fundamental.cc

Go to the documentation of this file.

// Copyright (c) 2007, 2008 libmv authors.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to
// deal in the Software without restriction, including without limitation the
// rights to use, copy, modify, merge, publish, distribute, sublicense, and/or
// sell copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
// IN THE SOFTWARE.
 
#include "libmv/multiview/fundamental.h"
 
#include "ceres/ceres.h"
#include "libmv/logging/logging.h"
#include "libmv/multiview/conditioning.h"
#include "libmv/multiview/projection.h"
#include "libmv/multiview/triangulation.h"
#include "libmv/numeric/numeric.h"
#include "libmv/numeric/poly.h"
 
namespace libmv {
 
static void EliminateRow(const Mat34& P, int row, Mat* X) {
  X->resize(2, 4);
 
  int first_row = (row + 1) % 3;
  int second_row = (row + 2) % 3;
 
  for (int i = 0; i < 4; ++i) {
    (*X)(0, i) = P(first_row, i);
    (*X)(1, i) = P(second_row, i);
  }
}
 
void ProjectionsFromFundamental(const Mat3& F, Mat34* P1, Mat34* P2) {
  *P1 << Mat3::Identity(), Vec3::Zero();
  Vec3 e2;
  Mat3 Ft = F.transpose();
  Nullspace(&Ft, &e2);
  *P2 << CrossProductMatrix(e2) * F, e2;
}
 
// Addapted from vgg_F_from_P.
void FundamentalFromProjections(const Mat34& P1, const Mat34& P2, Mat3* F) {
  Mat X[3];
  Mat Y[3];
  Mat XY;
 
  for (int i = 0; i < 3; ++i) {
    EliminateRow(P1, i, X + i);
    EliminateRow(P2, i, Y + i);
  }
 
  for (int i = 0; i < 3; ++i) {
    for (int j = 0; j < 3; ++j) {
      VerticalStack(X[j], Y[i], &XY);
      (*F)(i, j) = XY.determinant();
    }
  }
}
 
// HZ 11.1 pag.279 (x1 = x, x2 = x')
// http://www.cs.unc.edu/~marc/tutorial/node54.html
static double EightPointSolver(const Mat& x1, const Mat& x2, Mat3* F) {
  DCHECK_EQ(x1.rows(), 2);
  DCHECK_GE(x1.cols(), 8);
  DCHECK_EQ(x1.rows(), x2.rows());
  DCHECK_EQ(x1.cols(), x2.cols());
 
  int n = x1.cols();
  Mat A(n, 9);
  for (int i = 0; i < n; ++i) {
    A(i, 0) = x2(0, i) * x1(0, i);
    A(i, 1) = x2(0, i) * x1(1, i);
    A(i, 2) = x2(0, i);
    A(i, 3) = x2(1, i) * x1(0, i);
    A(i, 4) = x2(1, i) * x1(1, i);
    A(i, 5) = x2(1, i);
    A(i, 6) = x1(0, i);
    A(i, 7) = x1(1, i);
    A(i, 8) = 1;
  }
 
  Vec9 f;
  double smaller_singular_value = Nullspace(&A, &f);
  *F = Map<RMat3>(f.data());
  return smaller_singular_value;
}
 
// HZ 11.1.1 pag.280
void EnforceFundamentalRank2Constraint(Mat3* F) {
  Eigen::JacobiSVD<Mat3> USV(*F, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Vec3 d = USV.singularValues();
  d(2) = 0.0;
  *F = USV.matrixU() * d.asDiagonal() * USV.matrixV().transpose();
}
 
// HZ 11.2 pag.281 (x1 = x, x2 = x')
double NormalizedEightPointSolver(const Mat& x1, const Mat& x2, Mat3* F) {
  DCHECK_EQ(x1.rows(), 2);
  DCHECK_GE(x1.cols(), 8);
  DCHECK_EQ(x1.rows(), x2.rows());
  DCHECK_EQ(x1.cols(), x2.cols());
 
  // Normalize the data.
  Mat3 T1, T2;
  PreconditionerFromPoints(x1, &T1);
  PreconditionerFromPoints(x2, &T2);
  Mat x1_normalized, x2_normalized;
  ApplyTransformationToPoints(x1, T1, &x1_normalized);
  ApplyTransformationToPoints(x2, T2, &x2_normalized);
 
