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Merge pull request opencv#21702 from mlourakis:4.x
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Fixes and optimizations for the SQPnP solver

* Fixes and optimizations

- optimized the calculation of qa_sum by moving equal elements outside the loop
- unrolled copying of the lower triangle of omega
- substituted SVD with eigendecomposition in the factorization of omega (2-3 times faster)
- fixed the initialization of lambda in FOAM
- added a cheirality test that checks a solution on all 3D points rather than on their mean. The old test rejected valid poses in some cases
- fixed some typos & errors in comments

* reverted to SVD

Eigen decomposition seems to yield larger errors in certain tests, reverted to SVD

* nearestRotationMatrixSVD

Added nearestRotationMatrixSVD()
Previous nearestRotationMatrix() renamed to nearestRotationMatrixFOAM() and reverts to nearestRotationMatrixSVD() for singular matrices

* fixed checks order

Fixed the order of checks in PoseSolver::solveInternal()
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@mlourakis
mlourakis authored May 13, 2022
1 parent 602caa9 commit 8d0fbc6
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134 changes: 92 additions & 42 deletions modules/calib3d/src/sqpnp.cpp
View file
Original file line number Diff line number Diff line change
Expand Up @@ -118,7 +118,7 @@ void PoseSolver::solve(InputArray objectPoints, InputArray imagePoints, OutputAr
num_solutions_ = 0;

computeOmega(_objectPoints, _imagePoints);
solveInternal();
solveInternal(_objectPoints);

int depthRot = rvecs.fixedType() ? rvecs.depth() : CV_64F;
int depthTrans = tvecs.fixedType() ? tvecs.depth() : CV_64F;
Expand Down Expand Up @@ -194,37 +194,41 @@ void PoseSolver::computeOmega(InputArray objectPoints, InputArray imagePoints)
omega_(7, 7) += sq_norm * Y2; omega_(7, 8) += sq_norm * YZ;
omega_(8, 8) += sq_norm * Z2;

//Compute qa_sum
//Compute qa_sum. Certain pairs of elements are equal, so filling them outside the loop saves some operations
qa_sum(0, 0) += X; qa_sum(0, 1) += Y; qa_sum(0, 2) += Z;
qa_sum(1, 3) += X; qa_sum(1, 4) += Y; qa_sum(1, 5) += Z;

qa_sum(0, 6) += -x * X; qa_sum(0, 7) += -x * Y; qa_sum(0, 8) += -x * Z;
qa_sum(1, 6) += -y * X; qa_sum(1, 7) += -y * Y; qa_sum(1, 8) += -y * Z;

qa_sum(2, 0) += -x * X; qa_sum(2, 1) += -x * Y; qa_sum(2, 2) += -x * Z;
qa_sum(2, 3) += -y * X; qa_sum(2, 4) += -y * Y; qa_sum(2, 5) += -y * Z;

qa_sum(2, 6) += sq_norm * X; qa_sum(2, 7) += sq_norm * Y; qa_sum(2, 8) += sq_norm * Z;
}

//Complete qa_sum
qa_sum(1, 3) = qa_sum(0, 0); qa_sum(1, 4) = qa_sum(0, 1); qa_sum(1, 5) = qa_sum(0, 2);
qa_sum(2, 0) = qa_sum(0, 6); qa_sum(2, 1) = qa_sum(0, 7); qa_sum(2, 2) = qa_sum(0, 8);
qa_sum(2, 3) = qa_sum(1, 6); qa_sum(2, 4) = qa_sum(1, 7); qa_sum(2, 5) = qa_sum(1, 8);


//lower triangles of omega_'s off-diagonal blocks (0:2, 6:8), (3:5, 6:8) and (6:8, 6:8)
omega_(1, 6) = omega_(0, 7); omega_(2, 6) = omega_(0, 8); omega_(2, 7) = omega_(1, 8);
omega_(4, 6) = omega_(3, 7); omega_(5, 6) = omega_(3, 8); omega_(5, 7) = omega_(4, 8);
omega_(7, 6) = omega_(6, 7); omega_(8, 6) = omega_(6, 8); omega_(8, 7) = omega_(7, 8);


