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BoofCV is an open source Java library for real-time computer vision and robotics applications.
/*
* Copyright (c) 2021, Peter Abeles. All Rights Reserved.
*
* This file is part of BoofCV (http://boofcv.org).
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package boofcv.alg.geo.h;
import boofcv.alg.geo.LowLevelMultiViewOps;
import boofcv.alg.geo.NormalizationPoint2D;
import boofcv.struct.geo.AssociatedPair;
import org.ejml.data.DMatrix2x2;
import org.ejml.data.DMatrixRMaj;
import org.ejml.dense.fixed.CommonOps_DDF2;
import org.ejml.dense.row.CommonOps_DDRM;
import org.ejml.dense.row.linsol.svd.SolveNullSpaceSvd_DDRM;
import org.ejml.interfaces.SolveNullSpace;
import java.util.List;
/**
* Direct method for computing homography that is more computationally efficient and stable than DLT. Takes
* advantage of the sparse structure of the matrix found in {@link HomographyDirectLinearTransform DLT} to reduce
* the number of computations and EYM matrix approximation. See the paper [1] for details. Requires at least
* four points.
*
*
* [1] Harker, Matthew, and Paul O'Leary. "Computation of Homographies." BMVC. 2005.
*
* @author Peter Abeles
*/
public class HomographyTotalLeastSquares {
// Solver for null space
SolveNullSpace solverNull = new SolveNullSpaceSvd_DDRM();
DMatrixRMaj nullspace = new DMatrixRMaj(3, 1);
// Used to normalize input points
protected NormalizationPoint2D N1 = new NormalizationPoint2D();
protected NormalizationPoint2D N2 = new NormalizationPoint2D();
// X1 = (x[i],y[i]; ... ) and X2 = (x[i]',y[i]'; ... )
// X1 = hat{P} X2 = X' <-- paper notation
DMatrixRMaj X1 = new DMatrixRMaj(1, 1);
DMatrixRMaj X2 = new DMatrixRMaj(1, 1);
// Reduced matrix for solving for 6,7,8
DMatrixRMaj A = new DMatrixRMaj(1, 1);
// Storage for intermediate steps
private DMatrixRMaj P_plus = new DMatrixRMaj(1, 1);
private double XP_bar[] = new double[4];
/**
*
* Computes the homography matrix given a set of observed points in two images. A set of {@link AssociatedPair}
* is passed in. The computed homography 'H' is found such that the attributes 'p1' and 'p2' in {@link AssociatedPair}
* refers to x1 and x2, respectively, in the equation below:
* x2 = H*x1
*
*
* @param points A set of observed image points that are generated from a planar object. Minimum of 4 pairs required.
* @param foundH Output: Storage for the found solution. 3x3 matrix.
* @return true if the calculation was a success.
*/
public boolean process( List points, DMatrixRMaj foundH ) {
if (points.size() < 4)
throw new IllegalArgumentException("Must be at least 4 points.");
// Have to normalize the points. Being zero mean is essential in its derivation
LowLevelMultiViewOps.computeNormalization(points, N1, N2);
LowLevelMultiViewOps.applyNormalization(points, N1, N2, X1, X2);
// Construct the linear system which is used to solve for H[6] to H[8]
constructA678();
// Solve for those elements using the null space
if (!solverNull.process(A, 1, nullspace))
return false;
DMatrixRMaj H = foundH;
H.data[6] = nullspace.data[0];
H.data[7] = nullspace.data[1];
H.data[8] = nullspace.data[2];
// Determine H[0] to H[5]
H.data[2] = -(XP_bar[0]*H.data[6] + XP_bar[1]*H.data[7]);
H.data[5] = -(XP_bar[2]*H.data[6] + XP_bar[3]*H.data[7]);
backsubstitution0134(P_plus, X1, X2, H.data);
// Remove the normalization
HomographyDirectLinearTransform.undoNormalizationH(foundH, N1, N2);
CommonOps_DDRM.scale(1.0/foundH.get(2, 2), foundH);
return true;
}
/**
* Backsubstitution for solving for 0,1 and 3,4 using solution for 6,7,8
*/
static void backsubstitution0134( DMatrixRMaj P_plus, DMatrixRMaj P, DMatrixRMaj X,
double H[] ) {
final int N = P.numRows;
DMatrixRMaj tmp = new DMatrixRMaj(N*2, 1);
double H6 = H[6];
double H7 = H[7];
double H8 = H[8];
for (int i = 0, index = 0; i < N; i++) {
double x = -X.data[index], y = -X.data[index + 1];
double sum = P.data[index++]*H6 + P.data[index++]*H7 + H8;
tmp.data[i] = x*sum;
tmp.data[i + N] = y*sum;
}
double h0 = 0, h1 = 0, h3 = 0, h4 = 0;
for (int i = 0; i < N; i++) {
double P_pls_0 = P_plus.data[i];
double P_pls_1 = P_plus.data[i + N];
double tmp_i = tmp.data[i];
double tmp_j = tmp.data[i + N];
h0 += P_pls_0*tmp_i;
h1 += P_pls_1*tmp_i;
h3 += P_pls_0*tmp_j;
h4 += P_pls_1*tmp_j;
}
H[0] = -h0;
H[1] = -h1;
H[3] = -h3;
H[4] = -h4;
}
/**