  // Estimate the fundamental matrix.
  double smaller_singular_value =
      EightPointSolver(x1_normalized, x2_normalized, F);
  EnforceFundamentalRank2Constraint(F);
 
  // Denormalize the fundamental matrix.
  *F = T2.transpose() * (*F) * T1;
 
  return smaller_singular_value;
}
 
// Seven-point algorithm.
// http://www.cs.unc.edu/~marc/tutorial/node55.html
double FundamentalFrom7CorrespondencesLinear(const Mat& x1,
                                             const Mat& x2,
                                             std::vector<Mat3>* F) {
  DCHECK_EQ(x1.rows(), 2);
  DCHECK_EQ(x1.cols(), 7);
  DCHECK_EQ(x1.rows(), x2.rows());
  DCHECK_EQ(x2.cols(), x2.cols());
 
  // Build a 9 x n matrix from point matches, where each row is equivalent to
  // the equation x'T*F*x = 0 for a single correspondence pair (x', x). The
  // domain of the matrix is a 9 element vector corresponding to F. The
  // nullspace should be rank two; the two dimensions correspond to the set of
  // F matrices satisfying the epipolar geometry.
  Matrix<double, 7, 9> A;
  for (int ii = 0; ii < 7; ++ii) {
    A(ii, 0) = x1(0, ii) * x2(0, ii);  // 0 represents x coords,
    A(ii, 1) = x1(1, ii) * x2(0, ii);  // 1 represents y coords.
    A(ii, 2) = x2(0, ii);
    A(ii, 3) = x1(0, ii) * x2(1, ii);
    A(ii, 4) = x1(1, ii) * x2(1, ii);
    A(ii, 5) = x2(1, ii);
    A(ii, 6) = x1(0, ii);
    A(ii, 7) = x1(1, ii);
    A(ii, 8) = 1.0;
  }
 
  // Find the two F matrices in the nullspace of A.
  Vec9 f1, f2;
  double s = Nullspace2(&A, &f1, &f2);
  Mat3 F1 = Map<RMat3>(f1.data());
  Mat3 F2 = Map<RMat3>(f2.data());
 
  // Then, use the condition det(F) = 0 to determine F. In other words, solve
  // det(F1 + a*F2) = 0 for a.
  double a = F1(0, 0), j = F2(0, 0);
  double b = F1(0, 1), k = F2(0, 1);
  double c = F1(0, 2), l = F2(0, 2);
  double d = F1(1, 0), m = F2(1, 0);
  double e = F1(1, 1), n = F2(1, 1);
  double f = F1(1, 2), o = F2(1, 2);
  double g = F1(2, 0), p = F2(2, 0);
  double h = F1(2, 1), q = F2(2, 1);
  double i = F1(2, 2), r = F2(2, 2);
 
  // Run fundamental_7point_coeffs.py to get the below coefficients.
  // The coefficients are in ascending powers of alpha, i.e. P[N]*x^N.
  double P[4] = {
      a * e * i + b * f * g + c * d * h - a * f * h - b * d * i - c * e * g,
      a * e * r + a * i * n + b * f * p + b * g * o + c * d * q + c * h * m +
          d * h * l + e * i * j + f * g * k - a * f * q - a * h * o -
          b * d * r - b * i * m - c * e * p - c * g * n - d * i * k -
          e * g * l - f * h * j,
      a * n * r + b * o * p + c * m * q + d * l * q + e * j * r + f * k * p +
          g * k * o + h * l * m + i * j * n - a * o * q - b * m * r -
          c * n * p - d * k * r - e * l * p - f * j * q - g * l * n -
          h * j * o - i * k * m,
      j * n * r + k * o * p + l * m * q - j * o * q - k * m * r - l * n * p,
  };
 
  // Solve for the roots of P[3]*x^3 + P[2]*x^2 + P[1]*x + P[0] = 0.
  double roots[3];
  int num_roots = SolveCubicPolynomial(P, roots);
 
