Pose estimation for augmented reality
Pose from a non-linear minimization method

Table of Contents

Introduction

The "gold-standard" solution to the PnP consists in estimating the six parameters of the transformation cTw by minimizing the forward projection error using a Gauss-Newton approach. A complete derivation of this problem is given in [2].

Source code

The following source code that uses ViSP is also available in pose-gauss-newton-visp.cpp file. It allows to compute the pose of the camera from points.

#include <visp/vpColVector.h>
#include <visp/vpExponentialMap.h>
#include <visp/vpHomogeneousMatrix.h>
#include <visp/vpMatrix.h>
vpHomogeneousMatrix pose_gauss_newton(
#if VISP_VERSION_INT >= VP_VERSION_INT(2, 10, 0)
const
#endif
std::vector< vpColVector > &wX, const std::vector< vpColVector > &x, const vpHomogeneousMatrix &cTw)
{
int npoints = (int)wX.size();
vpMatrix J(2*npoints, 6), Jp;
vpColVector err, sd(2*npoints), s(2*npoints), xq(npoints*2), xn(npoints*2);
vpHomogeneousMatrix cTw_ = cTw;
double residual=0, residual_prev, lambda = 0.25;
// From input vector x = (x, y, 1)^T we create a new one xn = (x, y)^T to ease computation of e_q
for (int i = 0; i < x.size(); i ++) {
xn[i*2] = x[i][0]; // x
xn[i*2+1] = x[i][1]; // y
}
// Iterative Gauss-Newton minimization loop
do {
for (int i = 0; i < npoints; i++) {
vpColVector cX = cTw_ * wX[i]; // Update cX, cY, cZ
double Xi = cX[0];
double Yi = cX[1];
double Zi = cX[2];
double xi = Xi / Zi;
double yi = Yi / Zi;
// Update x(q)
xq[i*2] = xi; // x(q) = cX/cZ
xq[i*2+1] = yi; // y(q) = cY/cZ
// Update J using equation (11)
J[i*2][0] = -1 / Zi; // -1/cZ
J[i*2][1] = 0; // 0
J[i*2][2] = xi / Zi; // x/cZ
J[i*2][3] = xi * yi; // xy
J[i*2][4] = -(1 + xi * xi); // -(1+x^2)
J[i*2][5] = yi; // y
J[i*2+1][0] = 0; // 0
J[i*2+1][1] = -1 / Zi; // -1/cZ
J[i*2+1][2] = yi / Zi; // y/cZ
J[i*2+1][3] = 1 + yi * yi; // 1+y^2
J[i*2+1][4] = -xi * yi; // -xy
J[i*2+1][5] = -xi; // -x
}
vpColVector e_q = xq - xn; // Equation (7)
J.pseudoInverse(Jp); // Compute pseudo inverse of the Jacobian
vpColVector dq = -lambda * Jp * e_q; // Equation (10)
cTw_ = vpExponentialMap::direct(dq).inverse() * cTw_; // Update the pose
residual_prev = residual; // Memorize previous residual
residual = e_q.sumSquare(); // Compute the actual residual
} while (fabs(residual - residual_prev) > 0);
return cTw_;
}
int main()
{
int npoints = 4;
std::vector< vpColVector > wX(npoints);
std::vector< vpColVector > x(npoints);
for (int i = 0; i < npoints; i++) {
wX[i].resize(4);
x[i].resize(3);
}
// Ground truth pose used to generate the data
vpHomogeneousMatrix cTw_truth(-0.1, 0.1, 0.5, vpMath::rad(5), vpMath::rad(0), vpMath::rad(45));
// Input data: 3D coordinates of at least 4 points
double L = 0.2;
wX[0][0] = -L; wX[0][1] = -L; wX[0][2] = 0; wX[0][3] = 1; // wX_0 ( -L, -L, 0, 1)^T
wX[1][0] = 2*L; wX[1][1] = -L; wX[1][2] = 0; wX[1][3] = 1; // wX_1 (-2L, -L, 0, 1)^T
wX[2][0] = L; wX[2][1] = L; wX[2][2] = 0; wX[2][3] = 1; // wX_2 ( L, L, 0, 1)^T
wX[3][0] = -L; wX[3][1] = L; wX[3][2] = 0; wX[3][3] = 1; // wX_3 ( -L, L, 0, 1)^T
// Input data: 2D coordinates of the points on the image plane
for(int i = 0; i < npoints; i++) {
vpColVector cX = cTw_truth * wX[i]; // Update cX, cY, cZ
x[i][0] = cX[0] / cX[2]; // x = cX/cZ
x[i][1] = cX[1] / cX[2]; // y = cY/cZ
x[i][2] = 1;
}
// Initialize the pose to estimate near the solution
vpHomogeneousMatrix cTw(-0.05, 0.05, 0.45, vpMath::rad(1), vpMath::rad(0), vpMath::rad(35));
cTw = pose_gauss_newton(wX, x, cTw);
std::cout << "cTw (ground truth):\n" << cTw_truth << std::endl;
std::cout << "cTw (from non linear method):\n" << cTw << std::endl;
return 0;
}

