void
MetricBundleOptimizerBase::poseDerivatives(int i, int j, Vector3d& XX,
Matrix3x6d& d_dRT, Matrix3x3d& d_dX) const
{
XX = _cams[i].transformPointIntoCameraSpace(_Xs[j]);
// See Frank Dellaerts bundle adjustment tutorial.
// d(dR * R0 * X + t)/d omega = -[R0 * X]_x
Matrix3x3d J;
makeCrossProductMatrix(XX - _cams[i].getTranslation(), J);
scaleMatrixIP(-1.0, J);
// Now the transformation from world coords into camera space is xx = Rx + T
// Hence the derivative of x wrt. T is just the identity matrix.
makeIdentityMatrix(d_dRT);
copyMatrixSlice(J, 0, 0, 3, 3, d_dRT, 0, 3);
// The derivative of Rx+T wrt x is just R.
copyMatrix(_cams[i].getRotation(), d_dX);
} // end MetricBundleOptimizerBase::poseDerivatives()
void
computeConsistentRotations_LSQ(int const nIterations, int const nViews,
std::vector<Matrix3x3d> const& relativeRotations,
std::vector<std::pair<int, int> > const& viewPairs,
std::vector<Matrix3x3d>& rotations)
{
double const gamma = 1.0;
int const nRelPoses = relativeRotations.size();
rotations.resize(nViews);
Matrix3x3d zero3x3d;
makeZeroMatrix(zero3x3d);
Matrix4x4d zeroQuat;
makeZeroMatrix(zeroQuat); zeroQuat[0][0] = 1;
vector<double> denomQ(nViews, 1.0); // from the psd constraint
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
denomQ[i] += 1;
denomQ[j] += 1;
}
for (int i = 0; i < nViews; ++i) denomQ[i] = 1.0 / denomQ[i];
vector<Matrix4x4d> Q(nViews, zeroQuat);
vector<Matrix4x4d> Q1(nViews, zeroQuat);
vector<Matrix4x4d> ZQ1(nViews, zeroQuat);
vector<Matrix4x4d> Q2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> Q2j(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2j(nRelPoses, zeroQuat);
vector<Matrix3x3d> A(nRelPoses, zero3x3d);
vector<Matrix3x3d> A1(nRelPoses, zero3x3d);
vector<Matrix3x3d> A2(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA1(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA2(nRelPoses, zero3x3d);
for (int iter = 0; iter < nIterations; ++iter)
{
// Convex hull of rotation matrices
for (int i = 0; i < nViews; ++i)
{
Matrix4x4d q = Q[i] + ZQ1[i];
projectConvHull_SO3(q);
Q1[i] = q;
addMatricesIP(Q[i] - q, ZQ1[i]);
} // end for (i)
// Squared Frobenius term 0.5*sum_k |A_k|_F^2
for (int k = 0; k < nRelPoses; ++k)
{
Matrix3x3d a = A[k] + ZA1[k];
scaleMatrixIP(1.0 / (1.0 + gamma), a);
A1[k] = a;
addMatricesIP(A[k] - a, ZA1[k]);
} // end for (k)
// Enforce linear consistency
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
Matrix4x4d qi = Q[i] + ZQ2i[k];
Matrix4x4d qj = Q[j] + ZQ2j[k];
Matrix3x3d a = A[k] + ZA2[k];
proxConsistency(relativeRotations[k], qi, qj, a);
Q2i[k] = qi;
Q2j[k] = qj;
A2[k] = a;
addMatricesIP(Q[i] - qi, ZQ2i[k]);
addMatricesIP(Q[j] - qj, ZQ2j[k]);
addMatricesIP(A[k] - a, ZA2[k]);
} // end for (k)
// Averaging of the solutions
for (int i = 0; i < nViews; ++i) Q[i] = Q1[i] - ZQ1[i];
for (int k = 0; k < nRelPoses; ++k) A[k] = A1[k] - ZA1[k];
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
addMatricesIP(Q2i[k] - ZQ2i[k], Q[i]);