* Constructs equation for elements 6 to 8 in H
*/
void constructA678() {
final int N = X1.numRows;
// Pseudo-inverse of hat(p)
computePseudo(X1, P_plus);
DMatrixRMaj PPpXP = new DMatrixRMaj(1, 1);
DMatrixRMaj PPpYP = new DMatrixRMaj(1, 1);
computePPXP(X1, P_plus, X2, 0, PPpXP);
computePPXP(X1, P_plus, X2, 1, PPpYP);
DMatrixRMaj PPpX = new DMatrixRMaj(1, 1);
DMatrixRMaj PPpY = new DMatrixRMaj(1, 1);
computePPpX(X1, P_plus, X2, 0, PPpX);
computePPpX(X1, P_plus, X2, 1, PPpY);
//============ Equations 20
computeEq20(X2, X1, XP_bar);
//============ Equation 21
A.reshape(N*2, 3);
double XP_bar_x = XP_bar[0];
double XP_bar_y = XP_bar[1];
double YP_bar_x = XP_bar[2];
double YP_bar_y = XP_bar[3];
// Compute the top half of A
for (int i = 0, index = 0, indexA = 0; i < N; i++, index += 2) {
double x = -X2.data[i*2]; // X'
double P_hat_x = X1.data[index]; // hat{P}[0]
double P_hat_y = X1.data[index + 1]; // hat{P}[1]
// x'*hat{p} - bar{X*P} - PPpXP
A.data[indexA++] = x*P_hat_x - XP_bar_x - PPpXP.data[index];
A.data[indexA++] = x*P_hat_y - XP_bar_y - PPpXP.data[index + 1];
// X'*1 - PPx1
A.data[indexA++] = x - PPpX.data[i];
}
// Compute the bottom half of A
for (int i = 0, index = 0, indexA = N*3; i < N; i++, index += 2) {
double x = -X2.data[i*2 + 1];
double P_hat_x = X1.data[index];
double P_hat_y = X1.data[index + 1];
// x'*hat{p} - bar{X*P} - PPpXP
A.data[indexA++] = x*P_hat_x - YP_bar_x - PPpYP.data[index];
A.data[indexA++] = x*P_hat_y - YP_bar_y - PPpYP.data[index + 1];
// X'*1 - PPx1
A.data[indexA++] = x - PPpY.data[i];
}
}
static void computeEq20( DMatrixRMaj X, DMatrixRMaj P, double output[] ) {
final int N = X.numRows;
double a00 = 0, a01 = 0, a10 = 0, a11 = 0;
for (int i = 0, index = 0; i < N; i++, index += 2) {
double x1 = X.data[index];
double x2 = X.data[index + 1];
double p1 = P.data[index];
double p2 = P.data[index + 1];
a00 += x1*p1;
a01 += x1*p2;
a10 += x2*p1;
a11 += x2*p2;
}
output[0] = -a00/N;
output[1] = -a01/N;
output[2] = -a10/N;
output[3] = -a11/N;
}
/**
* Computes inv(AT*A)*AT
*/
static void computePseudo( DMatrixRMaj A, DMatrixRMaj output ) {
final int N = A.numRows;
DMatrix2x2 m = new DMatrix2x2();
DMatrix2x2 m_inv = new DMatrix2x2();
for (int i = 0, index = 0; i < N; i++) {
double a_i1 = A.data[index++];
double a_i2 = A.data[index++];
m.a11 += a_i1*a_i1;
m.a12 += a_i1*a_i2;
m.a22 += a_i2*a_i2;
}
m.a21 = m.a12;
CommonOps_DDF2.invert(m, m_inv);
output.reshape(2, N);
for (int i = 0, index = 0; i < N; i++) {
output.data[i] = A.data[index++]*m_inv.a11 + A.data[index++]*m_inv.a12;
}
int end = 2*N;
for (int i = N, A_index = 0; i < end; i++) {
output.data[i] = A.data[A_index++]*m_inv.a21 + A.data[A_index++]*m_inv.a22;
}
}
/**
* Computes P*P_plus*X*P. Takes in account the size of each matrix and does the multiplication in an order
* to minimize memory requirements. A naive implementation requires a temporary array of NxN
*
* @param X A diagonal matrix
*/
static void computePPXP( DMatrixRMaj P, DMatrixRMaj P_plus, DMatrixRMaj X, int offsetX, DMatrixRMaj output ) {
final int N = P.numRows;
output.reshape(N, 2);
// diag(X) * P <-- N x 2
for (int i = 0, index = 0; i < N; i++, index += 2) {
double x = -X.data[index + offsetX];
output.data[index] = x*P.data[index];
output.data[index + 1] = x*P.data[index + 1];
}
// A = P_plus*( diag(x)*P ) <-- 2 x 2
double a00 = 0, a01 = 0, a10 = 0, a11 = 0;
for (int i = 0, index = 0; i < N; i++, index += 2) {
a00 += P_plus.data[i]*output.data[index];
a01 += P_plus.data[i]*output.data[index + 1];
a10 += P_plus.data[i + N]*output.data[index];
a11 += P_plus.data[i + N]*output.data[index + 1];
}
// P*A <-- N x 2
for (int i = 0, index = 0; i < N; i++, index += 2) {
output.data[index] = P.data[index]*a00 + P.data[index + 1]*a10;
output.data[index + 1] = P.data[index]*a01 + P.data[index + 1]*a11;
}
}
/**
* P*P_plus*X*1
*/
static void computePPpX( DMatrixRMaj P, DMatrixRMaj P_plus, DMatrixRMaj X, int offsetX, DMatrixRMaj output ) {
final int N = P.numRows;
output.reshape(N, 1);
// A=P_plus * X <-- 2x1
double a00 = 0, a10 = 0;
for (int i = 0, indexX = offsetX; i < N; i++, indexX += 2) {
double x = -X.data[indexX];
a00 += x*P_plus.data[i];
a10 += x*P_plus.data[i + N];
}
// P*A
for (int i = 0, indexP = 0; i < N; i++) {
output.data[i] = a00*P.data[indexP++] + a10*P.data[indexP++];
}
}
}
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