  // Build the fundamental matrix for each solution.
  for (int kk = 0; kk < num_roots; ++kk) {
    F->push_back(F1 + roots[kk] * F2);
  }
  return s;
}
 
double FundamentalFromCorrespondences7Point(const Mat& x1,
                                            const Mat& x2,
                                            std::vector<Mat3>* F) {
  DCHECK_EQ(x1.rows(), 2);
  DCHECK_GE(x1.cols(), 7);
  DCHECK_EQ(x1.rows(), x2.rows());
  DCHECK_EQ(x1.cols(), x2.cols());
 
  // Normalize the data.
  Mat3 T1, T2;
  PreconditionerFromPoints(x1, &T1);
  PreconditionerFromPoints(x2, &T2);
  Mat x1_normalized, x2_normalized;
  ApplyTransformationToPoints(x1, T1, &x1_normalized);
  ApplyTransformationToPoints(x2, T2, &x2_normalized);
 
  // Estimate the fundamental matrix.
  double smaller_singular_value = FundamentalFrom7CorrespondencesLinear(
      x1_normalized, x2_normalized, &(*F));
 
  for (int k = 0; k < F->size(); ++k) {
    Mat3& Fmat = (*F)[k];
    // Denormalize the fundamental matrix.
    Fmat = T2.transpose() * Fmat * T1;
  }
  return smaller_singular_value;
}
 
void NormalizeFundamental(const Mat3& F, Mat3* F_normalized) {
  *F_normalized = F / FrobeniusNorm(F);
  if ((*F_normalized)(2, 2) < 0) {
    *F_normalized *= -1;
  }
}
 
double SampsonDistance(const Mat& F, const Vec2& x1, const Vec2& x2) {
  Vec3 x(x1(0), x1(1), 1.0);
  Vec3 y(x2(0), x2(1), 1.0);
 
  Vec3 F_x = F * x;
  Vec3 Ft_y = F.transpose() * y;
  double y_F_x = y.dot(F_x);
 
  return Square(y_F_x) /
         (F_x.head<2>().squaredNorm() + Ft_y.head<2>().squaredNorm());
}
 
double SymmetricEpipolarDistance(const Mat& F, const Vec2& x1, const Vec2& x2) {
  Vec3 x(x1(0), x1(1), 1.0);
  Vec3 y(x2(0), x2(1), 1.0);
 
  Vec3 F_x = F * x;
  Vec3 Ft_y = F.transpose() * y;
  double y_F_x = y.dot(F_x);
 
  return Square(y_F_x) *
         (1 / F_x.head<2>().squaredNorm() + 1 / Ft_y.head<2>().squaredNorm());
}
 
// HZ 9.6 pag 257 (formula 9.12)
void EssentialFromFundamental(const Mat3& F,
                              const Mat3& K1,
                              const Mat3& K2,
                              Mat3* E) {
  *E = K2.transpose() * F * K1;
}
 
// HZ 9.6 pag 257 (formula 9.12)
// Or http://ai.stanford.edu/~birch/projective/node20.html
void FundamentalFromEssential(const Mat3& E,
                              const Mat3& K1,
                              const Mat3& K2,
                              Mat3* F) {
  *F = K2.inverse().transpose() * E * K1.inverse();
}
 
void RelativeCameraMotion(const Mat3& R1,
                          const Vec3& t1,
                          const Mat3& R2,
                          const Vec3& t2,
                          Mat3* R,
                          Vec3* t) {
  *R = R2 * R1.transpose();
  *t = t2 - (*R) * t1;
}
 
// HZ 9.6 pag 257
void EssentialFromRt(
    const Mat3& R1, const Vec3& t1, const Mat3& R2, const Vec3& t2, Mat3* E) {
  Mat3 R;
  Vec3 t;
  RelativeCameraMotion(R1, t1, R2, t2, &R, &t);
  Mat3 Tx = CrossProductMatrix(t);
  *E = Tx * R;
}
 
// HZ 9.6 pag 259 (Result 9.19)
void MotionFromEssential(const Mat3& E,
                         std::vector<Mat3>* Rs,
                         std::vector<Vec3>* ts) {
  Eigen::JacobiSVD<Mat3> USV(E, Eigen::ComputeFullU | Eigen::ComputeFullV);
  Mat3 U = USV.matrixU();
  Mat3 Vt = USV.matrixV().transpose();
 
