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#include "relpose_5pt.h"
#include "PoseLib/misc/essential.h"
#include "PoseLib/misc/sturm.h"
#include <Eigen/Dense>
namespace poselib {
// a, b are first order polys [x,y,z,1]
// c is degree 2 poly with order
// [ x^2, x*y, x*z, x, y^2, y*z, y, z^2, z, 1]
inline void o1(const double a[4], const double b[4], double c[10]) {
c[0] = a[0] * b[0];
c[1] = a[0] * b[1] + a[1] * b[0];
c[2] = a[0] * b[2] + a[2] * b[0];
c[3] = a[0] * b[3] + a[3] * b[0];
c[4] = a[1] * b[1];
c[5] = a[1] * b[2] + a[2] * b[1];
c[6] = a[1] * b[3] + a[3] * b[1];
c[7] = a[2] * b[2];
c[8] = a[2] * b[3] + a[3] * b[2];
c[9] = a[3] * b[3];
}
inline void o1p(const double a[4], const double b[4], double c[10]) {
c[0] += a[0] * b[0];
c[1] += a[0] * b[1] + a[1] * b[0];
c[2] += a[0] * b[2] + a[2] * b[0];
c[3] += a[0] * b[3] + a[3] * b[0];
c[4] += a[1] * b[1];
c[5] += a[1] * b[2] + a[2] * b[1];
c[6] += a[1] * b[3] + a[3] * b[1];
c[7] += a[2] * b[2];
c[8] += a[2] * b[3] + a[3] * b[2];
c[9] += a[3] * b[3];
}
inline void o1m(const double a[4], const double b[4], double c[10]) {
c[0] -= a[0] * b[0];
c[1] -= a[0] * b[1] + a[1] * b[0];
c[2] -= a[0] * b[2] + a[2] * b[0];
c[3] -= a[0] * b[3] + a[3] * b[0];
c[4] -= a[1] * b[1];
c[5] -= a[1] * b[2] + a[2] * b[1];
c[6] -= a[1] * b[3] + a[3] * b[1];
c[7] -= a[2] * b[2];
c[8] -= a[2] * b[3] + a[3] * b[2];
c[9] -= a[3] * b[3];
}
// a is second degree poly with order
// [ x^2, x*y, x*z, x, y^2, y*z, y, z^2, z, 1]
// b is first degree with order
// [x y z 1]
// c is third degree with order (same as nister's paper)
// [ x^3, y^3, x^2*y, x*y^2, x^2*z, x^2, y^2*z, y^2, x*y*z, x*y, x*z^2, x*z, x, y*z^2, y*z, y, z^3, z^2, z, 1]
inline void o2(const double a[10], const double b[4], double c[20]) {
c[0] = a[0] * b[0];
c[1] = a[4] * b[1];
c[2] = a[0] * b[1] + a[1] * b[0];
c[3] = a[1] * b[1] + a[4] * b[0];
c[4] = a[0] * b[2] + a[2] * b[0];
c[5] = a[0] * b[3] + a[3] * b[0];
c[6] = a[4] * b[2] + a[5] * b[1];
c[7] = a[4] * b[3] + a[6] * b[1];
c[8] = a[1] * b[2] + a[2] * b[1] + a[5] * b[0];
c[9] = a[1] * b[3] + a[3] * b[1] + a[6] * b[0];
c[10] = a[2] * b[2] + a[7] * b[0];
c[11] = a[2] * b[3] + a[3] * b[2] + a[8] * b[0];
c[12] = a[3] * b[3] + a[9] * b[0];
c[13] = a[5] * b[2] + a[7] * b[1];
c[14] = a[5] * b[3] + a[6] * b[2] + a[8] * b[1];
c[15] = a[6] * b[3] + a[9] * b[1];
c[16] = a[7] * b[2];
c[17] = a[7] * b[3] + a[8] * b[2];
c[18] = a[8] * b[3] + a[9] * b[2];
c[19] = a[9] * b[3];
}
inline void o2p(const double a[10], const double b[4], double c[20]) {
c[0] += a[0] * b[0];
c[1] += a[4] * b[1];
c[2] += a[0] * b[1] + a[1] * b[0];
c[3] += a[1] * b[1] + a[4] * b[0];
c[4] += a[0] * b[2] + a[2] * b[0];
c[5] += a[0] * b[3] + a[3] * b[0];
c[6] += a[4] * b[2] + a[5] * b[1];
c[7] += a[4] * b[3] + a[6] * b[1];
c[8] += a[1] * b[2] + a[2] * b[1] + a[5] * b[0];