//upper triangle of omega_'s block (3:5, 3:5)
omega_(3, 3) = omega_(0, 0); omega_(3, 4) = omega_(0, 1); omega_(3, 5) = omega_(0, 2);
omega_(4, 4) = omega_(1, 1); omega_(4, 5) = omega_(1, 2);
omega_(5, 5) = omega_(2, 2);

//Mirror upper triangle to lower triangle
for (int r = 0; r < 9; r++)
{
for (int c = 0; c < r; c++)
{
omega_(r, c) = omega_(c, r);
}
}
omega_(4, 4) = omega_(1, 1); omega_(4, 5) = omega_(1, 2);
omega_(5, 5) = omega_(2, 2);

//Mirror omega_'s upper triangle to lower triangle
//Note that elements (7, 6), (8, 6) & (8, 7) have already been assigned above
omega_(1, 0) = omega_(0, 1);
omega_(2, 0) = omega_(0, 2); omega_(2, 1) = omega_(1, 2);
omega_(3, 0) = omega_(0, 3); omega_(3, 1) = omega_(1, 3); omega_(3, 2) = omega_(2, 3);
omega_(4, 0) = omega_(0, 4); omega_(4, 1) = omega_(1, 4); omega_(4, 2) = omega_(2, 4); omega_(4, 3) = omega_(3, 4);
omega_(5, 0) = omega_(0, 5); omega_(5, 1) = omega_(1, 5); omega_(5, 2) = omega_(2, 5); omega_(5, 3) = omega_(3, 5); omega_(5, 4) = omega_(4, 5);
omega_(6, 0) = omega_(0, 6); omega_(6, 1) = omega_(1, 6); omega_(6, 2) = omega_(2, 6); omega_(6, 3) = omega_(3, 6); omega_(6, 4) = omega_(4, 6); omega_(6, 5) = omega_(5, 6);
omega_(7, 0) = omega_(0, 7); omega_(7, 1) = omega_(1, 7); omega_(7, 2) = omega_(2, 7); omega_(7, 3) = omega_(3, 7); omega_(7, 4) = omega_(4, 7); omega_(7, 5) = omega_(5, 7);
omega_(8, 0) = omega_(0, 8); omega_(8, 1) = omega_(1, 8); omega_(8, 2) = omega_(2, 8); omega_(8, 3) = omega_(3, 8); omega_(8, 4) = omega_(4, 8); omega_(8, 5) = omega_(5, 8);

cv::Matx<double, 3, 3> q;
q(0, 0) = n; q(0, 1) = 0; q(0, 2) = -sum_img.x;
Expand All @@ -247,6 +251,11 @@ void PoseSolver::computeOmega(InputArray objectPoints, InputArray imagePoints)
cv::SVD omega_svd(omega_, cv::SVD::FULL_UV);
s_ = omega_svd.w;
u_ = cv::Mat(omega_svd.vt.t());
#if 0
// EVD equivalent of the SVD; less accurate
cv::eigen(omega_, s_, u_);
u_ = u_.t(); // eigenvectors were returned as rows
#endif

CV_Assert(s_(0) >= 1e-7);

Expand All @@ -257,7 +266,7 @@ void PoseSolver::computeOmega(InputArray objectPoints, InputArray imagePoints)
point_mean_ = cv::Vec3d(sum_obj.x / n, sum_obj.y / n, sum_obj.z / n);
}

void PoseSolver::solveInternal()
void PoseSolver::solveInternal(InputArray objectPoints)
{
double min_sq_err = std::numeric_limits<double>::max();
int num_eigen_points = num_null_vectors_ > 0 ? num_null_vectors_ : 1;
Expand All @@ -274,42 +283,39 @@ void PoseSolver::solveInternal()
{
solutions[0].r_hat = det3x3(e) * e;
solutions[0].t = p_ * solutions[0].r_hat;
checkSolution(solutions[0], min_sq_err);
checkSolution(solutions[0], objectPoints, min_sq_err);
}
else
{
Matx<double, 9, 1> r;
nearestRotationMatrix(e, r);
nearestRotationMatrixFOAM(e, r);
solutions[0] = runSQP(r);
solutions[0].t = p_ * solutions[0].r_hat;
checkSolution(solutions[0], min_sq_err);
checkSolution(solutions[0], objectPoints, min_sq_err);