addMatricesIP(Q2j[k] - ZQ2j[k], Q[j]);
addMatricesIP(A2[k] - ZA2[k], A[k]);
} // end for (k)
for (int i = 0; i < nViews; ++i) scaleMatrixIP(denomQ[i], Q[i]);
for (int k = 0; k < nRelPoses; ++k) scaleMatrixIP(0.5, A[k]);
if ((iter % 500) == 0)
{
double E = 0;
for (int k = 0; k < nRelPoses; ++k) E += sqrMatrixNormFrobenius(A[k]);
cout << "iter: " << iter << " E = " << E << endl;
}
} // end for (iter)
rotations.resize(nViews);
for (int i = 0; i < nViews; ++i)
rotations[i] = getRotationFromQuat(Q[i]);
} // end computeConsistentRotations_LSQ()
void
computeConsistentRotations_Linf(double const sigma, int const nIterations, int const nViews,
std::vector<Matrix3x3d> const& relativeRotations,
std::vector<std::pair<int, int> > const& viewPairs,
std::vector<Matrix3x3d>& rotations, std::vector<double>& zs)
{
double const gamma = 1.0;
int const nRelPoses = relativeRotations.size();
rotations.resize(nViews);
Matrix3x3d zero3x3d;
makeZeroMatrix(zero3x3d);
Matrix4x4d zeroQuat;
makeZeroMatrix(zeroQuat); zeroQuat[0][0] = 1;
double const denomT = 1.0 / (1.0 + nRelPoses);
vector<double> denomQ(nViews, 1.0); // from the psd constraint
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
denomQ[i] += 1;
denomQ[j] += 1;
}
for (int i = 0; i < nViews; ++i) denomQ[i] = 1.0 / denomQ[i];
double T = 0.0;
vector<double> T1(nRelPoses);
vector<double> ZT1(nRelPoses, 0.0);
double T2;
double ZT2 = 0;
vector<Matrix4x4d> Q(nViews, zeroQuat);
vector<Matrix4x4d> Q1(nViews, zeroQuat);
vector<Matrix4x4d> ZQ1(nViews, zeroQuat);
vector<Matrix4x4d> Q2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> Q2j(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2j(nRelPoses, zeroQuat);
vector<Matrix3x3d> A(nRelPoses, zero3x3d);
vector<Matrix3x3d> A1(nRelPoses, zero3x3d);
vector<Matrix3x3d> A2(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA1(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA2(nRelPoses, zero3x3d);
for (int iter = 0; iter < nIterations; ++iter)
{
// Convex hull of rotation matrices
for (int i = 0; i < nViews; ++i)
{
Matrix4x4d q = Q[i] + ZQ1[i];
if (i > 0)
projectConvHull_SO3(q);
else
{
makeZeroMatrix(q);
q[0][0] = 1;
}
Q1[i] = q;
addMatricesIP(Q[i] - q, ZQ1[i]);
} // end for (i)
// Shrinkage of T (we want to minimize T)
T2 = std::max(0.0, T + ZT2 - gamma);
ZT2 += T - T2;
// Cone constraint
for (int k = 0; k < nRelPoses; ++k)
{
double t = T + ZT1[k];
Matrix3x3d a = A[k] + ZA1[k];
proxDataResidual_Frobenius(sigma, t, a);
T1[k] = t;
ZT1[k] += T - t;
A1[k] = a;
addMatricesIP(A[k] - a, ZA1[k]);
} // end for (k)
// Enforce linear consistency
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
Matrix4x4d qi = Q[i] + ZQ2i[k];
Matrix4x4d qj = Q[j] + ZQ2j[k];
Matrix3x3d a = A[k] + ZA2[k];
proxConsistency(relativeRotations[k], qi, qj, a);
Q2i[k] = qi;
Q2j[k] = qj;
A2[k] = a;
addMatricesIP(Q[i] - qi, ZQ2i[k]);
addMatricesIP(Q[j] - qj, ZQ2j[k]);
addMatricesIP(A[k] - a, ZA2[k]);
} // end for (k)
// Averaging of the solutions
