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Commit 26aef8c7 authored by Alexander Shishkov's avatar Alexander Shishkov
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modules/calib3d/src/epnp.cpp
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#include <iostream>
using namespace std;
#include "precomp.hpp"
#include "epnp.h"
namespace cv
{
double ePnP( InputArray _opoints, InputArray _ipoints,
InputArray _cameraMatrix, InputArray _distCoeffs,
OutputArray _rvec, OutputArray _tvec)
{
Mat opoints = _opoints.getMat(), ipoints = _ipoints.getMat();
int npoints = std::max(opoints.checkVector(3, CV_32F), opoints.checkVector(3, CV_64F));
CV_Assert( npoints >= 0 && npoints == std::max(ipoints.checkVector(2, CV_32F), ipoints.checkVector(2, CV_64F)) );
Mat cameraMatrix = _cameraMatrix.getMat(), distCoeffs = _distCoeffs.getMat();
Mat undistortedPoints;
undistortPoints(ipoints, undistortedPoints, cameraMatrix, distCoeffs);
epnp PnP;
PnP.set_internal_parameters(cameraMatrix.at<double> (0, 2), cameraMatrix.at<double> (1, 2), cameraMatrix.at<double> (0, 0), cameraMatrix.at<double> (1, 1));
PnP.set_maximum_number_of_correspondences(npoints);
PnP.reset_correspondences();
for(int i = 0; i < npoints; i++) {
PnP.add_correspondence(opoints.at<Point3d>(0,i).x, opoints.at<Point3d>(0,i).y, opoints.at<Point3d>(0,i).z, undistortedPoints.at<Point2d>(0,i).x* cameraMatrix.at<double> (0, 0) + cameraMatrix.at<double> (0, 2),
undistortedPoints.at<Point2d>(0,i).y* cameraMatrix.at<double> (1, 1) + cameraMatrix.at<double> (1, 2));
}
double R_est[3][3], t_est[3];
double error = PnP.compute_pose(R_est, t_est);
_tvec.create(3,1,CV_64F);
_rvec.create(3,1,CV_64F);
Mat t = Mat(3, 1, CV_64FC1, t_est);
Mat tvec = _tvec.getMat();
t.copyTo(tvec);
Mat R = Mat(3, 3, CV_64FC1, R_est);
Mat rvec = _rvec.getMat();
Rodrigues(R, rvec);
return error;
}
}
epnp::epnp(void)
{
maximum_number_of_correspondences = 0;
number_of_correspondences = 0;
pws = 0;
us = 0;
alphas = 0;
pcs = 0;
}
epnp::~epnp()
{
delete [] pws;
delete [] us;
delete [] alphas;
delete [] pcs;
}
void epnp::set_internal_parameters(double uc, double vc, double fu, double fv)
{
this->uc = uc;
this->vc = vc;
this->fu = fu;
this->fv = fv;
}
void epnp::set_maximum_number_of_correspondences(int n)
{
if (maximum_number_of_correspondences < n) {
if (pws != 0) delete [] pws;
if (us != 0) delete [] us;
if (alphas != 0) delete [] alphas;
if (pcs != 0) delete [] pcs;
maximum_number_of_correspondences = n;
pws = new double[3 * maximum_number_of_correspondences];
us = new double[2 * maximum_number_of_correspondences];
alphas = new double[4 * maximum_number_of_correspondences];
pcs = new double[3 * maximum_number_of_correspondences];
}
}
void epnp::reset_correspondences(void)
{
number_of_correspondences = 0;
}
void epnp::add_correspondence(double X, double Y, double Z, double u, double v)
{
pws[3 * number_of_correspondences ] = X;
pws[3 * number_of_correspondences + 1] = Y;
pws[3 * number_of_correspondences + 2] = Z;
us[2 * number_of_correspondences ] = u;
us[2 * number_of_correspondences + 1] = v;
number_of_correspondences++;
}
void epnp::choose_control_points(void)
{
// Take C0 as the reference points centroid:
cws[0][0] = cws[0][1] = cws[0][2] = 0;
for(int i = 0; i < number_of_correspondences; i++)
for(int j = 0; j < 3; j++)
cws[0][j] += pws[3 * i + j];
for(int j = 0; j < 3; j++)
cws[0][j] /= number_of_correspondences;
// Take C1, C2, and C3 from PCA on the reference points:
CvMat * PW0 = cvCreateMat(number_of_correspondences, 3, CV_64F);
double pw0tpw0[3 * 3], dc[3], uct[3 * 3];
CvMat PW0tPW0 = cvMat(3, 3, CV_64F, pw0tpw0);
CvMat DC = cvMat(3, 1, CV_64F, dc);
CvMat UCt = cvMat(3, 3, CV_64F, uct);
for(int i = 0; i < number_of_correspondences; i++)
for(int j = 0; j < 3; j++)
PW0->data.db[3 * i + j] = pws[3 * i + j] - cws[0][j];
cvMulTransposed(PW0, &PW0tPW0, 1);
cvSVD(&PW0tPW0, &DC, &UCt, 0, CV_SVD_MODIFY_A | CV_SVD_U_T);
cvReleaseMat(&PW0);
for(int i = 1; i < 4; i++) {
double k = sqrt(dc[i - 1] / number_of_correspondences);
for(int j = 0; j < 3; j++)
cws[i][j] = cws[0][j] + k * uct[3 * (i - 1) + j];
}
}
void epnp::compute_barycentric_coordinates(void)
{
double cc[3 * 3], cc_inv[3 * 3];
