomogeneous(1,i)=pointsInCameraPlaneHomogeneous(1,i)/pointsInCameraPlaneHomogeneous(2,i);

}

pointsInCameraPlane=pointsInCameraPlaneHomogeneous.block(0,0,pointsInCameraPlaneHomogeneous.rows()-1,pointsInCameraPlaneHomogeneous.cols());

}

void transformPoints(Eigen::Matrix3Xd &points, Eigen::Matrix3d &rotationMatrix, Eigen::Vector3d &translationVector,Eigen::Matrix3Xd &transformedPoints)

{

transformedPoints=(rotationMatrix*points).colwise() + translationVector ;

}

void projectFromCameraCoordinateToCameraPlane_Example()

{

OpenCV coordinate system

▲

------------------------------► X

Y |

▼

int numberOfPixelInHeight,numberOfPixelInWidth;

double heightOfSensor, widthOfSensor;

double focalLength=1.5;

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;

double L = 0.2;

Eigen::Matrix3Xd controlPointsInWorldCoordinate(3,6);

Eigen::Matrix3Xd controlPointsInCameraCoordinate(3,6);

Eigen::Matrix3d cameraIntrinsicMatrix;

cameraIntrinsicMatrix<<focalLength*mx, 0, V0,

0,focalLength*my,U0,

0,0,1;

controlPointsInWorldCoordinate.col(0)= Eigen::Vector3d(-L, -L, 0);

controlPointsInWorldCoordinate.col(1)= Eigen::Vector3d(2 * L, -L, 0.2);

controlPointsInWorldCoordinate.col(2)= Eigen::Vector3d(L, L, 0.2);

controlPointsInWorldCoordinate.col(3)= Eigen::Vector3d(-L, L, 0);

controlPointsInWorldCoordinate.col(4)= Eigen::Vector3d(-2 * L, L, 0);

controlPointsInWorldCoordinate.col(5)= Eigen::Vector3d(0, 0, 0.5);

Eigen::Matrix3d rotationMatrix;

rotationMatrix<< 1,0,0

,0,1,0

,0,0,1;

Eigen::Vector3d translationVector(3,1);

translationVector<<-0.1, 0.1, 1.2;

transformPoints(controlPointsInWorldCoordinate, rotationMatrix, translationVector, controlPointsInCameraCoordinate );

Eigen::Matrix2Xd pointsInCameraPlane(2,controlPointsInCameraCoordinate.cols());

projectFromCameraCoordinateToCameraPlane(controlPointsInCameraCoordinate,cameraIntrinsicMatrix,pointsInCameraPlane);

std::vector<cv::Point3d> cvPointsInCameraCoordinate;

std::vector<cv::Point2d> cvpointsInCameraPlane;

cvPointsInCameraCoordinate.push_back(cv::Point3d(-L, -L, 0));

cvPointsInCameraCoordinate.push_back(cv::Point3d(2 * L, -L, 0.2));

cvPointsInCameraCoordinate.push_back(cv::Point3d(L, L, 0.2));

cvPointsInCameraCoordinate.push_back(cv::Point3d(-L, L, 0));

cvPointsInCameraCoordinate.push_back(cv::Point3d(-2 * L, L, 0));

cvPointsInCameraCoordinate.push_back(cv::Point3d(0, 0, 0.5));

cv::Mat cameraMatrix= (cv::Mat_<double>(3,3) <<

focalLength*mx, 0, V0,

0,focalLength*my,U0,

0,0,1);

cv::Mat rvec= cv::Mat::eye(3,3, CV_64F);

cv::Mat tvec=(cv::Mat_<double>(3,1)<<-0.1, 0.1, 1.2);

cv::projectPoints(cvPointsInCameraCoordinate,rvec,tvec,cameraMatrix,cv::noArray(),cvpointsInCameraPlane);

std::cout<<"cameraIntrinsicMatrix" <<std::endl;

std::cout<<cameraIntrinsicMatrix <<std::endl;

std::cout<<cameraMatrix <<std::endl;

Eigen::MatrixXd image(numberOfPixelInHeight,numberOfPixelInWidth);

image=MatrixXd::Zero(numberOfPixelInHeight,numberOfPixelInWidth);

int U,V;

for(int i=0;i<pointsInCameraPlane.cols();i++)

{

U=int(pointsInCameraPlane(0,i));

V=int(pointsInCameraPlane(1,i));

std::cout<<U<<","<<V <<std::endl;

image(U,V)=255;

std::cout<<cvpointsInCameraPlane.at(i)<<std::endl;

}

Mat dst;

eigen2cv(image, dst);

std::string fileName=std::string("eigen_camera_projection_result_focal_")+std::to_string(focalLength)+ std::string("_.jpg");

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