nearestRotationMatrix(-e, r);
nearestRotationMatrixFOAM(-e, r);
solutions[1] = runSQP(r);
solutions[1].t = p_ * solutions[1].r_hat;
checkSolution(solutions[1], min_sq_err);
checkSolution(solutions[1], objectPoints, min_sq_err);
}
}

int c = 1;

while (min_sq_err > 3 * s_[9 - num_eigen_points - c] && 9 - num_eigen_points - c > 0)
int index, c = 1;
while ((index = 9 - num_eigen_points - c) > 0 && min_sq_err > 3 * s_[index])
{
int index = 9 - num_eigen_points - c;

const cv::Matx<double, 9, 1> e = u_.col(index);
SQPSolution solutions[2];

Matx<double, 9, 1> r;
nearestRotationMatrix(e, r);
nearestRotationMatrixFOAM(e, r);
solutions[0] = runSQP(r);
solutions[0].t = p_ * solutions[0].r_hat;
checkSolution(solutions[0], min_sq_err);
checkSolution(solutions[0], objectPoints, min_sq_err);

nearestRotationMatrix(-e, r);
nearestRotationMatrixFOAM(-e, r);
solutions[1] = runSQP(r);
solutions[1].t = p_ * solutions[1].r_hat;
checkSolution(solutions[1], min_sq_err);
checkSolution(solutions[1], objectPoints, min_sq_err);

c++;
}
Expand Down Expand Up @@ -341,7 +347,7 @@ PoseSolver::SQPSolution PoseSolver::runSQP(const cv::Matx<double, 9, 1>& r0)

if (det_r > SQP_DET_THRESHOLD)
{
nearestRotationMatrix(r, solution.r_hat);
nearestRotationMatrixFOAM(r, solution.r_hat);
}
else
{
Expand Down Expand Up @@ -615,37 +621,59 @@ void PoseSolver::computeRowAndNullspace(const cv::Matx<double, 9, 1>& r,

}

// faster nearest rotation computation based on FOAM (see: http://users.ics.forth.gr/~lourakis/publ/2018_iros.pdf )
// if e = u*w*vt then r=u*diag([1, 1, det(u)*det(v)])*vt
void PoseSolver::nearestRotationMatrixSVD(const cv::Matx<double, 9, 1>& e,
cv::Matx<double, 9, 1>& r)
{
cv::Matx<double, 3, 3> e33 = e.reshape<3, 3>();
cv::SVD e33_svd(e33, cv::SVD::FULL_UV);
double detuv = cv::determinant(e33_svd.u)*cv::determinant(e33_svd.vt);
cv::Matx<double, 3, 3> diag = cv::Matx33d::eye();
diag(2, 2) = detuv;
cv::Matx<double, 3, 3> r33 = cv::Mat(e33_svd.u*diag*e33_svd.vt);
r = r33.reshape<9, 1>();
}

// Faster nearest rotation computation based on FOAM. See M. Lourakis: "An Efficient Solution to Absolute Orientation", ICPR 2016
// and M. Lourakis, G. Terzakis: "Efficient Absolute Orientation Revisited", IROS 2018.
/* Solve the nearest orthogonal approximation problem
* i.e., given e, find R minimizing ||R-e||_F
*
* The computation borrows from Markley's FOAM algorithm
* "Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm", J. Astronaut. Sci.
* "Attitude Determination Using Vector Observations: A Fast Optimal Matrix Algorithm", J. Astronaut. Sci. 1993.
*
* See also M. Lourakis: "An Efficient Solution to Absolute Orientation", ICPR 2016
*
* Copyright (C) 2019 Manolis Lourakis (lourakis **at** ics forth gr)
* Institute of Computer Science, Foundation for Research & Technology - Hellas
* Heraklion, Crete, Greece.
*/
void PoseSolver::nearestRotationMatrix(const cv::Matx<double, 9, 1>& e,
void PoseSolver::nearestRotationMatrixFOAM(const cv::Matx<double, 9, 1>& e,
cv::Matx<double, 9, 1>& r)
{
int i;
double l, lprev, det_e, e_sq, adj_e_sq, adj_e[9];