for (int i = 0; i < nViews; ++i) Q[i] = Q1[i] - ZQ1[i];
T = T2 - ZT2;
for (int k = 0; k < nRelPoses; ++k) T += T1[k] - ZT1[k];
T *= denomT;
T = std::max(0.0, T);
for (int k = 0; k < nRelPoses; ++k) A[k] = A1[k] - ZA1[k];
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
addMatricesIP(Q2i[k] - ZQ2i[k], Q[i]);
addMatricesIP(Q2j[k] - ZQ2j[k], Q[j]);
addMatricesIP(A2[k] - ZA2[k], A[k]);
} // end for (k)
for (int i = 0; i < nViews; ++i) scaleMatrixIP(denomQ[i], Q[i]);
for (int k = 0; k < nRelPoses; ++k) scaleMatrixIP(0.5, A[k]);
if ((iter % 500) == 0)
{
cout << "iter: " << iter << " t = " << T << endl;
cout << "T1 = "; displayVector(T1);
cout << "ZT1 = "; displayVector(ZT1);
cout << "T2 = " << T2 << " ZT2 = " << ZT2 << endl;
Matrix<double> ZZ(4, 4);
for (int i = 0; i < nViews; ++i)
{
copyMatrix(Q[i], ZZ);
SVD<double> svd(ZZ);
cout << "Q = "; displayMatrix(ZZ);
cout << "SV = "; displayVector(svd.getSingularValues());
//Matrix3x3d R = getRotationFromQuat(Q[i]);
//cout << "R = "; displayMatrix(R);
} // end for (i)
}
} // end for (iter)
rotations.resize(nViews);
for (int i = 0; i < nViews; ++i)
rotations[i] = getRotationFromQuat(Q[i]);
zs = ZT1;
} // end computeConsistentRotations_Linf()
void
computeConsistentRotations_L1(double const sigma, int const nIterations, int const nViews,
std::vector<Matrix3x3d> const& relativeRotations,
std::vector<std::pair<int, int> > const& viewPairs,
std::vector<Matrix3x3d>& rotations)
{
double const gamma = 1.0;
int const nRelPoses = relativeRotations.size();
rotations.resize(nViews);
Matrix3x3d zero3x3d;
makeZeroMatrix(zero3x3d);
Matrix4x4d zeroQuat;
makeZeroMatrix(zeroQuat); zeroQuat[0][0] = 1;
vector<double> denomQ(nViews, 1.0); // from the psd constraint
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
denomQ[i] += 1;
denomQ[j] += 1;
}
for (int i = 0; i < nViews; ++i) denomQ[i] = 1.0 / denomQ[i];
vector<double> T(nRelPoses, 0.0);
vector<double> T1(nRelPoses);
vector<double> ZT1(nRelPoses, 0.0);
vector<double> T2(nRelPoses);
vector<double> ZT2(nRelPoses, 0.0);
vector<Matrix4x4d> Q(nViews, zeroQuat);
vector<Matrix4x4d> Q1(nViews, zeroQuat);
vector<Matrix4x4d> ZQ1(nViews, zeroQuat);
vector<Matrix4x4d> Q2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> Q2j(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2i(nRelPoses, zeroQuat);
vector<Matrix4x4d> ZQ2j(nRelPoses, zeroQuat);
vector<Matrix3x3d> A(nRelPoses, zero3x3d);
vector<Matrix3x3d> A1(nRelPoses, zero3x3d);
vector<Matrix3x3d> A2(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA1(nRelPoses, zero3x3d);
vector<Matrix3x3d> ZA2(nRelPoses, zero3x3d);
for (int iter = 0; iter < nIterations; ++iter)
{
// Convex hull of rotation matrices
for (int i = 0; i < nViews; ++i)
{
Matrix4x4d q = Q[i] + ZQ1[i];
projectConvHull_SO3(q);
Q1[i] = q;
addMatricesIP(Q[i] - q, ZQ1[i]);
} // end for (i)
// Shrinkage of T (we want to minimize T)
for (int k = 0; k < nRelPoses; ++k)
{
T2[k] = std::max(0.0, T[k] + ZT2[k] - gamma);