CvMat CC = cvMat(3, 3, CV_64F, cc);
CvMat CC_inv = cvMat(3, 3, CV_64F, cc_inv);
for(int i = 0; i < 3; i++)
for(int j = 1; j < 4; j++)
cc[3 * i + j - 1] = cws[j][i] - cws[0][i];
cvInvert(&CC, &CC_inv, CV_SVD);
double * ci = cc_inv;
for(int i = 0; i < number_of_correspondences; i++) {
double * pi = pws + 3 * i;
double * a = alphas + 4 * i;
for(int j = 0; j < 3; j++)
a[1 + j] =
ci[3 * j ] * (pi[0] - cws[0][0]) +
ci[3 * j + 1] * (pi[1] - cws[0][1]) +
ci[3 * j + 2] * (pi[2] - cws[0][2]);
a[0] = 1.0f - a[1] - a[2] - a[3];
}
}
void epnp::fill_M(CvMat * M,
const int row, const double * as, const double u, const double v)
{
double * M1 = M->data.db + row * 12;
double * M2 = M1 + 12;
for(int i = 0; i < 4; i++) {
M1[3 * i ] = as[i] * fu;
M1[3 * i + 1] = 0.0;
M1[3 * i + 2] = as[i] * (uc - u);
M2[3 * i ] = 0.0;
M2[3 * i + 1] = as[i] * fv;
M2[3 * i + 2] = as[i] * (vc - v);
}
}
void epnp::compute_ccs(const double * betas, const double * ut)
{
for(int i = 0; i < 4; i++)
ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0f;
for(int i = 0; i < 4; i++) {
const double * v = ut + 12 * (11 - i);
for(int j = 0; j < 4; j++)
for(int k = 0; k < 3; k++)
ccs[j][k] += betas[i] * v[3 * j + k];
}
}
void epnp::compute_pcs(void)
{
for(int i = 0; i < number_of_correspondences; i++) {
double * a = alphas + 4 * i;
double * pc = pcs + 3 * i;
for(int j = 0; j < 3; j++)
pc[j] = a[0] * ccs[0][j] + a[1] * ccs[1][j] + a[2] * ccs[2][j] + a[3] * ccs[3][j];
}
}
double epnp::compute_pose(double R[3][3], double t[3])
{
choose_control_points();
compute_barycentric_coordinates();
CvMat * M = cvCreateMat(2 * number_of_correspondences, 12, CV_64F);
for(int i = 0; i < number_of_correspondences; i++)
fill_M(M, 2 * i, alphas + 4 * i, us[2 * i], us[2 * i + 1]);
double mtm[12 * 12], d[12], ut[12 * 12];
CvMat MtM = cvMat(12, 12, CV_64F, mtm);
CvMat D = cvMat(12, 1, CV_64F, d);
CvMat Ut = cvMat(12, 12, CV_64F, ut);
cvMulTransposed(M, &MtM, 1);
cvSVD(&MtM, &D, &Ut, 0, CV_SVD_MODIFY_A | CV_SVD_U_T);
cvReleaseMat(&M);
double l_6x10[6 * 10], rho[6];
CvMat L_6x10 = cvMat(6, 10, CV_64F, l_6x10);
CvMat Rho = cvMat(6, 1, CV_64F, rho);
compute_L_6x10(ut, l_6x10);
compute_rho(rho);
double Betas[4][4], rep_errors[4];
double Rs[4][3][3], ts[4][3];
find_betas_approx_1(&L_6x10, &Rho, Betas[1]);
gauss_newton(&L_6x10, &Rho, Betas[1]);
rep_errors[1] = compute_R_and_t(ut, Betas[1], Rs[1], ts[1]);
find_betas_approx_2(&L_6x10, &Rho, Betas[2]);
gauss_newton(&L_6x10, &Rho, Betas[2]);
rep_errors[2] = compute_R_and_t(ut, Betas[2], Rs[2], ts[2]);
find_betas_approx_3(&L_6x10, &Rho, Betas[3]);
gauss_newton(&L_6x10, &Rho, Betas[3]);
rep_errors[3] = compute_R_and_t(ut, Betas[3], Rs[3], ts[3]);
int N = 1;
if (rep_errors[2] < rep_errors[1]) N = 2;
if (rep_errors[3] < rep_errors[N]) N = 3;
copy_R_and_t(Rs[N], ts[N], R, t);
return rep_errors[N];
}
void epnp::copy_R_and_t(const double R_src[3][3], const double t_src[3],
double R_dst[3][3], double t_dst[3])
{
for(int i = 0; i < 3; i++) {
for(int j = 0; j < 3; j++)
R_dst[i][j] = R_src[i][j];
t_dst[i] = t_src[i];
}
}
double epnp::dist2(const double * p1, const double * p2)
{
return
(p1[0] - p2[0]) * (p1[0] - p2[0]) +
(p1[1] - p2[1]) * (p1[1] - p2[1]) +
(p1[2] - p2[2]) * (p1[2] - p2[2]);
}
double epnp::dot(const double * v1, const double * v2)
{
return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}
double epnp::reprojection_error(const double R[3][3], const double t[3])
{
double sum2 = 0.0;
for(int i = 0; i < number_of_correspondences; i++) {
double * pw = pws + 3 * i;
double Xc = dot(R[0], pw) + t[0];
double Yc = dot(R[1], pw) + t[1];
double inv_Zc = 1.0 / (dot(R[2], pw) + t[2]);
double ue = uc + fu * Xc * inv_Zc;
double ve = vc + fv * Yc * inv_Zc;
double u = us[2 * i], v = us[2 * i + 1];
sum2 += sqrt( (u - ue) * (u - ue) + (v - ve) * (v - ve) );
}
return sum2 / number_of_correspondences;
}
void epnp::estimate_R_and_t(double R[3][3], double t[3])
{
double pc0[3], pw0[3];
pc0[0] = pc0[1] = pc0[2] = 0.0;
pw0[0] = pw0[1] = pw0[2] = 0.0;
for(int i = 0; i < number_of_correspondences; i++) {
const double * pc = pcs + 3 * i;
const double * pw = pws + 3 * i;
for(int j = 0; j < 3; j++) {
pc0[j] += pc[j];
pw0[j] += pw[j];
}
}
for(int j = 0; j < 3; j++) {