c[9] += a[1] * b[3] + a[3] * b[1] + a[6] * b[0];
c[10] += a[2] * b[2] + a[7] * b[0];
c[11] += a[2] * b[3] + a[3] * b[2] + a[8] * b[0];
c[12] += a[3] * b[3] + a[9] * b[0];
c[13] += a[5] * b[2] + a[7] * b[1];
c[14] += a[5] * b[3] + a[6] * b[2] + a[8] * b[1];
c[15] += a[6] * b[3] + a[9] * b[1];
c[16] += a[7] * b[2];
c[17] += a[7] * b[3] + a[8] * b[2];
c[18] += a[8] * b[3] + a[9] * b[2];
c[19] += a[9] * b[3];
}
void compute_trace_constraints(const Eigen::Matrix<double, 4, 9> &N, Eigen::Matrix<double, 10, 20> &coeffs) {
double const *N_ptr = N.data();
#define EE(i, j) N_ptr + 4 * (3 * j + i)
double d[60];
// Determinant constraint
Eigen::Matrix<double, 1, 20> row;
double *c_data = row.data();
o1(EE(0, 1), EE(1, 2), d);
o1m(EE(0, 2), EE(1, 1), d);
o2(d, EE(2, 0), c_data);
o1(EE(0, 2), EE(1, 0), d);
o1m(EE(0, 0), EE(1, 2), d);
o2p(d, EE(2, 1), c_data);
o1(EE(0, 0), EE(1, 1), d);
o1m(EE(0, 1), EE(1, 0), d);
o2p(d, EE(2, 2), c_data);
coeffs.row(9) = row;
double *EET[3][3] = {{d, d + 10, d + 20}, {d + 10, d + 40, d + 30}, {d + 20, d + 30, d + 50}};
// Compute EE^T (equation 20 in paper)
for (int i = 0; i < 3; ++i) {
for (int j = i; j < 3; ++j) {
o1(EE(i, 0), EE(j, 0), EET[i][j]);
o1p(EE(i, 1), EE(j, 1), EET[i][j]);
o1p(EE(i, 2), EE(j, 2), EET[i][j]);
}
}
// Subtract trace (equation 22 in paper)
for (int i = 0; i < 10; ++i) {
double t = 0.5 * (EET[0][0][i] + EET[1][1][i] + EET[2][2][i]);
EET[0][0][i] -= t;
EET[1][1][i] -= t;
EET[2][2][i] -= t;
}
int cnt = 0;
for (int i = 0; i < 3; ++i) {
for (int j = 0; j < 3; ++j) {
o2(EET[i][0], EE(0, j), c_data);
o2p(EET[i][1], EE(1, j), c_data);
o2p(EET[i][2], EE(2, j), c_data);
coeffs.row(cnt++) = row;
}
}
#undef EE
}
int relpose_5pt(const std::vector<Eigen::Vector3d> &x1, const std::vector<Eigen::Vector3d> &x2,
std::vector<Eigen::Matrix3d> *essential_matrices) {
// Compute nullspace to epipolar constraints
Eigen::Matrix<double, 9, 5> epipolar_constraints;
for (int i = 0; i < 5; ++i) {
epipolar_constraints.col(i) << x1[i](0) * x2[i], x1[i](1) * x2[i], x1[i](2) * x2[i];
}
Eigen::Matrix<double, 9, 9> Q = epipolar_constraints.fullPivHouseholderQr().matrixQ();
Eigen::Matrix<double, 4, 9> N = Q.rightCols(4).transpose();
// Compute equation coefficients for the trace constraints + determinant
Eigen::Matrix<double, 10, 20> coeffs;
compute_trace_constraints(N, coeffs);
coeffs.block<10, 10>(0, 10) = coeffs.block<10, 10>(0, 0).partialPivLu().solve(coeffs.block<10, 10>(0, 10));
// Perform eliminations using the 6 bottom rows
Eigen::Matrix<double, 3, 13> A;
for (int i = 0; i < 3; ++i) {
A(i, 0) = 0.0;
A.block<1, 3>(i, 1) = coeffs.block<1, 3>(4 + 2 * i, 10);
A.block<1, 3>(i, 0) -= coeffs.block<1, 3>(5 + 2 * i, 10);
A(i, 4) = 0.0;
A.block<1, 3>(i, 5) = coeffs.block<1, 3>(4 + 2 * i, 13);
A.block<1, 3>(i, 4) -= coeffs.block<1, 3>(5 + 2 * i, 13);