ZT2[k] += T[k] - T2[k];
} // end for (k)
// Cone constraint
for (int k = 0; k < nRelPoses; ++k)
{
double t = T1[k] + ZT1[k];
Matrix3x3d a = A[k] + ZA1[k];
proxDataResidual_Frobenius(sigma, t, a);
T1[k] = t;
ZT1[k] += T[k] - t;
A1[k] = a;
addMatricesIP(A[k] - a, ZA1[k]);
} // end for (k)
// Enforce linear consistency
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
Matrix4x4d qi = Q[i] + ZQ2i[k];
Matrix4x4d qj = Q[j] + ZQ2j[k];
Matrix3x3d a = A[k] + ZA2[k];
proxConsistency(relativeRotations[k], qi, qj, a);
Q2i[k] = qi;
Q2j[k] = qj;
A2[k] = a;
addMatricesIP(Q[i] - qi, ZQ2i[k]);
addMatricesIP(Q[j] - qj, ZQ2j[k]);
addMatricesIP(A[k] - a, ZA2[k]);
} // end for (k)
// Averaging of the solutions
for (int i = 0; i < nViews; ++i) Q[i] = Q1[i] - ZQ1[i];
for (int k = 0; k < nRelPoses; ++k)
T[k] = std::max(0.0, 0.5 * (T1[k] - ZT1[k] + T2[k] - ZT2[k]));
for (int k = 0; k < nRelPoses; ++k) A[k] = A1[k] - ZA1[k];
for (int k = 0; k < nRelPoses; ++k)
{
int const i = viewPairs[k].first;
int const j = viewPairs[k].second;
addMatricesIP(Q2i[k] - ZQ2i[k], Q[i]);
addMatricesIP(Q2j[k] - ZQ2j[k], Q[j]);
addMatricesIP(A2[k] - ZA2[k], A[k]);
} // end for (k)
for (int i = 0; i < nViews; ++i) scaleMatrixIP(denomQ[i], Q[i]);
for (int k = 0; k < nRelPoses; ++k) scaleMatrixIP(0.5, A[k]);
if ((iter % 500) == 0)
{
cout << "iter: " << iter << endl;
cout << " T = "; displayVector(T);
}
} // end for (iter)
rotations.resize(nViews);
for (int i = 0; i < nViews; ++i)
rotations[i] = getRotationFromQuat(Q[i]);
} // end computeConsistentRotations_L1()
void
computeConsistentRotations(int const nViews,
std::vector<Matrix3x3d> const& relativeRotations,
std::vector<std::pair<int, int> > const& viewPairs,
std::vector<Matrix3x3d>& rotations, int method)
{
#if !defined(V3DLIB_ENABLE_ARPACK)
if (method == V3D_CONSISTENT_ROTATION_METHOD_SPARSE_EIG)
method = V3D_CONSISTENT_ROTATION_METHOD_EIG_ATA;
#endif
int const nRelPoses = relativeRotations.size();
rotations.resize(nViews);
switch (method)
{
case V3D_CONSISTENT_ROTATION_METHOD_SVD:
{
Matrix<double> A(3*nRelPoses, 3*nViews, 0.0);
Matrix3x3d I;
makeIdentityMatrix(I);
scaleMatrixIP(-1.0, I);
for (int i = 0; i < nRelPoses; ++i)
{
int const view1 = viewPairs[i].first;
int const view2 = viewPairs[i].second;
Matrix3x3d const& Rrel = relativeRotations[i];
copyMatrixSlice(Rrel, 0, 0, 3, 3, A, 3*i, 3*view1);
copyMatrixSlice(I, 0, 0, 3, 3, A, 3*i, 3*view2);
} // end for (i)
SVD<double> svd(A);
int const startColumn = A.num_cols()-3; // last columns of right sing. vec for SVD
Matrix<double> const& V = svd.getV();
for (int i = 0; i < nViews; ++i)
{
copyMatrixSlice(V, 3*i, startColumn, 3, 3, rotations[i], 0, 0);
enforceRotationMatrix(rotations[i]);
}
break;
}
case V3D_CONSISTENT_ROTATION_METHOD_SVD_ATA:
case V3D_CONSISTENT_ROTATION_METHOD_EIG_ATA:
case V3D_CONSISTENT_ROTATION_METHOD_SPARSE_EIG:
{
vector<pair<int, int> > nzA;
vector<double> valsA;
nzA.reserve(12*nRelPoses);
valsA.reserve(12*nRelPoses);