pc0[j] /= number_of_correspondences;
pw0[j] /= number_of_correspondences;
}
double abt[3 * 3], abt_d[3], abt_u[3 * 3], abt_v[3 * 3];
CvMat ABt = cvMat(3, 3, CV_64F, abt);
CvMat ABt_D = cvMat(3, 1, CV_64F, abt_d);
CvMat ABt_U = cvMat(3, 3, CV_64F, abt_u);
CvMat ABt_V = cvMat(3, 3, CV_64F, abt_v);
cvSetZero(&ABt);
for(int i = 0; i < number_of_correspondences; i++) {
double * pc = pcs + 3 * i;
double * pw = pws + 3 * i;
for(int j = 0; j < 3; j++) {
abt[3 * j ] += (pc[j] - pc0[j]) * (pw[0] - pw0[0]);
abt[3 * j + 1] += (pc[j] - pc0[j]) * (pw[1] - pw0[1]);
abt[3 * j + 2] += (pc[j] - pc0[j]) * (pw[2] - pw0[2]);
}
}
cvSVD(&ABt, &ABt_D, &ABt_U, &ABt_V, CV_SVD_MODIFY_A);
for(int i = 0; i < 3; i++)
for(int j = 0; j < 3; j++)
R[i][j] = dot(abt_u + 3 * i, abt_v + 3 * j);
const double det =
R[0][0] * R[1][1] * R[2][2] + R[0][1] * R[1][2] * R[2][0] + R[0][2] * R[1][0] * R[2][1] -
R[0][2] * R[1][1] * R[2][0] - R[0][1] * R[1][0] * R[2][2] - R[0][0] * R[1][2] * R[2][1];
if (det < 0) {
R[2][0] = -R[2][0];
R[2][1] = -R[2][1];
R[2][2] = -R[2][2];
}
t[0] = pc0[0] - dot(R[0], pw0);
t[1] = pc0[1] - dot(R[1], pw0);
t[2] = pc0[2] - dot(R[2], pw0);
}
void epnp::print_pose(const double R[3][3], const double t[3])
{
cout << R[0][0] << " " << R[0][1] << " " << R[0][2] << " " << t[0] << endl;
cout << R[1][0] << " " << R[1][1] << " " << R[1][2] << " " << t[1] << endl;
cout << R[2][0] << " " << R[2][1] << " " << R[2][2] << " " << t[2] << endl;
}
void epnp::solve_for_sign(void)
{
if (pcs[2] < 0.0) {
for(int i = 0; i < 4; i++)
for(int j = 0; j < 3; j++)
ccs[i][j] = -ccs[i][j];
for(int i = 0; i < number_of_correspondences; i++) {
pcs[3 * i ] = -pcs[3 * i];
pcs[3 * i + 1] = -pcs[3 * i + 1];
pcs[3 * i + 2] = -pcs[3 * i + 2];
}
}
}
double epnp::compute_R_and_t(const double * ut, const double * betas,
double R[3][3], double t[3])
{
compute_ccs(betas, ut);
compute_pcs();
solve_for_sign();
estimate_R_and_t(R, t);
return reprojection_error(R, t);
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_1 = [B11 B12 B13 B14]
void epnp::find_betas_approx_1(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x4[6 * 4], b4[4];
CvMat L_6x4 = cvMat(6, 4, CV_64F, l_6x4);
CvMat B4 = cvMat(4, 1, CV_64F, b4);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x4, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x4, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x4, i, 2, cvmGet(L_6x10, i, 3));
cvmSet(&L_6x4, i, 3, cvmGet(L_6x10, i, 6));
}
cvSolve(&L_6x4, Rho, &B4, CV_SVD);
if (b4[0] < 0) {
betas[0] = sqrt(-b4[0]);
betas[1] = -b4[1] / betas[0];
betas[2] = -b4[2] / betas[0];
betas[3] = -b4[3] / betas[0];
} else {
betas[0] = sqrt(b4[0]);
betas[1] = b4[1] / betas[0];
betas[2] = b4[2] / betas[0];
betas[3] = b4[3] / betas[0];
}
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_2 = [B11 B12 B22 ]
void epnp::find_betas_approx_2(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x3[6 * 3], b3[3];
CvMat L_6x3 = cvMat(6, 3, CV_64F, l_6x3);
CvMat B3 = cvMat(3, 1, CV_64F, b3);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x3, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x3, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x3, i, 2, cvmGet(L_6x10, i, 2));
}
cvSolve(&L_6x3, Rho, &B3, CV_SVD);
if (b3[0] < 0) {
betas[0] = sqrt(-b3[0]);
betas[1] = (b3[2] < 0) ? sqrt(-b3[2]) : 0.0;
} else {
betas[0] = sqrt(b3[0]);
betas[1] = (b3[2] > 0) ? sqrt(b3[2]) : 0.0;
}
if (b3[1] < 0) betas[0] = -betas[0];
betas[2] = 0.0;
betas[3] = 0.0;
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_3 = [B11 B12 B22 B13 B23 ]
void epnp::find_betas_approx_3(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x5[6 * 5], b5[5];
CvMat L_6x5 = cvMat(6, 5, CV_64F, l_6x5);
CvMat B5 = cvMat(5, 1, CV_64F, b5);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x5, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x5, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x5, i, 2, cvmGet(L_6x10, i, 2));
cvmSet(&L_6x5, i, 3, cvmGet(L_6x10, i, 3));
cvmSet(&L_6x5, i, 4, cvmGet(L_6x10, i, 4));
}
cvSolve(&L_6x5, Rho, &B5, CV_SVD);
if (b5[0] < 0) {
betas[0] = sqrt(-b5[0]);
betas[1] = (b5[2] < 0) ? sqrt(-b5[2]) : 0.0;
} else {
betas[0] = sqrt(b5[0]);