A(i, 8) = 0.0;
A.block<1, 4>(i, 9) = coeffs.block<1, 4>(4 + 2 * i, 16);
A.block<1, 4>(i, 8) -= coeffs.block<1, 4>(5 + 2 * i, 16);
}
// Compute degree 10 poly representing determinant (equation 14 in the paper)
double c[11];
c[0] = A(0, 12) * A(1, 3) * A(2, 7) - A(0, 12) * A(1, 7) * A(2, 3) - A(0, 3) * A(2, 7) * A(1, 12) +
A(0, 7) * A(2, 3) * A(1, 12) + A(0, 3) * A(1, 7) * A(2, 12) - A(0, 7) * A(1, 3) * A(2, 12);
c[1] = A(0, 11) * A(1, 3) * A(2, 7) - A(0, 11) * A(1, 7) * A(2, 3) + A(0, 12) * A(1, 2) * A(2, 7) +
A(0, 12) * A(1, 3) * A(2, 6) - A(0, 12) * A(1, 6) * A(2, 3) - A(0, 12) * A(1, 7) * A(2, 2) -
A(0, 2) * A(2, 7) * A(1, 12) - A(0, 3) * A(2, 6) * A(1, 12) - A(0, 3) * A(2, 7) * A(1, 11) +
A(0, 6) * A(2, 3) * A(1, 12) + A(0, 7) * A(2, 2) * A(1, 12) + A(0, 7) * A(2, 3) * A(1, 11) +
A(0, 2) * A(1, 7) * A(2, 12) + A(0, 3) * A(1, 6) * A(2, 12) + A(0, 3) * A(1, 7) * A(2, 11) -
A(0, 6) * A(1, 3) * A(2, 12) - A(0, 7) * A(1, 2) * A(2, 12) - A(0, 7) * A(1, 3) * A(2, 11);
c[2] = A(0, 10) * A(1, 3) * A(2, 7) - A(0, 10) * A(1, 7) * A(2, 3) + A(0, 11) * A(1, 2) * A(2, 7) +
A(0, 11) * A(1, 3) * A(2, 6) - A(0, 11) * A(1, 6) * A(2, 3) - A(0, 11) * A(1, 7) * A(2, 2) +
A(1, 1) * A(0, 12) * A(2, 7) + A(0, 12) * A(1, 2) * A(2, 6) + A(0, 12) * A(1, 3) * A(2, 5) -
A(0, 12) * A(1, 5) * A(2, 3) - A(0, 12) * A(1, 6) * A(2, 2) - A(0, 12) * A(1, 7) * A(2, 1) -
A(0, 1) * A(2, 7) * A(1, 12) - A(0, 2) * A(2, 6) * A(1, 12) - A(0, 2) * A(2, 7) * A(1, 11) -
A(0, 3) * A(2, 5) * A(1, 12) - A(0, 3) * A(2, 6) * A(1, 11) - A(0, 3) * A(2, 7) * A(1, 10) +
A(0, 5) * A(2, 3) * A(1, 12) + A(0, 6) * A(2, 2) * A(1, 12) + A(0, 6) * A(2, 3) * A(1, 11) +
A(0, 7) * A(2, 1) * A(1, 12) + A(0, 7) * A(2, 2) * A(1, 11) + A(0, 7) * A(2, 3) * A(1, 10) +
A(0, 1) * A(1, 7) * A(2, 12) + A(0, 2) * A(1, 6) * A(2, 12) + A(0, 2) * A(1, 7) * A(2, 11) +
A(0, 3) * A(1, 5) * A(2, 12) + A(0, 3) * A(1, 6) * A(2, 11) + A(0, 3) * A(1, 7) * A(2, 10) -
A(0, 5) * A(1, 3) * A(2, 12) - A(0, 6) * A(1, 2) * A(2, 12) - A(0, 6) * A(1, 3) * A(2, 11) -
A(0, 7) * A(1, 1) * A(2, 12) - A(0, 7) * A(1, 2) * A(2, 11) - A(0, 7) * A(1, 3) * A(2, 10);
c[3] = A(0, 3) * A(1, 7) * A(2, 9) - A(0, 3) * A(1, 9) * A(2, 7) - A(0, 7) * A(1, 3) * A(2, 9) +
A(0, 7) * A(1, 9) * A(2, 3) + A(0, 9) * A(1, 3) * A(2, 7) - A(0, 9) * A(1, 7) * A(2, 3) +
A(0, 10) * A(1, 2) * A(2, 7) + A(0, 10) * A(1, 3) * A(2, 6) - A(0, 10) * A(1, 6) * A(2, 3) -
A(0, 10) * A(1, 7) * A(2, 2) + A(1, 0) * A(0, 12) * A(2, 7) + A(0, 11) * A(1, 1) * A(2, 7) +
A(0, 11) * A(1, 2) * A(2, 6) + A(0, 11) * A(1, 3) * A(2, 5) - A(0, 11) * A(1, 5) * A(2, 3) -
A(0, 11) * A(1, 6) * A(2, 2) - A(0, 11) * A(1, 7) * A(2, 1) + A(1, 1) * A(0, 12) * A(2, 6) +
A(0, 12) * A(1, 2) * A(2, 5) + A(0, 12) * A(1, 3) * A(2, 4) - A(0, 12) * A(1, 4) * A(2, 3) -
A(0, 12) * A(1, 5) * A(2, 2) - A(0, 12) * A(1, 6) * A(2, 1) - A(0, 12) * A(1, 7) * A(2, 0) -