  // Last column of U is undetermined since d = (a a 0).
  if (U.determinant() < 0) {
    U.col(2) *= -1;
  }
  // Last row of Vt is undetermined since d = (a a 0).
  if (Vt.determinant() < 0) {
    Vt.row(2) *= -1;
  }
 
  Mat3 W;
  // clang-format off
  W << 0, -1,  0,
       1,  0,  0,
       0,  0,  1;
  // clang-format on
 
  Mat3 U_W_Vt = U * W * Vt;
  Mat3 U_Wt_Vt = U * W.transpose() * Vt;
 
  Rs->resize(4);
  (*Rs)[0] = U_W_Vt;
  (*Rs)[1] = U_W_Vt;
  (*Rs)[2] = U_Wt_Vt;
  (*Rs)[3] = U_Wt_Vt;
 
  ts->resize(4);
  (*ts)[0] = U.col(2);
  (*ts)[1] = -U.col(2);
  (*ts)[2] = U.col(2);
  (*ts)[3] = -U.col(2);
}
 
int MotionFromEssentialChooseSolution(const std::vector<Mat3>& Rs,
                                      const std::vector<Vec3>& ts,
                                      const Mat3& K1,
                                      const Vec2& x1,
                                      const Mat3& K2,
                                      const Vec2& x2) {
  DCHECK_EQ(4, Rs.size());
  DCHECK_EQ(4, ts.size());
 
  Mat34 P1, P2;
  Mat3 R1;
  Vec3 t1;
  R1.setIdentity();
  t1.setZero();
  P_From_KRt(K1, R1, t1, &P1);
  for (int i = 0; i < 4; ++i) {
    const Mat3& R2 = Rs[i];
    const Vec3& t2 = ts[i];
    P_From_KRt(K2, R2, t2, &P2);
    Vec3 X;
    TriangulateDLT(P1, x1, P2, x2, &X);
    double d1 = Depth(R1, t1, X);
    double d2 = Depth(R2, t2, X);
    // Test if point is front to the two cameras.
    if (d1 > 0 && d2 > 0) {
      return i;
    }
  }
  return -1;
}
 
bool MotionFromEssentialAndCorrespondence(const Mat3& E,
                                          const Mat3& K1,
                                          const Vec2& x1,
                                          const Mat3& K2,
                                          const Vec2& x2,
                                          Mat3* R,
                                          Vec3* t) {
  std::vector<Mat3> Rs;
  std::vector<Vec3> ts;
  MotionFromEssential(E, &Rs, &ts);
  int solution = MotionFromEssentialChooseSolution(Rs, ts, K1, x1, K2, x2);
  if (solution >= 0) {
    *R = Rs[solution];
    *t = ts[solution];
    return true;
  } else {
    return false;
  }
}
 
void FundamentalToEssential(const Mat3& F, Mat3* E) {
  Eigen::JacobiSVD<Mat3> svd(F, Eigen::ComputeFullU | Eigen::ComputeFullV);
 
  // See Hartley & Zisserman page 294, result 11.1, which shows how to get the
  // closest essential matrix to a matrix that is "almost" an essential matrix.
  double a = svd.singularValues()(0);
  double b = svd.singularValues()(1);
  double s = (a + b) / 2.0;
 
  LG << "Initial reconstruction's rotation is non-euclidean by "
     << (((a - b) / std::max(a, b)) * 100)
     << "%; singular values:" << svd.singularValues().transpose();
 
  Vec3 diag;
  diag << s, s, 0;
 
  *E = svd.matrixU() * diag.asDiagonal() * svd.matrixV().transpose();
}
 
// Default settings for fundamental estimation which should be suitable
// for a wide range of use cases.
EstimateFundamentalOptions::EstimateFundamentalOptions()
    : max_num_iterations(50), expected_average_symmetric_distance(1e-16) {
}
 
namespace {
// Cost functor which computes symmetric epipolar distance
// used for fundamental matrix refinement.
class FundamentalSymmetricEpipolarCostFunctor {
 public:
  FundamentalSymmetricEpipolarCostFunctor(const Vec2& x, const Vec2& y)
      : x_(x), y_(y) {}
 
  template <typename T>
  bool operator()(const T* fundamental_parameters, T* residuals) const {
    typedef Eigen::Matrix<T, 3, 3> Mat3;
    typedef Eigen::Matrix<T, 3, 1> Vec3;
 