betas[1] = (b5[2] > 0) ? sqrt(b5[2]) : 0.0;
}
if (b5[1] < 0) betas[0] = -betas[0];
betas[2] = b5[3] / betas[0];
betas[3] = 0.0;
}
void epnp::compute_L_6x10(const double * ut, double * l_6x10)
{
const double * v[4];
v[0] = ut + 12 * 11;
v[1] = ut + 12 * 10;
v[2] = ut + 12 * 9;
v[3] = ut + 12 * 8;
double dv[4][6][3];
for(int i = 0; i < 4; i++) {
int a = 0, b = 1;
for(int j = 0; j < 6; j++) {
dv[i][j][0] = v[i][3 * a ] - v[i][3 * b];
dv[i][j][1] = v[i][3 * a + 1] - v[i][3 * b + 1];
dv[i][j][2] = v[i][3 * a + 2] - v[i][3 * b + 2];
b++;
if (b > 3) {
a++;
b = a + 1;
}
}
}
for(int i = 0; i < 6; i++) {
double * row = l_6x10 + 10 * i;
row[0] = dot(dv[0][i], dv[0][i]);
row[1] = 2.0f * dot(dv[0][i], dv[1][i]);
row[2] = dot(dv[1][i], dv[1][i]);
row[3] = 2.0f * dot(dv[0][i], dv[2][i]);
row[4] = 2.0f * dot(dv[1][i], dv[2][i]);
row[5] = dot(dv[2][i], dv[2][i]);
row[6] = 2.0f * dot(dv[0][i], dv[3][i]);
row[7] = 2.0f * dot(dv[1][i], dv[3][i]);
row[8] = 2.0f * dot(dv[2][i], dv[3][i]);
row[9] = dot(dv[3][i], dv[3][i]);
}
}
void epnp::compute_rho(double * rho)
{
rho[0] = dist2(cws[0], cws[1]);
rho[1] = dist2(cws[0], cws[2]);
rho[2] = dist2(cws[0], cws[3]);
rho[3] = dist2(cws[1], cws[2]);
rho[4] = dist2(cws[1], cws[3]);
rho[5] = dist2(cws[2], cws[3]);
}
void epnp::compute_A_and_b_gauss_newton(const double * l_6x10, const double * rho,
double betas[4], CvMat * A, CvMat * b)
{
for(int i = 0; i < 6; i++) {
const double * rowL = l_6x10 + i * 10;
double * rowA = A->data.db + i * 4;
rowA[0] = 2 * rowL[0] * betas[0] + rowL[1] * betas[1] + rowL[3] * betas[2] + rowL[6] * betas[3];
rowA[1] = rowL[1] * betas[0] + 2 * rowL[2] * betas[1] + rowL[4] * betas[2] + rowL[7] * betas[3];
rowA[2] = rowL[3] * betas[0] + rowL[4] * betas[1] + 2 * rowL[5] * betas[2] + rowL[8] * betas[3];
rowA[3] = rowL[6] * betas[0] + rowL[7] * betas[1] + rowL[8] * betas[2] + 2 * rowL[9] * betas[3];
cvmSet(b, i, 0, rho[i] -
(
rowL[0] * betas[0] * betas[0] +
rowL[1] * betas[0] * betas[1] +
rowL[2] * betas[1] * betas[1] +
rowL[3] * betas[0] * betas[2] +
rowL[4] * betas[1] * betas[2] +
rowL[5] * betas[2] * betas[2] +
rowL[6] * betas[0] * betas[3] +
rowL[7] * betas[1] * betas[3] +
rowL[8] * betas[2] * betas[3] +
rowL[9] * betas[3] * betas[3]
));
}
}
void epnp::gauss_newton(const CvMat * L_6x10, const CvMat * Rho,
double betas[4])
{
const int iterations_number = 5;
double a[6*4], b[6], x[4];
CvMat A = cvMat(6, 4, CV_64F, a);
CvMat B = cvMat(6, 1, CV_64F, b);
CvMat X = cvMat(4, 1, CV_64F, x);
for(int k = 0; k < iterations_number; k++) {
compute_A_and_b_gauss_newton(L_6x10->data.db, Rho->data.db,
betas, &A, &B);
qr_solve(&A, &B, &X);
for(int i = 0; i < 4; i++)
betas[i] += x[i];
}
}
void epnp::qr_solve(CvMat * A, CvMat * b, CvMat * X)
{
static int max_nr = 0;
static double * A1, * A2;
const int nr = A->rows;
const int nc = A->cols;
if (max_nr != 0 && max_nr < nr) {
delete [] A1;
delete [] A2;
}
if (max_nr < nr) {
max_nr = nr;
A1 = new double[nr];
A2 = new double[nr];
}
double * pA = A->data.db, * ppAkk = pA;
for(int k = 0; k < nc; k++) {
double * ppAik = ppAkk, eta = fabs(*ppAik);
for(int i = k + 1; i < nr; i++) {
double elt = fabs(*ppAik);
if (eta < elt) eta = elt;
ppAik += nc;
}
if (eta == 0) {
A1[k] = A2[k] = 0.0;
cerr << "God damnit, A is singular, this shouldn't happen." << endl;
return;
} else {
double * ppAik = ppAkk, sum = 0.0, inv_eta = 1. / eta;
for(int i = k; i < nr; i++) {
*ppAik *= inv_eta;
sum += *ppAik * *ppAik;
ppAik += nc;
}
double sigma = sqrt(sum);
if (*ppAkk < 0)
sigma = -sigma;
*ppAkk += sigma;
A1[k] = sigma * *ppAkk;
A2[k] = -eta * sigma;
for(int j = k + 1; j < nc; j++) {
double * ppAik = ppAkk, sum = 0;
for(int i = k; i < nr; i++) {
sum += *ppAik * ppAik[j - k];
ppAik += nc;
}
double tau = sum / A1[k];
ppAik = ppAkk;
for(int i = k; i < nr; i++) {
ppAik[j - k] -= tau * *ppAik;
ppAik += nc;
}
}
}
ppAkk += nc + 1;
}
// b <- Qt b
double * ppAjj = pA, * pb = b->data.db;
for(int j = 0; j < nc; j++) {
double * ppAij = ppAjj, tau = 0;
for(int i = j; i < nr; i++) {
tau += *ppAij * pb[i];
ppAij += nc;
}
tau /= A1[j];
ppAij = ppAjj;
for(int i = j; i < nr; i++) {
pb[i] -= tau * *ppAij;
ppAij += nc;
}
ppAjj += nc + 1;
}
// X = R-1 b
double * pX = X->data.db;
pX[nc - 1] = pb[nc - 1] / A2[nc - 1];