// det(e)
det_e = e(0) * e(4) * e(8) - e(0) * e(5) * e(7) - e(1) * e(3) * e(8) + e(2) * e(3) * e(7) + e(1) * e(6) * e(5) - e(2) * e(6) * e(4);
if (fabs(det_e) < 1E-04) { // singular, handle it with SVD
PoseSolver::nearestRotationMatrixSVD(e, r);
return;
}

// e's adjoint
adj_e[0] = e(4) * e(8) - e(5) * e(7); adj_e[1] = e(2) * e(7) - e(1) * e(8); adj_e[2] = e(1) * e(5) - e(2) * e(4);
adj_e[3] = e(5) * e(6) - e(3) * e(8); adj_e[4] = e(0) * e(8) - e(2) * e(6); adj_e[5] = e(2) * e(3) - e(0) * e(5);
adj_e[6] = e(3) * e(7) - e(4) * e(6); adj_e[7] = e(1) * e(6) - e(0) * e(7); adj_e[8] = e(0) * e(4) - e(1) * e(3);

// det(e), ||e||^2, ||adj(e)||^2
det_e = e(0) * e(4) * e(8) - e(0) * e(5) * e(7) - e(1) * e(3) * e(8) + e(2) * e(3) * e(7) + e(1) * e(6) * e(5) - e(2) * e(6) * e(4);
// ||e||^2, ||adj(e)||^2
e_sq = e(0) * e(0) + e(1) * e(1) + e(2) * e(2) + e(3) * e(3) + e(4) * e(4) + e(5) * e(5) + e(6) * e(6) + e(7) * e(7) + e(8) * e(8);
adj_e_sq = adj_e[0] * adj_e[0] + adj_e[1] * adj_e[1] + adj_e[2] * adj_e[2] + adj_e[3] * adj_e[3] + adj_e[4] * adj_e[4] + adj_e[5] * adj_e[5] + adj_e[6] * adj_e[6] + adj_e[7] * adj_e[7] + adj_e[8] * adj_e[8];

// compute l_max with Newton-Raphson from FOAM's characteristic polynomial, i.e. eq.(23) - (26)
for (i = 200, l = 2.0, lprev = 0.0; fabs(l - lprev) > 1E-12 * fabs(lprev) && i > 0; --i) {
l = 0.5*(e_sq + 3.0); // 1/2*(trace(mat(e)*mat(e)') + trace(eye(3)))
if (det_e < 0.0) l = -l;
for (i = 15, lprev = 0.0; fabs(l - lprev) > 1E-12 * fabs(lprev) && i > 0; --i) {
double tmp, p, pp;

tmp = (l * l - e_sq);
Expand Down Expand Up @@ -719,9 +747,31 @@ inline bool PoseSolver::positiveDepth(const SQPSolution& solution) const
return (r(6) * mean(0) + r(7) * mean(1) + r(8) * mean(2) + t(2) > 0);
}

void PoseSolver::checkSolution(SQPSolution& solution, double& min_error)
inline bool PoseSolver::positiveMajorityDepths(const SQPSolution& solution, InputArray objectPoints) const
{
const cv::Matx<double, 9, 1>& r = solution.r_hat;
const cv::Matx<double, 3, 1>& t = solution.t;
int npos = 0, nneg = 0;