    Mat3 F(fundamental_parameters);
 
    Vec3 x(T(x_(0)), T(x_(1)), T(1.0));
    Vec3 y(T(y_(0)), T(y_(1)), T(1.0));
 
    Vec3 F_x = F * x;
    Vec3 Ft_y = F.transpose() * y;
    T y_F_x = y.dot(F_x);
 
    residuals[0] = y_F_x * T(1) / F_x.head(2).norm();
    residuals[1] = y_F_x * T(1) / Ft_y.head(2).norm();
 
    return true;
  }
 
  const Mat x_;
  const Mat y_;
};
 
// Termination checking callback used for fundamental estimation.
// It finished the minimization as soon as actual average of
// symmetric epipolar distance is less or equal to the expected
// average value.
class TerminationCheckingCallback : public ceres::IterationCallback {
 public:
  TerminationCheckingCallback(const Mat& x1,
                              const Mat& x2,
                              const EstimateFundamentalOptions& options,
                              Mat3* F)
      : options_(options), x1_(x1), x2_(x2), F_(F) {}
 
  virtual ceres::CallbackReturnType operator()(
      const ceres::IterationSummary& summary) {
    // If the step wasn't successful, there's nothing to do.
    if (!summary.step_is_successful) {
      return ceres::SOLVER_CONTINUE;
    }
 
    // Calculate average of symmetric epipolar distance.
    double average_distance = 0.0;
    for (int i = 0; i < x1_.cols(); i++) {
      average_distance = SymmetricEpipolarDistance(*F_, x1_.col(i), x2_.col(i));
    }
    average_distance /= x1_.cols();
 
    if (average_distance <= options_.expected_average_symmetric_distance) {
      return ceres::SOLVER_TERMINATE_SUCCESSFULLY;
    }
 
    return ceres::SOLVER_CONTINUE;
  }
 
 private:
  const EstimateFundamentalOptions& options_;
  const Mat& x1_;
  const Mat& x2_;
  Mat3* F_;
};
}  // namespace
 
/* Fundamental transformation estimation. */
bool EstimateFundamentalFromCorrespondences(
    const Mat& x1,
    const Mat& x2,
    const EstimateFundamentalOptions& options,
    Mat3* F) {
  // Step 1: Algebraic fundamental estimation.
 
  // Assume algebraic estiation always succeeds,
  NormalizedEightPointSolver(x1, x2, F);
 
  LG << "Estimated matrix after algebraic estimation:\n" << *F;
 
  // Step 2: Refine matrix using Ceres minimizer.
  ceres::Problem problem;
  for (int i = 0; i < x1.cols(); i++) {
    FundamentalSymmetricEpipolarCostFunctor*
        fundamental_symmetric_epipolar_cost_function =
            new FundamentalSymmetricEpipolarCostFunctor(x1.col(i), x2.col(i));
 
    problem.AddResidualBlock(
        new ceres::AutoDiffCostFunction<FundamentalSymmetricEpipolarCostFunctor,
                                        2,  // num_residuals
                                        9>(
            fundamental_symmetric_epipolar_cost_function),
        NULL,
        F->data());
  }
 
  // Configure the solve.
  ceres::Solver::Options solver_options;
  solver_options.linear_solver_type = ceres::DENSE_QR;
  solver_options.max_num_iterations = options.max_num_iterations;
  solver_options.update_state_every_iteration = true;
 
  // Terminate if the average symmetric distance is good enough.
  TerminationCheckingCallback callback(x1, x2, options, F);
  solver_options.callbacks.push_back(&callback);
 
  // Run the solve.
  ceres::Solver::Summary summary;
  ceres::Solve(solver_options, &problem, &summary);
 
  VLOG(1) << "Summary:\n" << summary.FullReport();
 
  LG << "Final refined matrix:\n" << *F;
 
  return summary.IsSolutionUsable();
}
 
}  // namespace libmv