for(int i = nc - 2; i >= 0; i--) {
double * ppAij = pA + i * nc + (i + 1), sum = 0;
for(int j = i + 1; j < nc; j++) {
sum += *ppAij * pX[j];
ppAij++;
}
pX[i] = (pb[i] - sum) / A2[i];
}
}
void epnp::relative_error(double & rot_err, double & transl_err,
const double Rtrue[3][3], const double ttrue[3],
const double Rest[3][3], const double test[3])
{
double qtrue[4], qest[4];
mat_to_quat(Rtrue, qtrue);
mat_to_quat(Rest, qest);
double rot_err1 = sqrt((qtrue[0] - qest[0]) * (qtrue[0] - qest[0]) +
(qtrue[1] - qest[1]) * (qtrue[1] - qest[1]) +
(qtrue[2] - qest[2]) * (qtrue[2] - qest[2]) +
(qtrue[3] - qest[3]) * (qtrue[3] - qest[3]) ) /
sqrt(qtrue[0] * qtrue[0] + qtrue[1] * qtrue[1] + qtrue[2] * qtrue[2] + qtrue[3] * qtrue[3]);
double rot_err2 = sqrt((qtrue[0] + qest[0]) * (qtrue[0] + qest[0]) +
(qtrue[1] + qest[1]) * (qtrue[1] + qest[1]) +
(qtrue[2] + qest[2]) * (qtrue[2] + qest[2]) +
(qtrue[3] + qest[3]) * (qtrue[3] + qest[3]) ) /
sqrt(qtrue[0] * qtrue[0] + qtrue[1] * qtrue[1] + qtrue[2] * qtrue[2] + qtrue[3] * qtrue[3]);
rot_err = min(rot_err1, rot_err2);
transl_err =
sqrt((ttrue[0] - test[0]) * (ttrue[0] - test[0]) +
(ttrue[1] - test[1]) * (ttrue[1] - test[1]) +
(ttrue[2] - test[2]) * (ttrue[2] - test[2])) /
sqrt(ttrue[0] * ttrue[0] + ttrue[1] * ttrue[1] + ttrue[2] * ttrue[2]);
}
void epnp::mat_to_quat(const double R[3][3], double q[4])
{
double tr = R[0][0] + R[1][1] + R[2][2];
double n4;
if (tr > 0.0f) {
q[0] = R[1][2] - R[2][1];
q[1] = R[2][0] - R[0][2];
q[2] = R[0][1] - R[1][0];
q[3] = tr + 1.0f;
n4 = q[3];
} else if ( (R[0][0] > R[1][1]) && (R[0][0] > R[2][2]) ) {
q[0] = 1.0f + R[0][0] - R[1][1] - R[2][2];
q[1] = R[1][0] + R[0][1];
q[2] = R[2][0] + R[0][2];
q[3] = R[1][2] - R[2][1];
n4 = q[0];
} else if (R[1][1] > R[2][2]) {
q[0] = R[1][0] + R[0][1];
q[1] = 1.0f + R[1][1] - R[0][0] - R[2][2];
q[2] = R[2][1] + R[1][2];
q[3] = R[2][0] - R[0][2];
n4 = q[1];
} else {
q[0] = R[2][0] + R[0][2];
q[1] = R[2][1] + R[1][2];
q[2] = 1.0f + R[2][2] - R[0][0] - R[1][1];
q[3] = R[0][1] - R[1][0];
n4 = q[2];
}
double scale = 0.5f / double(sqrt(n4));
q[0] *= scale;
q[1] *= scale;
q[2] *= scale;
q[3] *= scale;
}
#include <iostream>
using namespace std;
#include "precomp.hpp"
#include "epnp.h"
epnp::epnp(const cv::Mat& cameraMatrix, const cv::Mat& opoints, const cv::Mat& ipoints)
{
if (cameraMatrix.depth() == CV_32F)
init_camera_parameters<float>(cameraMatrix);
else
init_camera_parameters<double>(cameraMatrix);
number_of_correspondences = std::max(opoints.checkVector(3, CV_32F), opoints.checkVector(3, CV_64F));
pws.resize(3 * number_of_correspondences);
us.resize(2 * number_of_correspondences);
if (opoints.depth() == ipoints.depth())
{
if (opoints.depth() == CV_32F)
init_points<cv::Point3f,cv::Point2f>(opoints, ipoints);
else
init_points<cv::Point3d,cv::Point2d>(opoints, ipoints);
}
else if (opoints.depth() == CV_32F)
init_points<cv::Point3f,cv::Point2d>(opoints, ipoints);
else
init_points<cv::Point3d,cv::Point2f>(opoints, ipoints);
alphas.resize(4 * number_of_correspondences);
pcs.resize(3 * number_of_correspondences);
}
epnp::~epnp()
{
}
void epnp::choose_control_points(void)
{
// Take C0 as the reference points centroid:
cws[0][0] = cws[0][1] = cws[0][2] = 0;
for(int i = 0; i < number_of_correspondences; i++)
for(int j = 0; j < 3; j++)
cws[0][j] += pws[3 * i + j];
for(int j = 0; j < 3; j++)
cws[0][j] /= number_of_correspondences;
// Take C1, C2, and C3 from PCA on the reference points:
CvMat * PW0 = cvCreateMat(number_of_correspondences, 3, CV_64F);
double pw0tpw0[3 * 3], dc[3], uct[3 * 3];
CvMat PW0tPW0 = cvMat(3, 3, CV_64F, pw0tpw0);
CvMat DC = cvMat(3, 1, CV_64F, dc);
CvMat UCt = cvMat(3, 3, CV_64F, uct);
for(int i = 0; i < number_of_correspondences; i++)
for(int j = 0; j < 3; j++)
PW0->data.db[3 * i + j] = pws[3 * i + j] - cws[0][j];
cvMulTransposed(PW0, &PW0tPW0, 1);
cvSVD(&PW0tPW0, &DC, &UCt, 0, CV_SVD_MODIFY_A | CV_SVD_U_T);
cvReleaseMat(&PW0);
for(int i = 1; i < 4; i++) {
double k = sqrt(dc[i - 1] / number_of_correspondences);
for(int j = 0; j < 3; j++)
cws[i][j] = cws[0][j] + k * uct[3 * (i - 1) + j];
}
}
void epnp::compute_barycentric_coordinates(void)
{
double cc[3 * 3], cc_inv[3 * 3];