A(0, 0) * A(2, 7) * A(1, 12) - A(0, 1) * A(2, 6) * A(1, 12) - A(0, 1) * A(2, 7) * A(1, 11) -
A(0, 2) * A(2, 5) * A(1, 12) - A(0, 2) * A(2, 6) * A(1, 11) - A(0, 2) * A(2, 7) * A(1, 10) -
A(0, 3) * A(2, 4) * A(1, 12) - A(0, 3) * A(2, 5) * A(1, 11) - A(0, 3) * A(2, 6) * A(1, 10) +
A(0, 4) * A(2, 3) * A(1, 12) + A(0, 5) * A(2, 2) * A(1, 12) + A(0, 5) * A(2, 3) * A(1, 11) +
A(0, 6) * A(2, 1) * A(1, 12) + A(0, 6) * A(2, 2) * A(1, 11) + A(0, 6) * A(2, 3) * A(1, 10) +
A(0, 7) * A(2, 0) * A(1, 12) + A(0, 7) * A(2, 1) * A(1, 11) + A(0, 7) * A(2, 2) * A(1, 10) +
A(0, 0) * A(1, 7) * A(2, 12) + A(0, 1) * A(1, 6) * A(2, 12) + A(0, 1) * A(1, 7) * A(2, 11) +
A(0, 2) * A(1, 5) * A(2, 12) + A(0, 2) * A(1, 6) * A(2, 11) + A(0, 2) * A(1, 7) * A(2, 10) +
A(0, 3) * A(1, 4) * A(2, 12) + A(0, 3) * A(1, 5) * A(2, 11) + A(0, 3) * A(1, 6) * A(2, 10) -
A(0, 4) * A(1, 3) * A(2, 12) - A(0, 5) * A(1, 2) * A(2, 12) - A(0, 5) * A(1, 3) * A(2, 11) -
A(0, 6) * A(1, 1) * A(2, 12) - A(0, 6) * A(1, 2) * A(2, 11) - A(0, 6) * A(1, 3) * A(2, 10) -
A(0, 7) * A(1, 0) * A(2, 12) - A(0, 7) * A(1, 1) * A(2, 11) - A(0, 7) * A(1, 2) * A(2, 10);
c[4] = A(0, 2) * A(1, 7) * A(2, 9) - A(0, 2) * A(1, 9) * A(2, 7) + A(0, 3) * A(1, 6) * A(2, 9) +
A(0, 3) * A(1, 7) * A(2, 8) - A(0, 3) * A(1, 8) * A(2, 7) - A(0, 3) * A(1, 9) * A(2, 6) -
A(0, 6) * A(1, 3) * A(2, 9) + A(0, 6) * A(1, 9) * A(2, 3) - A(0, 7) * A(1, 2) * A(2, 9) -
A(0, 7) * A(1, 3) * A(2, 8) + A(0, 7) * A(1, 8) * A(2, 3) + A(0, 7) * A(1, 9) * A(2, 2) +
A(0, 8) * A(1, 3) * A(2, 7) - A(0, 8) * A(1, 7) * A(2, 3) + A(0, 9) * A(1, 2) * A(2, 7) +
A(0, 9) * A(1, 3) * A(2, 6) - A(0, 9) * A(1, 6) * A(2, 3) - A(0, 9) * A(1, 7) * A(2, 2) +
A(0, 10) * A(1, 1) * A(2, 7) + A(0, 10) * A(1, 2) * A(2, 6) + A(0, 10) * A(1, 3) * A(2, 5) -
A(0, 10) * A(1, 5) * A(2, 3) - A(0, 10) * A(1, 6) * A(2, 2) - A(0, 10) * A(1, 7) * A(2, 1) +
A(1, 0) * A(0, 11) * A(2, 7) + A(1, 0) * A(0, 12) * A(2, 6) + A(0, 11) * A(1, 1) * A(2, 6) +
A(0, 11) * A(1, 2) * A(2, 5) + A(0, 11) * A(1, 3) * A(2, 4) - A(0, 11) * A(1, 4) * A(2, 3) -
A(0, 11) * A(1, 5) * A(2, 2) - A(0, 11) * A(1, 6) * A(2, 1) - A(0, 11) * A(1, 7) * A(2, 0) +
A(1, 1) * A(0, 12) * A(2, 5) + A(0, 12) * A(1, 2) * A(2, 4) - A(0, 12) * A(1, 4) * A(2, 2) -
A(0, 12) * A(1, 5) * A(2, 1) - A(0, 12) * A(1, 6) * A(2, 0) - A(0, 0) * A(2, 6) * A(1, 12) -
A(0, 0) * A(2, 7) * A(1, 11) - A(0, 1) * A(2, 5) * A(1, 12) - A(0, 1) * A(2, 6) * A(1, 11) -
A(0, 1) * A(2, 7) * A(1, 10) - A(0, 2) * A(2, 4) * A(1, 12) - A(0, 2) * A(2, 5) * A(1, 11) -
A(0, 2) * A(2, 6) * A(1, 10) - A(0, 3) * A(2, 4) * A(1, 11) - A(0, 3) * A(2, 5) * A(1, 10) +
A(0, 4) * A(2, 2) * A(1, 12) + A(0, 4) * A(2, 3) * A(1, 11) + A(0, 5) * A(2, 1) * A(1, 12) +
A(0, 5) * A(2, 2) * A(1, 11) + A(0, 5) * A(2, 3) * A(1, 10) + A(0, 6) * A(2, 0) * A(1, 12) +
