Similarity3.cpp

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/* ----------------------------------------------------------------------------

 * GTSAM Copyright 2010, Georgia Tech Research Corporation,
 * Atlanta, Georgia 30332-0415
 * All Rights Reserved
 * Authors: Frank Dellaert, et al. (see THANKS for the full author list)

 * See LICENSE for the license information

 * -------------------------------------------------------------------------- */

#include <gtsam/geometry/Similarity3.h>

#include <gtsam/geometry/Pose3.h>
#include <gtsam/base/Manifold.h>
#include <gtsam/slam/KarcherMeanFactor-inl.h>

namespace gtsam {

using std::vector;

namespace internal {
static Point3Pairs subtractCentroids(const Point3Pairs &abPointPairs,
                                    const Point3Pair &centroids) {
  Point3Pairs d_abPointPairs;
  for (const auto& [a, b] : abPointPairs) {
    Point3 da = a - centroids.first;
    Point3 db = b - centroids.second;
    d_abPointPairs.emplace_back(da, db);
  }
  return d_abPointPairs;
}

// We force the scale to be a non-negative quantity
// (see Section 10.1 of https://ethaneade.com/lie_groups.pdf)
static double calculateScale(const Point3Pairs &d_abPointPairs,
                             const Rot3 &aRb) {
  double x = 0, y = 0;
  Point3 da, db;
  for (const auto& [da, db] : d_abPointPairs) {
    const Vector3 da_prime = aRb * db;
    y += da.transpose() * da_prime;
    x += da_prime.transpose() * da_prime;
  }
  const double s = std::fabs(y / x);
  return s;
}

static Matrix3 calculateH(const Point3Pairs &d_abPointPairs) {
  Matrix3 H = Z_3x3;
  for (const auto& [da, db] : d_abPointPairs) {
    H += da * db.transpose();
  }
  return H;
}

// given a known or estimated rotation and point centroids.
static Similarity3 align(const Point3Pairs &d_abPointPairs, const Rot3 &aRb,
                         const Point3Pair &centroids) {
  const double s = calculateScale(d_abPointPairs, aRb);
  // dividing aTb by s is required because the registration cost function
  // minimizes ||a - sRb - t||, whereas Sim(3) computes s(Rb + t)
  const Point3 aTb = (centroids.first - s * (aRb * centroids.second)) / s;
  return Similarity3(aRb, aTb, s);
}

// Refer to: http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2005/Zinsser05-PSR.pdf Chapter 3
static Similarity3 alignGivenR(const Point3Pairs &abPointPairs,
                               const Rot3 &aRb) {
  auto centroids = means(abPointPairs);
  auto d_abPointPairs = internal::subtractCentroids(abPointPairs, centroids);
  return align(d_abPointPairs, aRb, centroids);
}
}  // namespace internal

Similarity3::Similarity3() :
    t_(0,0,0), s_(1) {
}

Similarity3::Similarity3(double s) :
    t_(0,0,0), s_(s) {
}

Similarity3::Similarity3(const Rot3& R, const Point3& t, double s) :
    R_(R), t_(t), s_(s) {
}

Similarity3::Similarity3(const Matrix3& R, const Vector3& t, double s) :
    R_(R), t_(t), s_(s) {
}

Similarity3::Similarity3(const Matrix4& T) :
    R_(T.topLeftCorner<3, 3>()), t_(T.topRightCorner<3, 1>()), s_(1.0 / T(3, 3)) {
}

bool Similarity3::equals(const Similarity3& other, double tol) const {
  return R_.equals(other.R_, tol) && traits<Point3>::Equals(t_, other.t_, tol)
      && s_ < (other.s_ + tol) && s_ > (other.s_ - tol);
}

bool Similarity3::operator==(const Similarity3& other) const {
  return R_.matrix() == other.R_.matrix() && t_ == other.t_ && s_ == other.s_;
}

void Similarity3::print(const std::string& s) const {
  std::cout << std::endl;
  std::cout << s;
  rotation().print("\nR:\n");
  std::cout << "t: " << translation().transpose() << " s: " << scale() << std::endl;
}

Similarity3 Similarity3::Identity() {
  return Similarity3();
}
Similarity3 Similarity3::operator*(const Similarity3& S) const {
  return Similarity3(R_ * S.R_, ((1.0 / S.s_) * t_) + R_ * S.t_, s_ * S.s_);
}

Similarity3 Similarity3::inverse() const {
  const Rot3 Rt = R_.inverse();
  const Point3 sRt = Rt * (-s_ * t_);
  return Similarity3(Rt, sRt, 1.0 / s_);
}

Point3 Similarity3::transformFrom(const Point3& p, //
    OptionalJacobian<3, 7> H1, OptionalJacobian<3, 3> H2) const {
  const Point3 q = R_ * p + t_;
  if (H1) {
    // For this derivative, see LieGroups.pdf
    const Matrix3 sR = s_ * R_.matrix();
    const Matrix3 DR = sR * skewSymmetric(-p.x(), -p.y(), -p.z());
    *H1 << DR, sR, sR * p;
  }
  if (H2)
    *H2 = s_ * R_.matrix(); // just 3*3 sub-block of matrix()
  return s_ * q;
}