CvMat CC = cvMat(3, 3, CV_64F, cc);
CvMat CC_inv = cvMat(3, 3, CV_64F, cc_inv);
for(int i = 0; i < 3; i++)
for(int j = 1; j < 4; j++)
cc[3 * i + j - 1] = cws[j][i] - cws[0][i];
cvInvert(&CC, &CC_inv, CV_SVD);
double * ci = cc_inv;
for(int i = 0; i < number_of_correspondences; i++) {
double * pi = pws.data() + 3 * i;
double * a = alphas.data() + 4 * i;
for(int j = 0; j < 3; j++)
a[1 + j] =
ci[3 * j ] * (pi[0] - cws[0][0]) +
ci[3 * j + 1] * (pi[1] - cws[0][1]) +
ci[3 * j + 2] * (pi[2] - cws[0][2]);
a[0] = 1.0f - a[1] - a[2] - a[3];
}
}
void epnp::fill_M(CvMat * M,
const int row, const double * as, const double u, const double v)
{
double * M1 = M->data.db + row * 12;
double * M2 = M1 + 12;
for(int i = 0; i < 4; i++) {
M1[3 * i ] = as[i] * fu;
M1[3 * i + 1] = 0.0;
M1[3 * i + 2] = as[i] * (uc - u);
M2[3 * i ] = 0.0;
M2[3 * i + 1] = as[i] * fv;
M2[3 * i + 2] = as[i] * (vc - v);
}
}
void epnp::compute_ccs(const double * betas, const double * ut)
{
for(int i = 0; i < 4; i++)
ccs[i][0] = ccs[i][1] = ccs[i][2] = 0.0f;
for(int i = 0; i < 4; i++) {
const double * v = ut + 12 * (11 - i);
for(int j = 0; j < 4; j++)
for(int k = 0; k < 3; k++)
ccs[j][k] += betas[i] * v[3 * j + k];
}
}
void epnp::compute_pcs(void)
{
for(int i = 0; i < number_of_correspondences; i++) {
double * a = alphas.data() + 4 * i;
double * pc = pcs.data() + 3 * i;
for(int j = 0; j < 3; j++)
pc[j] = a[0] * ccs[0][j] + a[1] * ccs[1][j] + a[2] * ccs[2][j] + a[3] * ccs[3][j];
}
}
void epnp::compute_pose(cv::Mat& R, cv::Mat& t)
{
choose_control_points();
compute_barycentric_coordinates();
CvMat * M = cvCreateMat(2 * number_of_correspondences, 12, CV_64F);
for(int i = 0; i < number_of_correspondences; i++)
fill_M(M, 2 * i, alphas.data() + 4 * i, us[2 * i], us[2 * i + 1]);
double mtm[12 * 12], d[12], ut[12 * 12];
CvMat MtM = cvMat(12, 12, CV_64F, mtm);
CvMat D = cvMat(12, 1, CV_64F, d);
CvMat Ut = cvMat(12, 12, CV_64F, ut);
cvMulTransposed(M, &MtM, 1);
cvSVD(&MtM, &D, &Ut, 0, CV_SVD_MODIFY_A | CV_SVD_U_T);
cvReleaseMat(&M);
double l_6x10[6 * 10], rho[6];
CvMat L_6x10 = cvMat(6, 10, CV_64F, l_6x10);
CvMat Rho = cvMat(6, 1, CV_64F, rho);
compute_L_6x10(ut, l_6x10);
compute_rho(rho);
double Betas[4][4], rep_errors[4];
double Rs[4][3][3], ts[4][3];
find_betas_approx_1(&L_6x10, &Rho, Betas[1]);
gauss_newton(&L_6x10, &Rho, Betas[1]);
rep_errors[1] = compute_R_and_t(ut, Betas[1], Rs[1], ts[1]);
find_betas_approx_2(&L_6x10, &Rho, Betas[2]);
gauss_newton(&L_6x10, &Rho, Betas[2]);
rep_errors[2] = compute_R_and_t(ut, Betas[2], Rs[2], ts[2]);
find_betas_approx_3(&L_6x10, &Rho, Betas[3]);
gauss_newton(&L_6x10, &Rho, Betas[3]);
rep_errors[3] = compute_R_and_t(ut, Betas[3], Rs[3], ts[3]);
int N = 1;
if (rep_errors[2] < rep_errors[1]) N = 2;
if (rep_errors[3] < rep_errors[N]) N = 3;
cv::Mat(3, 1, CV_64F, ts[N]).copyTo(t);
cv::Mat(3, 3, CV_64F, Rs[N]).copyTo(R);
}
void epnp::copy_R_and_t(const double R_src[3][3], const double t_src[3],
double R_dst[3][3], double t_dst[3])
{
for(int i = 0; i < 3; i++) {
for(int j = 0; j < 3; j++)
R_dst[i][j] = R_src[i][j];
t_dst[i] = t_src[i];
}
}
double epnp::dist2(const double * p1, const double * p2)
{
return
(p1[0] - p2[0]) * (p1[0] - p2[0]) +
(p1[1] - p2[1]) * (p1[1] - p2[1]) +
(p1[2] - p2[2]) * (p1[2] - p2[2]);
}
double epnp::dot(const double * v1, const double * v2)
{
return v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];
}
void epnp::estimate_R_and_t(double R[3][3], double t[3])
{
double pc0[3], pw0[3];
pc0[0] = pc0[1] = pc0[2] = 0.0;
pw0[0] = pw0[1] = pw0[2] = 0.0;
for(int i = 0; i < number_of_correspondences; i++) {
const double * pc = &pcs[3 * i];
const double * pw = &pws[3 * i];
for(int j = 0; j < 3; j++) {
pc0[j] += pc[j];
pw0[j] += pw[j];
}
}
for(int j = 0; j < 3; j++) {
pc0[j] /= number_of_correspondences;
pw0[j] /= number_of_correspondences;
}
double abt[3 * 3], abt_d[3], abt_u[3 * 3], abt_v[3 * 3];
CvMat ABt = cvMat(3, 3, CV_64F, abt);
CvMat ABt_D = cvMat(3, 1, CV_64F, abt_d);
CvMat ABt_U = cvMat(3, 3, CV_64F, abt_u);
CvMat ABt_V = cvMat(3, 3, CV_64F, abt_v);
cvSetZero(&ABt);
for(int i = 0; i < number_of_correspondences; i++) {
double * pc = &pcs[3 * i];
double * pw = &pws[3 * i];
for(int j = 0; j < 3; j++) {