A(0, 6) * A(2, 1) * A(1, 11) + A(0, 6) * A(2, 2) * A(1, 10) + A(0, 7) * A(2, 0) * A(1, 11) +
A(0, 7) * A(2, 1) * A(1, 10) + A(0, 0) * A(1, 6) * A(2, 12) + A(0, 0) * A(1, 7) * A(2, 11) +
A(0, 1) * A(1, 5) * A(2, 12) + A(0, 1) * A(1, 6) * A(2, 11) + A(0, 1) * A(1, 7) * A(2, 10) +
A(0, 2) * A(1, 4) * A(2, 12) + A(0, 2) * A(1, 5) * A(2, 11) + A(0, 2) * A(1, 6) * A(2, 10) +
A(0, 3) * A(1, 4) * A(2, 11) + A(0, 3) * A(1, 5) * A(2, 10) - A(0, 4) * A(1, 2) * A(2, 12) -
A(0, 4) * A(1, 3) * A(2, 11) - A(0, 5) * A(1, 1) * A(2, 12) - A(0, 5) * A(1, 2) * A(2, 11) -
A(0, 5) * A(1, 3) * A(2, 10) - A(0, 6) * A(1, 0) * A(2, 12) - A(0, 6) * A(1, 1) * A(2, 11) -
A(0, 6) * A(1, 2) * A(2, 10) - A(0, 7) * A(1, 0) * A(2, 11) - A(0, 7) * A(1, 1) * A(2, 10);
c[5] = A(0, 1) * A(1, 7) * A(2, 9) - A(0, 1) * A(1, 9) * A(2, 7) + A(0, 2) * A(1, 6) * A(2, 9) +
A(0, 2) * A(1, 7) * A(2, 8) - A(0, 2) * A(1, 8) * A(2, 7) - A(0, 2) * A(1, 9) * A(2, 6) +
A(0, 3) * A(1, 5) * A(2, 9) + A(0, 3) * A(1, 6) * A(2, 8) - A(0, 3) * A(1, 8) * A(2, 6) -
A(0, 3) * A(1, 9) * A(2, 5) - A(0, 5) * A(1, 3) * A(2, 9) + A(0, 5) * A(1, 9) * A(2, 3) -
A(0, 6) * A(1, 2) * A(2, 9) - A(0, 6) * A(1, 3) * A(2, 8) + A(0, 6) * A(1, 8) * A(2, 3) +
A(0, 6) * A(1, 9) * A(2, 2) - A(0, 7) * A(1, 1) * A(2, 9) - A(0, 7) * A(1, 2) * A(2, 8) +
A(0, 7) * A(1, 8) * A(2, 2) + A(0, 7) * A(1, 9) * A(2, 1) + A(0, 8) * A(1, 2) * A(2, 7) +
A(0, 8) * A(1, 3) * A(2, 6) - A(0, 8) * A(1, 6) * A(2, 3) - A(0, 8) * A(1, 7) * A(2, 2) +
A(0, 9) * A(1, 1) * A(2, 7) + A(0, 9) * A(1, 2) * A(2, 6) + A(0, 9) * A(1, 3) * A(2, 5) -
A(0, 9) * A(1, 5) * A(2, 3) - A(0, 9) * A(1, 6) * A(2, 2) - A(0, 9) * A(1, 7) * A(2, 1) +
A(0, 10) * A(1, 0) * A(2, 7) + A(0, 10) * A(1, 1) * A(2, 6) + A(0, 10) * A(1, 2) * A(2, 5) +
A(0, 10) * A(1, 3) * A(2, 4) - A(0, 10) * A(1, 4) * A(2, 3) - A(0, 10) * A(1, 5) * A(2, 2) -
A(0, 10) * A(1, 6) * A(2, 1) - A(0, 10) * A(1, 7) * A(2, 0) + A(1, 0) * A(0, 11) * A(2, 6) +
A(1, 0) * A(0, 12) * A(2, 5) + A(0, 11) * A(1, 1) * A(2, 5) + A(0, 11) * A(1, 2) * A(2, 4) -
A(0, 11) * A(1, 4) * A(2, 2) - A(0, 11) * A(1, 5) * A(2, 1) - A(0, 11) * A(1, 6) * A(2, 0) +
A(1, 1) * A(0, 12) * A(2, 4) - A(0, 12) * A(1, 4) * A(2, 1) - A(0, 12) * A(1, 5) * A(2, 0) -
A(0, 0) * A(2, 5) * A(1, 12) - A(0, 0) * A(2, 6) * A(1, 11) - A(0, 0) * A(2, 7) * A(1, 10) -
A(0, 1) * A(2, 4) * A(1, 12) - A(0, 1) * A(2, 5) * A(1, 11) - A(0, 1) * A(2, 6) * A(1, 10) -
A(0, 2) * A(2, 4) * A(1, 11) - A(0, 2) * A(2, 5) * A(1, 10) - A(0, 3) * A(2, 4) * A(1, 10) +
A(0, 4) * A(2, 1) * A(1, 12) + A(0, 4) * A(2, 2) * A(1, 11) + A(0, 4) * A(2, 3) * A(1, 10) +
A(0, 5) * A(2, 0) * A(1, 12) + A(0, 5) * A(2, 1) * A(1, 11) + A(0, 5) * A(2, 2) * A(1, 10) +
A(0, 6) * A(2, 0) * A(1, 11) + A(0, 6) * A(2, 1) * A(1, 10) + A(0, 7) * A(2, 0) * A(1, 10) +
A(0, 0) * A(1, 5) * A(2, 12) + A(0, 0) * A(1, 6) * A(2, 11) + A(0, 0) * A(1, 7) * A(2, 10) +