for (int i = 0; i < nRelPoses; ++i)
{
int const view1 = viewPairs[i].first;
int const view2 = viewPairs[i].second;
Matrix3x3d const& Rrel = relativeRotations[i];
nzA.push_back(make_pair(3*i+0, 3*view1+0)); valsA.push_back(Rrel[0][0]);
nzA.push_back(make_pair(3*i+0, 3*view1+1)); valsA.push_back(Rrel[0][1]);
nzA.push_back(make_pair(3*i+0, 3*view1+2)); valsA.push_back(Rrel[0][2]);
nzA.push_back(make_pair(3*i+1, 3*view1+0)); valsA.push_back(Rrel[1][0]);
nzA.push_back(make_pair(3*i+1, 3*view1+1)); valsA.push_back(Rrel[1][1]);
nzA.push_back(make_pair(3*i+1, 3*view1+2)); valsA.push_back(Rrel[1][2]);
nzA.push_back(make_pair(3*i+2, 3*view1+0)); valsA.push_back(Rrel[2][0]);
nzA.push_back(make_pair(3*i+2, 3*view1+1)); valsA.push_back(Rrel[2][1]);
nzA.push_back(make_pair(3*i+2, 3*view1+2)); valsA.push_back(Rrel[2][2]);
nzA.push_back(make_pair(3*i+0, 3*view2+0)); valsA.push_back(-1.0);
nzA.push_back(make_pair(3*i+1, 3*view2+1)); valsA.push_back(-1.0);
nzA.push_back(make_pair(3*i+2, 3*view2+2)); valsA.push_back(-1.0);
} // end for (i)
CCS_Matrix<double> A(3*nRelPoses, 3*nViews, nzA, valsA);
if (method == V3D_CONSISTENT_ROTATION_METHOD_SPARSE_EIG)
{
#if defined(V3DLIB_ENABLE_ARPACK)
Vector<double> sigma;
Matrix<double> V;
SparseSymmetricEigConfig cfg;
cfg.maxArnoldiIterations = 100000;
computeSparseSVD(A, V3D_ARPACK_SMALLEST_MAGNITUDE_EIGENVALUES, 3, sigma, V, cfg);
//computeSparseSVD(A, V3D_ARPACK_SMALLEST_EIGENVALUES, 3, sigma, V, cfg);
for (int i = 0; i < nViews; ++i)
{
copyMatrixSlice(V, 3*i, 0, 3, 1, rotations[i], 0, 2);
copyMatrixSlice(V, 3*i, 1, 3, 1, rotations[i], 0, 1);
copyMatrixSlice(V, 3*i, 2, 3, 1, rotations[i], 0, 0);
}
#endif
}
else
{
Matrix<double> AtA(3*nViews, 3*nViews);
multiply_At_A_SparseDense(A, AtA);
if (method == V3D_CONSISTENT_ROTATION_METHOD_SVD_ATA)
{
SVD<double> svd(AtA);
int const startColumn = A.num_cols()-3; // last columns of right sing. vec for SVD
Matrix<double> const& V = svd.getV();
for (int i = 0; i < nViews; ++i)
copyMatrixSlice(V, 3*i, startColumn, 3, 3, rotations[i], 0, 0);
}
else
{
Eigenvalue<double> svd(AtA);
int const startColumn = 0; // first columns of eigenvector matrix
Matrix<double> const& V = svd.getV();
for (int i = 0; i < nViews; ++i)
copyMatrixSlice(V, 3*i, startColumn, 3, 3, rotations[i], 0, 0);
} // end if
} // end if
break;
}
default:
throwV3DErrorHere("Unknown method argument for computeConsistentRotations().");
} // end switch
for (int i = 0; i < nViews; ++i)
enforceRotationMatrix(rotations[i]);
// Remove gauge freedom by setting R[0] = I.
Matrix3x3d const R0t = rotations[0].transposed();
for (int i = 0; i < nViews; ++i)
rotations[i] = rotations[i] * R0t;
// Note: it seems, that either all Rs have det(R)=1 or all have det(R)=-1.
// Since we remove the gauge freedem by multiplying all rotations with R_0^t,
// we always end up with det(R)=1 and any code to enforce a positive determinant
// is not necessary.
} // end computeConsistentRotations()
