public static Matrix Homography(List<Matrix> worldPoints, List<System.Drawing.PointF> imagePoints)
{
int n = worldPoints.Count;
// normalize image coordinates
var mu = new Matrix(2, 1);
for (int i = 0; i < n; i++)
{
mu[0] += imagePoints[i].X;
mu[1] += imagePoints[i].Y;
}
mu.Scale(1.0 / n);
var muAbs = new Matrix(2, 1);
for (int i = 0; i < n; i++)
{
muAbs[0] += Math.Abs(imagePoints[i].X - mu[0]);
muAbs[1] += Math.Abs(imagePoints[i].Y - mu[1]);
}
muAbs.Scale(1.0 / n);
var Hnorm = Matrix.Identity(3, 3);
Hnorm[0, 0] = 1 / muAbs[0];
Hnorm[1, 1] = 1 / muAbs[1];
Hnorm[0, 2] = -mu[0] / muAbs[0];
Hnorm[1, 2] = -mu[1] / muAbs[1];
var invHnorm = Matrix.Identity(3, 3);
invHnorm[0, 0] = muAbs[0];
invHnorm[1, 1] = muAbs[1];
invHnorm[0, 2] = mu[0];
invHnorm[1, 2] = mu[1];
var A = Matrix.Zero(2 * n, 9);
for (int i = 0; i < n; i++)
{
var X = worldPoints[i];
var imagePoint = imagePoints[i];
var x = new Matrix(3, 1);
x[0] = imagePoint.X;
x[1] = imagePoint.Y;
x[2] = 1;
var xn = new Matrix(3, 1);
xn.Mult(Hnorm, x);
// Zhang's formulation; Hartley's is similar
int ii = 2 * i;
A[ii, 0] = X[0];
A[ii, 1] = X[1];
A[ii, 2] = 1;
A[ii, 6] = -xn[0] * X[0];
A[ii, 7] = -xn[0] * X[1];
A[ii, 8] = -xn[0];
ii++; // next row
A[ii, 3] = X[0];
A[ii, 4] = X[1];
A[ii, 5] = 1;
A[ii, 6] = -xn[1] * X[0];
A[ii, 7] = -xn[1] * X[1];
A[ii, 8] = -xn[1];
}
// h is the eigenvector of ATA with the smallest eignvalue
var h = new Matrix(9, 1);
{
var ATA = new Matrix(9, 9);
ATA.MultATA(A, A);
var V = new Matrix(9, 9);
var ww = new Matrix(9, 1);
ATA.Eig(V, ww);
h.CopyCol(V, 0);
}
var Hn = new Matrix(3, 3);
Hn.Reshape(h);
var H = new Matrix(3, 3);
H.Mult(invHnorm, Hn);
return H;
}
// 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);
}
static public void PlaneFit(IList<Matrix> X, out Matrix R, out Matrix t, out Matrix d2)
{
int n = X.Count;
var mu = new Matrix(3, 1);
for (int i = 0; i < n; i++)
mu.Add(X[i]);
mu.Scale(1f / (float)n);
var A = new Matrix(3, 3);
var xc = new Matrix(3, 1);
var M = new Matrix(3, 3);
for (int i = 0; i < X.Count; i++)
{
var x = X[i];
xc.Sub(x, mu);
M.Outer(xc, xc);
A.Add(M);
}
var V = new Matrix(3, 3);
var d = new Matrix(3, 1);
A.Eig(V, d); // eigenvalues in ascending order
// arrange in descending order so that z = 0
var V2 = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
{
V2[i, 2] = V[i, 0];
V2[i, 1] = V[i, 1];
V2[i, 0] = V[i, 2];
}
d2 = new Matrix(3, 1);
d2[2] = d[0];
d2[1] = d[1];
d2[0] = d[2];
R = new Matrix(3, 3);
R.Transpose(V2);
if (R.Det3x3() < 0)
R.Scale(-1);
t = new Matrix(3, 1);
t.Mult(R, mu);
t.Scale(-1);
// eigenvalues are the sum of squared distances in each direction
// i.e., min eigenvalue is the sum of squared distances to the plane = d2[2]
// compute the distance to the plane by transforming to the plane and take z-coordinate:
// xPlane = R*x + t; distance = xPlane[2]
}
// 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 Matrix Homography(List <Matrix> worldPoints, List <System.Drawing.PointF> imagePoints)
{
int n = worldPoints.Count;
// normalize image coordinates
var mu = new Matrix(2, 1);
for (int i = 0; i < n; i++)
{
mu[0] += imagePoints[i].X;
mu[1] += imagePoints[i].Y;
}
mu.Scale(1.0 / n);
var muAbs = new Matrix(2, 1);
for (int i = 0; i < n; i++)
{
muAbs[0] += Math.Abs(imagePoints[i].X - mu[0]);
muAbs[1] += Math.Abs(imagePoints[i].Y - mu[1]);
}
muAbs.Scale(1.0 / n);