A(0, 1) * A(1, 4) * A(2, 12) + A(0, 1) * A(1, 5) * A(2, 11) + A(0, 1) * A(1, 6) * A(2, 10) +
A(0, 2) * A(1, 4) * A(2, 11) + A(0, 2) * A(1, 5) * A(2, 10) + A(0, 3) * A(1, 4) * A(2, 10) -
A(0, 4) * A(1, 1) * A(2, 12) - A(0, 4) * A(1, 2) * A(2, 11) - A(0, 4) * A(1, 3) * A(2, 10) -
A(0, 5) * A(1, 0) * A(2, 12) - A(0, 5) * A(1, 1) * A(2, 11) - A(0, 5) * A(1, 2) * A(2, 10) -
A(0, 6) * A(1, 0) * A(2, 11) - A(0, 6) * A(1, 1) * A(2, 10) - A(0, 7) * A(1, 0) * A(2, 10);
c[6] = A(0, 0) * A(1, 7) * A(2, 9) - A(0, 0) * A(1, 9) * A(2, 7) + A(0, 1) * A(1, 6) * A(2, 9) +
A(0, 1) * A(1, 7) * A(2, 8) - A(0, 1) * A(1, 8) * A(2, 7) - A(0, 1) * A(1, 9) * A(2, 6) +
A(0, 2) * A(1, 5) * A(2, 9) + A(0, 2) * A(1, 6) * A(2, 8) - A(0, 2) * A(1, 8) * A(2, 6) -
A(0, 2) * A(1, 9) * A(2, 5) + A(0, 3) * A(1, 4) * A(2, 9) + A(0, 3) * A(1, 5) * A(2, 8) -
A(0, 3) * A(1, 8) * A(2, 5) - A(0, 3) * A(1, 9) * A(2, 4) - A(0, 4) * A(1, 3) * A(2, 9) +
A(0, 4) * A(1, 9) * A(2, 3) - A(0, 5) * A(1, 2) * A(2, 9) - A(0, 5) * A(1, 3) * A(2, 8) +
A(0, 5) * A(1, 8) * A(2, 3) + A(0, 5) * A(1, 9) * A(2, 2) - A(0, 6) * A(1, 1) * A(2, 9) -
A(0, 6) * A(1, 2) * A(2, 8) + A(0, 6) * A(1, 8) * A(2, 2) + A(0, 6) * A(1, 9) * A(2, 1) -
A(0, 7) * A(1, 0) * A(2, 9) - A(0, 7) * A(1, 1) * A(2, 8) + A(0, 7) * A(1, 8) * A(2, 1) +
A(0, 7) * A(1, 9) * A(2, 0) + A(0, 8) * A(1, 1) * A(2, 7) + A(0, 8) * A(1, 2) * A(2, 6) +
A(0, 8) * A(1, 3) * A(2, 5) - A(0, 8) * A(1, 5) * A(2, 3) - A(0, 8) * A(1, 6) * A(2, 2) -
A(0, 8) * A(1, 7) * A(2, 1) + A(0, 9) * A(1, 0) * A(2, 7) + A(0, 9) * A(1, 1) * A(2, 6) +
A(0, 9) * A(1, 2) * A(2, 5) + A(0, 9) * A(1, 3) * A(2, 4) - A(0, 9) * A(1, 4) * A(2, 3) -
A(0, 9) * A(1, 5) * A(2, 2) - A(0, 9) * A(1, 6) * A(2, 1) - A(0, 9) * A(1, 7) * A(2, 0) +
A(0, 10) * A(1, 0) * A(2, 6) + A(0, 10) * A(1, 1) * A(2, 5) + A(0, 10) * A(1, 2) * A(2, 4) -
A(0, 10) * A(1, 4) * A(2, 2) - A(0, 10) * A(1, 5) * A(2, 1) - A(0, 10) * A(1, 6) * A(2, 0) +
A(1, 0) * A(0, 11) * A(2, 5) + A(1, 0) * A(0, 12) * A(2, 4) + A(0, 11) * A(1, 1) * A(2, 4) -
A(0, 11) * A(1, 4) * A(2, 1) - A(0, 11) * A(1, 5) * A(2, 0) - A(0, 12) * A(1, 4) * A(2, 0) -
A(0, 0) * A(2, 4) * A(1, 12) - A(0, 0) * A(2, 5) * A(1, 11) - A(0, 0) * A(2, 6) * A(1, 10) -
A(0, 1) * A(2, 4) * A(1, 11) - A(0, 1) * A(2, 5) * A(1, 10) - A(0, 2) * A(2, 4) * A(1, 10) +
A(0, 4) * A(2, 0) * A(1, 12) + A(0, 4) * A(2, 1) * A(1, 11) + A(0, 4) * A(2, 2) * A(1, 10) +
A(0, 5) * A(2, 0) * A(1, 11) + A(0, 5) * A(2, 1) * A(1, 10) + A(0, 6) * A(2, 0) * A(1, 10) +
A(0, 0) * A(1, 4) * A(2, 12) + A(0, 0) * A(1, 5) * A(2, 11) + A(0, 0) * A(1, 6) * A(2, 10) +
A(0, 1) * A(1, 4) * A(2, 11) + A(0, 1) * A(1, 5) * A(2, 10) + A(0, 2) * A(1, 4) * A(2, 10) -
A(0, 4) * A(1, 0) * A(2, 12) - A(0, 4) * A(1, 1) * A(2, 11) - A(0, 4) * A(1, 2) * A(2, 10) -
A(0, 5) * A(1, 0) * A(2, 11) - A(0, 5) * A(1, 1) * A(2, 10) - A(0, 6) * A(1, 0) * A(2, 10);