Source code explained

First of all we include ViSP headers that are requested to manipulate vectors and matrices and also to compute the exponential map.

#include <visp/vpColVector.h>
#include <visp/vpExponentialMap.h>
#include <visp/vpHomogeneousMatrix.h>
#include <visp/vpMatrix.h>

Then we introduce the function that does the pose estimation. It takes as input parameters ${^w}{\bf X} = (X,Y,Z,1)^T$ the 3D coordinates of the points in the world frame and ${\bf x} = (x,y,1)^T$ their normalized coordinates in the image plane. It returns the estimated pose as an homogeneous matrix transformation.

vpHomogeneousMatrix pose_gauss_newton(
#if VISP_VERSION_INT >= VP_VERSION_INT(2, 10, 0)
const
#endif
std::vector< vpColVector > &wX, const std::vector< vpColVector > &x, const vpHomogeneousMatrix &cTw)

The implementation of the iterative Gauss-Newton minimization process is done next. First, we create the variables used by the algorithm. Since the algorithm needs an initialization, we set an initial value in cTw not to far from the solution. Such an initialization could be done using Pose from Direct Linear Transform method of Pose from Dementhon's POSIT method approaches. Finally, the estimation is iterated.

int npoints = (int)wX.size();
vpMatrix J(2*npoints, 6), Jp;
vpColVector err, sd(2*npoints), s(2*npoints), xq(npoints*2), xn(npoints*2);
vpHomogeneousMatrix cTw_ = cTw;
double residual=0, residual_prev, lambda = 0.25;
// From input vector x = (x, y, 1)^T we create a new one xn = (x, y)^T to ease computation of e_q
for (int i = 0; i < x.size(); i ++) {
xn[i*2] = x[i][0]; // x
xn[i*2+1] = x[i][1]; // y
}
// Iterative Gauss-Newton minimization loop
do {
for (int i = 0; i < npoints; i++) {
vpColVector cX = cTw_ * wX[i]; // Update cX, cY, cZ
double Xi = cX[0];
double Yi = cX[1];
double Zi = cX[2];
double xi = Xi / Zi;
double yi = Yi / Zi;
// Update x(q)
xq[i*2] = xi; // x(q) = cX/cZ
xq[i*2+1] = yi; // y(q) = cY/cZ
// Update J using equation (11)
J[i*2][0] = -1 / Zi; // -1/cZ
J[i*2][1] = 0; // 0
J[i*2][2] = xi / Zi; // x/cZ
J[i*2][3] = xi * yi; // xy
J[i*2][4] = -(1 + xi * xi); // -(1+x^2)
J[i*2][5] = yi; // y
J[i*2+1][0] = 0; // 0
J[i*2+1][1] = -1 / Zi; // -1/cZ
J[i*2+1][2] = yi / Zi; // y/cZ
J[i*2+1][3] = 1 + yi * yi; // 1+y^2
J[i*2+1][4] = -xi * yi; // -xy
J[i*2+1][5] = -xi; // -x
}
vpColVector e_q = xq - xn; // Equation (7)
J.pseudoInverse(Jp); // Compute pseudo inverse of the Jacobian
vpColVector dq = -lambda * Jp * e_q; // Equation (10)
cTw_ = vpExponentialMap::direct(dq).inverse() * cTw_; // Update the pose
residual_prev = residual; // Memorize previous residual
residual = e_q.sumSquare(); // Compute the actual residual
} while (fabs(residual - residual_prev) > 0);