var Hnorm = Matrix.Identity(3, 3);
Hnorm[0, 0] = 1 / muAbs[0];
Hnorm[1, 1] = 1 / muAbs[1];
Hnorm[0, 2] = -mu[0] / muAbs[0];
Hnorm[1, 2] = -mu[1] / muAbs[1];
var invHnorm = Matrix.Identity(3, 3);
invHnorm[0, 0] = muAbs[0];
invHnorm[1, 1] = muAbs[1];
invHnorm[0, 2] = mu[0];
invHnorm[1, 2] = mu[1];
var A = Matrix.Zero(2 * n, 9);
for (int i = 0; i < n; i++)
{
var X = worldPoints[i];
var imagePoint = imagePoints[i];
var x = new Matrix(3, 1);
x[0] = imagePoint.X;
x[1] = imagePoint.Y;
x[2] = 1;
var xn = new Matrix(3, 1);
xn.Mult(Hnorm, x);
// Zhang's formulation; Hartley's is similar
int ii = 2 * i;
A[ii, 0] = X[0];
A[ii, 1] = X[1];
A[ii, 2] = 1;
A[ii, 6] = -xn[0] * X[0];
A[ii, 7] = -xn[0] * X[1];
A[ii, 8] = -xn[0];
ii++; // next row
A[ii, 3] = X[0];
A[ii, 4] = X[1];
A[ii, 5] = 1;
A[ii, 6] = -xn[1] * X[0];
A[ii, 7] = -xn[1] * X[1];
A[ii, 8] = -xn[1];
}
// h is the eigenvector of ATA with the smallest eignvalue
var h = new Matrix(9, 1);
{
var ATA = new Matrix(9, 9);
ATA.MultATA(A, A);
var V = new Matrix(9, 9);
var ww = new Matrix(9, 1);
ATA.Eig(V, ww);
h.CopyCol(V, 0);
}
var Hn = new Matrix(3, 3);
Hn.Reshape(h);
var H = new Matrix(3, 3);
H.Mult(invHnorm, Hn);
return(H);
}
static public void PlaneFit(IList <Matrix> X, out Matrix R, out Matrix t, out Matrix d2)
{
int n = X.Count;
var mu = new Matrix(3, 1);
for (int i = 0; i < n; i++)
{
mu.Add(X[i]);
}
mu.Scale(1f / (float)n);
var A = new Matrix(3, 3);
var xc = new Matrix(3, 1);
var M = new Matrix(3, 3);
for (int i = 0; i < X.Count; i++)
{
var x = X[i];
xc.Sub(x, mu);
M.Outer(xc, xc);
A.Add(M);
}
var V = new Matrix(3, 3);
var d = new Matrix(3, 1);
A.Eig(V, d); // eigenvalues in ascending order
// arrange in descending order so that z = 0
var V2 = new Matrix(3, 3);
for (int i = 0; i < 3; i++)
{
V2[i, 2] = V[i, 0];
V2[i, 1] = V[i, 1];
V2[i, 0] = V[i, 2];
}
d2 = new Matrix(3, 1);
d2[2] = d[0];
d2[1] = d[1];
d2[0] = d[2];
R = new Matrix(3, 3);
R.Transpose(V2);
if (R.Det3x3() < 0)
{
R.Scale(-1);
}
t = new Matrix(3, 1);
t.Mult(R, mu);
t.Scale(-1);
// eigenvalues are the sum of squared distances in each direction
// i.e., min eigenvalue is the sum of squared distances to the plane = d2[2]
// compute the distance to the plane by transforming to the plane and take z-coordinate:
// xPlane = R*x + t; distance = xPlane[2]
}
public static double PlaneFit(IList<Matrix> points, out Matrix X, out double D)
{
X = new Matrix(3, 1);
var mu = new RoomAliveToolkit.Matrix(3, 1);
for (int i = 0; i < points.Count; i++)
mu.Add(points[i]);
mu.Scale(1f / (float)points.Count);
var A = new RoomAliveToolkit.Matrix(3, 3);
var pc = new RoomAliveToolkit.Matrix(3, 1);
var M = new RoomAliveToolkit.Matrix(3, 3);
for (int i = 0; i < points.Count; i++)
{
var p = points[i];
pc.Sub(p, mu);
M.Outer(pc, pc);
A.Add(M);
}
var V = new RoomAliveToolkit.Matrix(3, 3);
var d = new RoomAliveToolkit.Matrix(3, 1);
A.Eig(V, d); // TODO: replace with 3x3 version?
//Console.WriteLine("------");
//Console.WriteLine(A);
//Console.WriteLine(V);
//Console.WriteLine(d);
double minEigenvalue = Double.MaxValue;
int minEigenvaluei = 0;
for (int i = 0; i < 3; i++)
if (d[i] < minEigenvalue)
{
minEigenvalue = d[i];
minEigenvaluei = i;
}
X.CopyCol(V, minEigenvaluei);
D = -X.Dot(mu);
// min eigenvalue is the sum of squared distances to the plane
// signed distance is: double distance = X.Dot(point) + D;
return minEigenvalue;
}