c[7] = A(0, 0) * A(1, 6) * A(2, 9) + A(0, 0) * A(1, 7) * A(2, 8) - A(0, 0) * A(1, 8) * A(2, 7) -
A(0, 0) * A(1, 9) * A(2, 6) + A(0, 1) * A(1, 5) * A(2, 9) + A(0, 1) * A(1, 6) * A(2, 8) -
A(0, 1) * A(1, 8) * A(2, 6) - A(0, 1) * A(1, 9) * A(2, 5) + A(0, 2) * A(1, 4) * A(2, 9) +
A(0, 2) * A(1, 5) * A(2, 8) - A(0, 2) * A(1, 8) * A(2, 5) - A(0, 2) * A(1, 9) * A(2, 4) +
A(0, 3) * A(1, 4) * A(2, 8) - A(0, 3) * A(1, 8) * A(2, 4) - A(0, 4) * A(1, 2) * A(2, 9) -
A(0, 4) * A(1, 3) * A(2, 8) + A(0, 4) * A(1, 8) * A(2, 3) + A(0, 4) * A(1, 9) * A(2, 2) -
A(0, 5) * A(1, 1) * A(2, 9) - A(0, 5) * A(1, 2) * A(2, 8) + A(0, 5) * A(1, 8) * A(2, 2) +
A(0, 5) * A(1, 9) * A(2, 1) - A(0, 6) * A(1, 0) * A(2, 9) - A(0, 6) * A(1, 1) * A(2, 8) +
A(0, 6) * A(1, 8) * A(2, 1) + A(0, 6) * A(1, 9) * A(2, 0) - A(0, 7) * A(1, 0) * A(2, 8) +
A(0, 7) * A(1, 8) * A(2, 0) + A(0, 8) * A(1, 0) * A(2, 7) + A(0, 8) * A(1, 1) * A(2, 6) +
A(0, 8) * A(1, 2) * A(2, 5) + A(0, 8) * A(1, 3) * A(2, 4) - A(0, 8) * A(1, 4) * A(2, 3) -
A(0, 8) * A(1, 5) * A(2, 2) - A(0, 8) * A(1, 6) * A(2, 1) - A(0, 8) * A(1, 7) * A(2, 0) +
A(0, 9) * A(1, 0) * A(2, 6) + A(0, 9) * A(1, 1) * A(2, 5) + A(0, 9) * A(1, 2) * A(2, 4) -
A(0, 9) * A(1, 4) * A(2, 2) - A(0, 9) * A(1, 5) * A(2, 1) - A(0, 9) * A(1, 6) * A(2, 0) +
A(0, 10) * A(1, 0) * A(2, 5) + A(0, 10) * A(1, 1) * A(2, 4) - A(0, 10) * A(1, 4) * A(2, 1) -
A(0, 10) * A(1, 5) * A(2, 0) + A(1, 0) * A(0, 11) * A(2, 4) - A(0, 11) * A(1, 4) * A(2, 0) -
A(0, 0) * A(2, 4) * A(1, 11) - A(0, 0) * A(2, 5) * A(1, 10) - A(0, 1) * A(2, 4) * A(1, 10) +
A(0, 4) * A(2, 0) * A(1, 11) + A(0, 4) * A(2, 1) * A(1, 10) + A(0, 5) * A(2, 0) * A(1, 10) +
A(0, 0) * A(1, 4) * A(2, 11) + A(0, 0) * A(1, 5) * A(2, 10) + A(0, 1) * A(1, 4) * A(2, 10) -
A(0, 4) * A(1, 0) * A(2, 11) - A(0, 4) * A(1, 1) * A(2, 10) - A(0, 5) * A(1, 0) * A(2, 10);
c[8] = A(0, 0) * A(1, 5) * A(2, 9) + A(0, 0) * A(1, 6) * A(2, 8) - A(0, 0) * A(1, 8) * A(2, 6) -
A(0, 0) * A(1, 9) * A(2, 5) + A(0, 1) * A(1, 4) * A(2, 9) + A(0, 1) * A(1, 5) * A(2, 8) -
A(0, 1) * A(1, 8) * A(2, 5) - A(0, 1) * A(1, 9) * A(2, 4) + A(0, 2) * A(1, 4) * A(2, 8) -
A(0, 2) * A(1, 8) * A(2, 4) - A(0, 4) * A(1, 1) * A(2, 9) - A(0, 4) * A(1, 2) * A(2, 8) +
A(0, 4) * A(1, 8) * A(2, 2) + A(0, 4) * A(1, 9) * A(2, 1) - A(0, 5) * A(1, 0) * A(2, 9) -
A(0, 5) * A(1, 1) * A(2, 8) + A(0, 5) * A(1, 8) * A(2, 1) + A(0, 5) * A(1, 9) * A(2, 0) -
A(0, 6) * A(1, 0) * A(2, 8) + A(0, 6) * A(1, 8) * A(2, 0) + A(0, 8) * A(1, 0) * A(2, 6) +
A(0, 8) * A(1, 1) * A(2, 5) + A(0, 8) * A(1, 2) * A(2, 4) - A(0, 8) * A(1, 4) * A(2, 2) -
A(0, 8) * A(1, 5) * A(2, 1) - A(0, 8) * A(1, 6) * A(2, 0) + A(0, 9) * A(1, 0) * A(2, 5) +
A(0, 9) * A(1, 1) * A(2, 4) - A(0, 9) * A(1, 4) * A(2, 1) - A(0, 9) * A(1, 5) * A(2, 0) +