When the residual obtained between two successive iterations can be neglected, we can exit the while loop and return the estimated value of the pose.

return cTw_;

Finally we define the main function in which we will initialize the input data before calling the previous function and computing the pose using non linear Gauss-Newton approach.

int main()

First in the main we create the data structures that will contain the 3D points coordinates wX in the world frame, their coordinates in the camera frame cX and their coordinates in the image plane x obtained after prerspective projection. Note here that at least 4 non coplanar points are requested to estimate the pose.

int npoints = 4;
std::vector< vpColVector > wX(npoints);
std::vector< vpColVector > x(npoints);
for (int i = 0; i < npoints; i++) {
wX[i].resize(4);
x[i].resize(3);
}

For our simulation we then initialize the input data from a ground truth pose cTw_truth. For each point we set in wX[i] the 3D coordinates in the world frame (wX, wY, wZ, 1) and compute in cX their 3D coordinates (cX, cY, cZ, 1) in the camera frame. Then in x[i] we update their coordinates (x, y) in the image plane, obtained by perspective projection.

// Ground truth pose used to generate the data
vpHomogeneousMatrix cTw_truth(-0.1, 0.1, 0.5, vpMath::rad(5), vpMath::rad(0), vpMath::rad(45));
// Input data: 3D coordinates of at least 4 points
double L = 0.2;
wX[0][0] = -L; wX[0][1] = -L; wX[0][2] = 0; wX[0][3] = 1; // wX_0 ( -L, -L, 0, 1)^T
wX[1][0] = 2*L; wX[1][1] = -L; wX[1][2] = 0; wX[1][3] = 1; // wX_1 (-2L, -L, 0, 1)^T
wX[2][0] = L; wX[2][1] = L; wX[2][2] = 0; wX[2][3] = 1; // wX_2 ( L, L, 0, 1)^T
wX[3][0] = -L; wX[3][1] = L; wX[3][2] = 0; wX[3][3] = 1; // wX_3 ( -L, L, 0, 1)^T
// Input data: 2D coordinates of the points on the image plane
for(int i = 0; i < npoints; i++) {
vpColVector cX = cTw_truth * wX[i]; // Update cX, cY, cZ
x[i][0] = cX[0] / cX[2]; // x = cX/cZ
x[i][1] = cX[1] / cX[2]; // y = cY/cZ
x[i][2] = 1;
}

From here we have initialized ${^w}{\bf X} = (X,Y,Z,1)^T$ and ${\bf x} = (x,y,1)^T$. Since the non-linear minimization method requires a initial value of the pose to estimate we initialize ${^c}{\bf T}_w$ not so far from the solution.

// Initialize the pose to estimate near the solution
vpHomogeneousMatrix cTw(-0.05, 0.05, 0.45, vpMath::rad(1), vpMath::rad(0), vpMath::rad(35));
Note
In a real application this initialization has to be done using:

We are now ready to call the function that does the pose estimation.

cTw = pose_gauss_newton(wX, x, cTw);

Resulting pose estimation

If you run the previous code, it we produce the following result that shows that the estimated pose is equal to the ground truth one used to generate the input data:

cTw (ground truth):
0.7072945484 -0.706170438 0.03252282796 -0.1
0.706170438 0.7036809008 -0.078463382 0.1
0.03252282796 0.078463382 0.9963863524 0.5
0 0 0 1
cTw (from non linear method):
0.7072945484 -0.706170438 0.03252282796 -0.1
0.706170438 0.7036809008 -0.078463382 0.1
0.03252282796 0.078463382 0.9963863524 0.5
0 0 0 1
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