Mat _objectPoints = objectPoints.getMat();

int n = _objectPoints.cols * _objectPoints.rows;

for (int i = 0; i < n; i++)
{
const cv::Point3d& obj_pt = _objectPoints.at<cv::Point3d>(i);
if (r(6) * obj_pt.x + r(7) * obj_pt.y + r(8) * obj_pt.z + t(2) > 0) ++npos;
else ++nneg;
}

return npos >= nneg;
}


void PoseSolver::checkSolution(SQPSolution& solution, InputArray objectPoints, double& min_error)
{
if (positiveDepth(solution))
bool cheirok = positiveDepth(solution) || positiveMajorityDepths(solution, objectPoints); // check the majority if the check with centroid fails
if (cheirok)
{
solution.sq_error = (omega_ * solution.r_hat).ddot(solution.r_hat);
if (fabs(min_error - solution.sq_error) > EQUAL_SQUARED_ERRORS_DIFF)
Expand Down
47 changes: 33 additions & 14 deletions modules/calib3d/src/sqpnp.hpp
View file
Original file line number Diff line number Diff line change
Expand Up @@ -85,45 +85,64 @@ class PoseSolver {

/*
* @brief Computes the 9x9 PSD Omega matrix and supporting matrices.
* @param objectPoints The 3D points in object coordinates.
*/
void solveInternal();
void solveInternal(InputArray objectPoints);

/*
* @brief Produces the distance from being orthogonal for a given 3x3 matrix
* in row-major form.
* @param e The vector to test representing a 3x3 matrix in row major form.
* in row-major order.
* @param e The vector to test representing a 3x3 matrix in row-major order.
* @return The distance the matrix is from being orthogonal.
*/
static double orthogonalityError(const cv::Matx<double, 9, 1>& e);

/*
* @brief Processes a solution and sorts it by error.
* @param solution The solution to evaluate.
* @param min_error The current minimum error.
* @param objectPoints The 3D points in object coordinates.
* @param min_error The current minimum error.
*/
void checkSolution(SQPSolution& solution, double& min_error);
void checkSolution(SQPSolution& solution, InputArray objectPoints, double& min_error);

/*
* @brief Computes the determinant of a matrix stored in row-major format.
* @param e Vector representing a 3x3 matrix stored in row-major format.
* @brief Computes the determinant of a matrix stored in row-major order.
* @param e Vector representing a 3x3 matrix stored in row-major order.
* @return The determinant of the matrix.
*/
static double det3x3(const cv::Matx<double, 9, 1>& e);

/*
* @brief Tests the cheirality for a given solution.
* @brief Tests the cheirality on the mean object point for a given solution.
* @param solution The solution to evaluate.
*/
inline bool positiveDepth(const SQPSolution& solution) const;

/*
* @brief Determines the nearest rotation matrix to a given rotaiton matrix.
* Input and output are 9x1 vector representing a vector stored in row-major
* form.
* @param e The input 3x3 matrix stored in a vector in row-major form.
* @param r The nearest rotation matrix to the input e (again in row-major form).
* @brief Tests the cheirality on all object points for a given solution.
* @param solution The solution to evaluate.
* @param objectPoints The 3D points in object coordinates.
*/
inline bool positiveMajorityDepths(const SQPSolution& solution, InputArray objectPoints) const;

/*
* @brief Determines the nearest rotation matrix to a given rotation matrix using SVD.
* Input and output are 9x1 vector representing a matrix stored in row-major
* order.
* @param e The input 3x3 matrix stored in a vector in row-major order.
* @param r The nearest rotation matrix to the input e (again in row-major order).
*/
static void nearestRotationMatrixSVD(const cv::Matx<double, 9, 1>& e,
cv::Matx<double, 9, 1>& r);

/*
* @brief Determines the nearest rotation matrix to a given rotation matrix using the FOAM algorithm.
* Input and output are 9x1 vector representing a matrix stored in row-major
* order.
* @param e The input 3x3 matrix stored in a vector in row-major order.
* @param r The nearest rotation matrix to the input e (again in row-major order).
*/
static void nearestRotationMatrix(const cv::Matx<double, 9, 1>& e,
static void nearestRotationMatrixFOAM(const cv::Matx<double, 9, 1>& e,
cv::Matx<double, 9, 1>& r);

/*
Expand Down

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