A(0, 10) * A(1, 0) * A(2, 4) - A(0, 10) * A(1, 4) * A(2, 0) - A(0, 0) * A(2, 4) * A(1, 10) +
A(0, 4) * A(2, 0) * A(1, 10) + A(0, 0) * A(1, 4) * A(2, 10) - A(0, 4) * A(1, 0) * A(2, 10);
c[9] = A(0, 0) * A(1, 4) * A(2, 9) + A(0, 0) * A(1, 5) * A(2, 8) - A(0, 0) * A(1, 8) * A(2, 5) -
A(0, 0) * A(1, 9) * A(2, 4) + A(0, 1) * A(1, 4) * A(2, 8) - A(0, 1) * A(1, 8) * A(2, 4) -
A(0, 4) * A(1, 0) * A(2, 9) - A(0, 4) * A(1, 1) * A(2, 8) + A(0, 4) * A(1, 8) * A(2, 1) +
A(0, 4) * A(1, 9) * A(2, 0) - A(0, 5) * A(1, 0) * A(2, 8) + A(0, 5) * A(1, 8) * A(2, 0) +
A(0, 8) * A(1, 0) * A(2, 5) + A(0, 8) * A(1, 1) * A(2, 4) - A(0, 8) * A(1, 4) * A(2, 1) -
A(0, 8) * A(1, 5) * A(2, 0) + A(0, 9) * A(1, 0) * A(2, 4) - A(0, 9) * A(1, 4) * A(2, 0);
c[10] = A(0, 0) * A(1, 4) * A(2, 8) - A(0, 0) * A(1, 8) * A(2, 4) - A(0, 4) * A(1, 0) * A(2, 8) +
A(0, 4) * A(1, 8) * A(2, 0) + A(0, 8) * A(1, 0) * A(2, 4) - A(0, 8) * A(1, 4) * A(2, 0);
// Solve for the roots using sturm bracketing
double roots[10];
int n_sols = poselib::sturm::bisect_sturm<10>(c, roots);
// Back substitution to recover essential matrices
Eigen::Matrix<double, 3, 2> B;
Eigen::Matrix<double, 3, 1> b;
Eigen::Matrix<double, 2, 1> xz;
Eigen::Matrix<double, 3, 3> E;
Eigen::Map<Eigen::Matrix<double, 1, 9>> e(E.data());
essential_matrices->reserve(n_sols);
for (int i = 0; i < n_sols; ++i) {
const double z = roots[i];
const double z2 = z * z;
const double z3 = z2 * z;
const double z4 = z2 * z2;
B.col(0) = A.block<3, 1>(0, 0) * z3 + A.block<3, 1>(0, 1) * z2 + A.block<3, 1>(0, 2) * z + A.block<3, 1>(0, 3);
B.col(1) = A.block<3, 1>(0, 4) * z3 + A.block<3, 1>(0, 5) * z2 + A.block<3, 1>(0, 6) * z + A.block<3, 1>(0, 7);
b = A.block<3, 1>(0, 8) * z4 + A.block<3, 1>(0, 9) * z3 + A.block<3, 1>(0, 10) * z2 + A.block<3, 1>(0, 11) * z +
A.block<3, 1>(0, 12);
// We try to solve using top two rows
xz = B.block<2, 2>(0, 0).inverse() * b.block<2, 1>(0, 0);
// If this fails we revert to more expensive QR solver using all three rows
if (std::abs(B.row(2) * xz - b(2)) > 1e-6) {
xz = B.colPivHouseholderQr().solve(b);
}
const double x = -xz(0), y = -xz(1);
e = N.row(0) * x + N.row(1) * y + N.row(2) * z + N.row(3);
// Since the rows of N are orthogonal unit vectors, we can normalize the coefficients instead
const double inv_norm = 1.0 / std::sqrt(x * x + y * y + z * z + 1.0);
e *= inv_norm;
essential_matrices->push_back(E);
}
return n_sols;
}
int relpose_5pt(const std::vector<Eigen::Vector3d> &x1, const std::vector<Eigen::Vector3d> &x2,
std::vector<CameraPose> *output) {
std::vector<Eigen::Matrix3d> essential_matrices;
int n_sols = relpose_5pt(x1, x2, &essential_matrices);
output->clear();
output->reserve(n_sols);
for (int i = 0; i < n_sols; ++i) {
motion_from_essential(essential_matrices[i], x1, x2, output);
}
return output->size();
}
} // namespace poselib
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