abt[3 * j ] += (pc[j] - pc0[j]) * (pw[0] - pw0[0]);
abt[3 * j + 1] += (pc[j] - pc0[j]) * (pw[1] - pw0[1]);
abt[3 * j + 2] += (pc[j] - pc0[j]) * (pw[2] - pw0[2]);
}
}
cvSVD(&ABt, &ABt_D, &ABt_U, &ABt_V, CV_SVD_MODIFY_A);
for(int i = 0; i < 3; i++)
for(int j = 0; j < 3; j++)
R[i][j] = dot(abt_u + 3 * i, abt_v + 3 * j);
const double det =
R[0][0] * R[1][1] * R[2][2] + R[0][1] * R[1][2] * R[2][0] + R[0][2] * R[1][0] * R[2][1] -
R[0][2] * R[1][1] * R[2][0] - R[0][1] * R[1][0] * R[2][2] - R[0][0] * R[1][2] * R[2][1];
if (det < 0) {
R[2][0] = -R[2][0];
R[2][1] = -R[2][1];
R[2][2] = -R[2][2];
}
t[0] = pc0[0] - dot(R[0], pw0);
t[1] = pc0[1] - dot(R[1], pw0);
t[2] = pc0[2] - dot(R[2], pw0);
}
void epnp::solve_for_sign(void)
{
if (pcs[2] < 0.0) {
for(int i = 0; i < 4; i++)
for(int j = 0; j < 3; j++)
ccs[i][j] = -ccs[i][j];
for(int i = 0; i < number_of_correspondences; i++) {
pcs[3 * i ] = -pcs[3 * i];
pcs[3 * i + 1] = -pcs[3 * i + 1];
pcs[3 * i + 2] = -pcs[3 * i + 2];
}
}
}
double epnp::compute_R_and_t(const double * ut, const double * betas,
double R[3][3], double t[3])
{
compute_ccs(betas, ut);
compute_pcs();
solve_for_sign();
estimate_R_and_t(R, t);
return reprojection_error(R, t);
}
double epnp::reprojection_error(const double R[3][3], const double t[3])
{
double sum2 = 0.0;
for(int i = 0; i < number_of_correspondences; i++) {
double * pw = &pws[3 * i];
double Xc = dot(R[0], pw) + t[0];
double Yc = dot(R[1], pw) + t[1];
double inv_Zc = 1.0 / (dot(R[2], pw) + t[2]);
double ue = uc + fu * Xc * inv_Zc;
double ve = vc + fv * Yc * inv_Zc;
double u = us[2 * i], v = us[2 * i + 1];
sum2 += sqrt( (u - ue) * (u - ue) + (v - ve) * (v - ve) );
}
return sum2 / number_of_correspondences;
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_1 = [B11 B12 B13 B14]
void epnp::find_betas_approx_1(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x4[6 * 4], b4[4];
CvMat L_6x4 = cvMat(6, 4, CV_64F, l_6x4);
CvMat B4 = cvMat(4, 1, CV_64F, b4);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x4, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x4, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x4, i, 2, cvmGet(L_6x10, i, 3));
cvmSet(&L_6x4, i, 3, cvmGet(L_6x10, i, 6));
}
cvSolve(&L_6x4, Rho, &B4, CV_SVD);
if (b4[0] < 0) {
betas[0] = sqrt(-b4[0]);
betas[1] = -b4[1] / betas[0];
betas[2] = -b4[2] / betas[0];
betas[3] = -b4[3] / betas[0];
} else {
betas[0] = sqrt(b4[0]);
betas[1] = b4[1] / betas[0];
betas[2] = b4[2] / betas[0];
betas[3] = b4[3] / betas[0];
}
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_2 = [B11 B12 B22 ]
void epnp::find_betas_approx_2(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x3[6 * 3], b3[3];
CvMat L_6x3 = cvMat(6, 3, CV_64F, l_6x3);
CvMat B3 = cvMat(3, 1, CV_64F, b3);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x3, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x3, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x3, i, 2, cvmGet(L_6x10, i, 2));
}
cvSolve(&L_6x3, Rho, &B3, CV_SVD);
if (b3[0] < 0) {
betas[0] = sqrt(-b3[0]);
betas[1] = (b3[2] < 0) ? sqrt(-b3[2]) : 0.0;
} else {
betas[0] = sqrt(b3[0]);
betas[1] = (b3[2] > 0) ? sqrt(b3[2]) : 0.0;
}
if (b3[1] < 0) betas[0] = -betas[0];
betas[2] = 0.0;
betas[3] = 0.0;
}
// betas10 = [B11 B12 B22 B13 B23 B33 B14 B24 B34 B44]
// betas_approx_3 = [B11 B12 B22 B13 B23 ]
void epnp::find_betas_approx_3(const CvMat * L_6x10, const CvMat * Rho,
double * betas)
{
double l_6x5[6 * 5], b5[5];
CvMat L_6x5 = cvMat(6, 5, CV_64F, l_6x5);
CvMat B5 = cvMat(5, 1, CV_64F, b5);
for(int i = 0; i < 6; i++) {
cvmSet(&L_6x5, i, 0, cvmGet(L_6x10, i, 0));
cvmSet(&L_6x5, i, 1, cvmGet(L_6x10, i, 1));
cvmSet(&L_6x5, i, 2, cvmGet(L_6x10, i, 2));
cvmSet(&L_6x5, i, 3, cvmGet(L_6x10, i, 3));
cvmSet(&L_6x5, i, 4, cvmGet(L_6x10, i, 4));
}
cvSolve(&L_6x5, Rho, &B5, CV_SVD);
if (b5[0] < 0) {
betas[0] = sqrt(-b5[0]);
betas[1] = (b5[2] < 0) ? sqrt(-b5[2]) : 0.0;
} else {
betas[0] = sqrt(b5[0]);
betas[1] = (b5[2] > 0) ? sqrt(b5[2]) : 0.0;
}
if (b5[1] < 0) betas[0] = -betas[0];
betas[2] = b5[3] / betas[0];
betas[3] = 0.0;
}
void epnp::compute_L_6x10(const double * ut, double * l_6x10)
{
const double * v[4];
v[0] = ut + 12 * 11;
v[1] = ut + 12 * 10;
v[2] = ut + 12 * 9;
