public static void LeastSquares(Matrix x, Matrix A, Matrix b)
{
// use svd
// for overdetermined systems A*x = b
// x = V * diag(1/wj) * U T * b
// NRC p. 66
int m = A.m;
int n = A.n;
Matrix U = new Matrix(m, n), V = new Matrix(n, n), w = new Matrix(n, 1), W = new Matrix(n, n);
A.SVD(U, w, V);
w.Reciprocal();
W.Diag(w);
Matrix M = new Matrix(n, n);
M.Mult(V, W);
Matrix N = new Matrix(n, m);
N.MultAAT(M, U);
x.Mult(N, b);
}
// Use DLT to obtain estimate of calibration rig pose; in our case this is the pose of the Kinect camera.
// This pose estimate will provide a good initial estimate for subsequent projector calibration.
// Note for a full PnP solution we should probably refine with Levenberg-Marquardt.
// DLT is described in Hartley and Zisserman p. 178
public static void DLT(Matrix cameraMatrix, Matrix distCoeffs, List <Matrix> worldPoints, List <System.Drawing.PointF> imagePoints, out Matrix R, out Matrix t)
{
int n = worldPoints.Count;
var A = Matrix.Zero(2 * n, 12);
for (int j = 0; j < n; j++)
{
var X = worldPoints[j];
var imagePoint = imagePoints[j];
double x, y;
Undistort(cameraMatrix, distCoeffs, imagePoint.X, imagePoint.Y, out x, out y);
int ii = 2 * j;
A[ii, 4] = -X[0];
A[ii, 5] = -X[1];
A[ii, 6] = -X[2];
A[ii, 7] = -1;
A[ii, 8] = y * X[0];
A[ii, 9] = y * X[1];
A[ii, 10] = y * X[2];
A[ii, 11] = y;
ii++; // next row
A[ii, 0] = X[0];
A[ii, 1] = X[1];
A[ii, 2] = X[2];
A[ii, 3] = 1;
A[ii, 8] = -x * X[0];
A[ii, 9] = -x * X[1];
A[ii, 10] = -x * X[2];
A[ii, 11] = -x;
}
// Pcolumn is the eigenvector of ATA with the smallest eignvalue
var Pcolumn = new Matrix(12, 1);
{
var ATA = new Matrix(12, 12);
ATA.MultATA(A, A);
var V = new Matrix(12, 12);
var ww = new Matrix(12, 1);
ATA.Eig(V, ww);
Pcolumn.CopyCol(V, 0);
}
// reshape into 3x4 projection matrix
var P = new Matrix(3, 4);
P.Reshape(Pcolumn);
R = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
{
for (int j = 0; j < 3; j++)
{
R[i, j] = P[i, j];
}
}
if (R.Det3x3() < 0)
{
R.Scale(-1);
P.Scale(-1);
}
// orthogonalize R
{
var U = new Matrix(3, 3);
var V = new Matrix(3, 3);
var ww = new Matrix(3, 1);
R.SVD(U, ww, V);
R.MultAAT(U, V);
}
// determine scale factor
var RP = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
{
for (int j = 0; j < 3; j++)
{
RP[i, j] = P[i, j];
}
}
double s = RP.Norm() / R.Norm();
t = new Matrix(3, 1);
for (int i = 0; i < 3; i++)
{
t[i] = P[i, 3];
}
t.Scale(1.0 / s);
}
// Use DLT to obtain estimate of calibration rig pose; in our case this is the pose of the Kinect camera.
// This pose estimate will provide a good initial estimate for subsequent projector calibration.
// Note for a full PnP solution we should probably refine with Levenberg-Marquardt.
// DLT is described in Hartley and Zisserman p. 178
public static void DLT(Matrix cameraMatrix, Matrix distCoeffs, List<Matrix> worldPoints, List<System.Drawing.PointF> imagePoints, out Matrix R, out Matrix t)
{
int n = worldPoints.Count;
var A = Matrix.Zero(2 * n, 12);
for (int j = 0; j < n; j++)
{
var X = worldPoints[j];
var imagePoint = imagePoints[j];
double x, y;
Undistort(cameraMatrix, distCoeffs, imagePoint.X, imagePoint.Y, out x, out y);
int ii = 2 * j;
A[ii, 4] = -X[0];
A[ii, 5] = -X[1];
A[ii, 6] = -X[2];
A[ii, 7] = -1;
A[ii, 8] = y * X[0];
A[ii, 9] = y * X[1];
A[ii, 10] = y * X[2];
A[ii, 11] = y;
ii++; // next row
A[ii, 0] = X[0];
A[ii, 1] = X[1];
A[ii, 2] = X[2];
A[ii, 3] = 1;
A[ii, 8] = -x * X[0];
A[ii, 9] = -x * X[1];
A[ii, 10] = -x * X[2];
A[ii, 11] = -x;
}
// Pcolumn is the eigenvector of ATA with the smallest eignvalue
var Pcolumn = new Matrix(12, 1);
{
var ATA = new Matrix(12, 12);
ATA.MultATA(A, A);
var V = new Matrix(12, 12);
var ww = new Matrix(12, 1);
ATA.Eig(V, ww);
Pcolumn.CopyCol(V, 0);
}
// reshape into 3x4 projection matrix
var P = new Matrix(3, 4);
P.Reshape(Pcolumn);
R = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
R[i, j] = P[i, j];
if (R.Det3x3() < 0)
{
R.Scale(-1);
P.Scale(-1);
}
// orthogonalize R
{
var U = new Matrix(3, 3);
var V = new Matrix(3, 3);
var ww = new Matrix(3, 1);
R.SVD(U, ww, V);
R.MultAAT(U, V);
}
// determine scale factor
var RP = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
for (int j = 0; j < 3; j++)
RP[i, j] = P[i, j];
double s = RP.Norm() / R.Norm();
t = new Matrix(3, 1);
for (int i = 0; i < 3; i++)
t[i] = P[i, 3];
t.Scale(1.0 / s);
}
public static void LeastSquares(Matrix x, Matrix A, Matrix b)
{
// use svd
// for overdetermined systems A*x = b
// x = V * diag(1/wj) * U T * b
// NRC p. 66
int m = A.m;
int n = A.n;
Matrix U = new Matrix(m, n), V = new Matrix(n, n), w = new Matrix(n, 1), W = new Matrix(n, n);
A.SVD(U, w, V);
w.Reciprocal();
W.Diag(w);
Matrix M = new Matrix(n, n);
M.Mult(V, W);
Matrix N = new Matrix(n, m);
N.MultAAT(M, U);
x.Mult(N, b);
}