v[3] = ut + 12 * 8;
double dv[4][6][3];
for(int i = 0; i < 4; i++) {
int a = 0, b = 1;
for(int j = 0; j < 6; j++) {
dv[i][j][0] = v[i][3 * a ] - v[i][3 * b];
dv[i][j][1] = v[i][3 * a + 1] - v[i][3 * b + 1];
dv[i][j][2] = v[i][3 * a + 2] - v[i][3 * b + 2];
b++;
if (b > 3) {
a++;
b = a + 1;
}
}
}
for(int i = 0; i < 6; i++) {
double * row = l_6x10 + 10 * i;
row[0] = dot(dv[0][i], dv[0][i]);
row[1] = 2.0f * dot(dv[0][i], dv[1][i]);
row[2] = dot(dv[1][i], dv[1][i]);
row[3] = 2.0f * dot(dv[0][i], dv[2][i]);
row[4] = 2.0f * dot(dv[1][i], dv[2][i]);
row[5] = dot(dv[2][i], dv[2][i]);
row[6] = 2.0f * dot(dv[0][i], dv[3][i]);
row[7] = 2.0f * dot(dv[1][i], dv[3][i]);
row[8] = 2.0f * dot(dv[2][i], dv[3][i]);
row[9] = dot(dv[3][i], dv[3][i]);
}
}
void epnp::compute_rho(double * rho)
{
rho[0] = dist2(cws[0], cws[1]);
rho[1] = dist2(cws[0], cws[2]);
rho[2] = dist2(cws[0], cws[3]);
rho[3] = dist2(cws[1], cws[2]);
rho[4] = dist2(cws[1], cws[3]);
rho[5] = dist2(cws[2], cws[3]);
}
void epnp::compute_A_and_b_gauss_newton(const double * l_6x10, const double * rho,
const double betas[4], CvMat * A, CvMat * b)
{
for(int i = 0; i < 6; i++) {
const double * rowL = l_6x10 + i * 10;
double * rowA = A->data.db + i * 4;
rowA[0] = 2 * rowL[0] * betas[0] + rowL[1] * betas[1] + rowL[3] * betas[2] + rowL[6] * betas[3];
rowA[1] = rowL[1] * betas[0] + 2 * rowL[2] * betas[1] + rowL[4] * betas[2] + rowL[7] * betas[3];
rowA[2] = rowL[3] * betas[0] + rowL[4] * betas[1] + 2 * rowL[5] * betas[2] + rowL[8] * betas[3];
rowA[3] = rowL[6] * betas[0] + rowL[7] * betas[1] + rowL[8] * betas[2] + 2 * rowL[9] * betas[3];
cvmSet(b, i, 0, rho[i] -
(
rowL[0] * betas[0] * betas[0] +
rowL[1] * betas[0] * betas[1] +
rowL[2] * betas[1] * betas[1] +
rowL[3] * betas[0] * betas[2] +
rowL[4] * betas[1] * betas[2] +
rowL[5] * betas[2] * betas[2] +
rowL[6] * betas[0] * betas[3] +
rowL[7] * betas[1] * betas[3] +
rowL[8] * betas[2] * betas[3] +
rowL[9] * betas[3] * betas[3]
));
}
}
void epnp::gauss_newton(const CvMat * L_6x10, const CvMat * Rho,
double betas[4])
{
const int iterations_number = 5;
double a[6*4], b[6], x[4];
CvMat A = cvMat(6, 4, CV_64F, a);
CvMat B = cvMat(6, 1, CV_64F, b);
CvMat X = cvMat(4, 1, CV_64F, x);
for(int k = 0; k < iterations_number; k++) {
compute_A_and_b_gauss_newton(L_6x10->data.db, Rho->data.db,
betas, &A, &B);
qr_solve(&A, &B, &X);
for(int i = 0; i < 4; i++)
betas[i] += x[i];
}
}
void epnp::qr_solve(CvMat * A, CvMat * b, CvMat * X)
{
static int max_nr = 0;
static double * A1, * A2;
const int nr = A->rows;
const int nc = A->cols;
if (max_nr != 0 && max_nr < nr) {
delete [] A1;
delete [] A2;
}
if (max_nr < nr) {
max_nr = nr;
A1 = new double[nr];
A2 = new double[nr];
}
double * pA = A->data.db, * ppAkk = pA;
for(int k = 0; k < nc; k++) {
double * ppAik = ppAkk, eta = fabs(*ppAik);
for(int i = k + 1; i < nr; i++) {
double elt = fabs(*ppAik);
if (eta < elt) eta = elt;
ppAik += nc;
}
if (eta == 0) {
A1[k] = A2[k] = 0.0;
//cerr << "God damnit, A is singular, this shouldn't happen." << endl;
return;
} else {
double * ppAik = ppAkk, sum = 0.0, inv_eta = 1. / eta;
for(int i = k; i < nr; i++) {
*ppAik *= inv_eta;
sum += *ppAik * *ppAik;
ppAik += nc;
}
double sigma = sqrt(sum);
if (*ppAkk < 0)
sigma = -sigma;
*ppAkk += sigma;
A1[k] = sigma * *ppAkk;
A2[k] = -eta * sigma;
for(int j = k + 1; j < nc; j++) {
double * ppAik = ppAkk, sum = 0;
for(int i = k; i < nr; i++) {
sum += *ppAik * ppAik[j - k];
ppAik += nc;
}
double tau = sum / A1[k];
ppAik = ppAkk;
for(int i = k; i < nr; i++) {
ppAik[j - k] -= tau * *ppAik;
ppAik += nc;
}
}
}
ppAkk += nc + 1;
}
// b <- Qt b
double * ppAjj = pA, * pb = b->data.db;
for(int j = 0; j < nc; j++) {
double * ppAij = ppAjj, tau = 0;
for(int i = j; i < nr; i++) {
tau += *ppAij * pb[i];
ppAij += nc;
}
tau /= A1[j];
ppAij = ppAjj;
for(int i = j; i < nr; i++) {
pb[i] -= tau * *ppAij;
ppAij += nc;
}
ppAjj += nc + 1;
}
// X = R-1 b
double * pX = X->data.db;
pX[nc - 1] = pb[nc - 1] / A2[nc - 1];
for(int i = nc - 2; i >= 0; i--) {
double * ppAij = pA + i * nc + (i + 1), sum = 0;
for(int j = i + 1; j < nc; j++) {
sum += *ppAij * pX[j];
ppAij++;
}
pX[i] = (pb[i] - sum) / A2[i];
}
}
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