Pose3 Similarity3::transformFrom(const Pose3& T) const {
  Rot3 R = R_.compose(T.rotation());
  Point3 t = Point3(s_ * (R_ * T.translation() + t_));
  return Pose3(R, t);
}

Point3 Similarity3::operator*(const Point3& p) const {
  return transformFrom(p);
}

Similarity3 Similarity3::Align(const Point3Pairs &abPointPairs) {
  // Refer to Chapter 3 of
  // http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2005/Zinsser05-PSR.pdf
  if (abPointPairs.size() < 3)
    throw std::runtime_error("input should have at least 3 pairs of points");
  auto centroids = means(abPointPairs);
  auto d_abPointPairs = internal::subtractCentroids(abPointPairs, centroids);
  Matrix3 H = internal::calculateH(d_abPointPairs);
  // ClosestTo finds rotation matrix closest to H in Frobenius sense
  Rot3 aRb = Rot3::ClosestTo(H);
  return internal::align(d_abPointPairs, aRb, centroids);
}

Similarity3 Similarity3::Align(const vector<Pose3Pair> &abPosePairs) {
  const size_t n = abPosePairs.size();
  if (n < 2)
    throw std::runtime_error("input should have at least 2 pairs of poses");

  // calculate rotation
  vector<Rot3> rotations;
  Point3Pairs abPointPairs;
  rotations.reserve(n);
  abPointPairs.reserve(n);
  // Below denotes the pose of the i'th object/camera/etc in frame "a" or frame "b"
  Pose3 aTi, bTi;
  for (const auto &[aTi, bTi] : abPosePairs) {
    const Rot3 aRb = aTi.rotation().compose(bTi.rotation().inverse());
    rotations.emplace_back(aRb);
    abPointPairs.emplace_back(aTi.translation(), bTi.translation());
  }
  const Rot3 aRb_estimate = FindKarcherMean<Rot3>(rotations);

  return internal::alignGivenR(abPointPairs, aRb_estimate);
}

Matrix4 Similarity3::wedge(const Vector7 &xi) {
  // http://www.ethaneade.org/latex2html/lie/node29.html
  const auto w = xi.head<3>();
  const auto u = xi.segment<3>(3);
  const double lambda = xi[6];
  Matrix4 W;
  W << skewSymmetric(w), u, 0, 0, 0, -lambda;
  return W;
}

Matrix7 Similarity3::AdjointMap() const {
  // http://www.ethaneade.org/latex2html/lie/node30.html
  const Matrix3 R = R_.matrix();
  const Vector3 t = t_;
  const Matrix3 A = s_ * skewSymmetric(t) * R;
  Matrix7 adj;
  adj << R, Z_3x3, Matrix31::Zero(), // 3*7
  A, s_ * R, -s_ * t, // 3*7
  Matrix16::Zero(), 1; // 1*7
  return adj;
}

Matrix3 Similarity3::GetV(Vector3 w, double lambda) {
  // http://www.ethaneade.org/latex2html/lie/node29.html
  const double theta2 = w.transpose() * w;
  double Y, Z, W;
  if (theta2 > 1e-9) {
    const double theta = sqrt(theta2);
    const double X = sin(theta) / theta;
    Y = (1 - cos(theta)) / theta2;
    Z = (1 - X) / theta2;
    W = (0.5 - Y) / theta2;
  } else {
    // Taylor series expansion for theta=0, X not needed (as is 1)
    Y = 0.5 - theta2 / 24.0;
    Z = 1.0 / 6.0 - theta2 / 120.0;
    W = 1.0 / 24.0 - theta2 / 720.0;
  }
  const double lambda2 = lambda * lambda, lambda3 = lambda2 * lambda;
  const double expMinLambda = exp(-lambda);
  double A, alpha = 0.0, beta, mu;
  if (lambda2 > 1e-9) {
    A = (1.0 - expMinLambda) / lambda;
    alpha = 1.0 / (1.0 + theta2 / lambda2);
    beta = (expMinLambda - 1 + lambda) / lambda2;
    mu = (1 - lambda + (0.5 * lambda2) - expMinLambda) / lambda3;
  } else {
    A = 1.0 - lambda / 2.0 + lambda2 / 6.0;
    beta = 0.5 - lambda / 6.0 + lambda2 / 24.0 - lambda3 / 120.0;
    mu = 1.0 / 6.0 - lambda / 24.0 + lambda2 / 120.0 - lambda3 / 720.0;
  }
  const double gamma = Y - (lambda * Z), upsilon = Z - (lambda * W);
  const double B = alpha * (beta - gamma) + gamma;
  const double C = alpha * (mu - upsilon) + upsilon;
  const Matrix3 Wx = skewSymmetric(w[0], w[1], w[2]);
  return A * I_3x3 + B * Wx + C * Wx * Wx;
}

Vector7 Similarity3::Logmap(const Similarity3& T, OptionalJacobian<7, 7> Hm) {
  // To get the logmap, calculate w and lambda, then solve for u as shown by Ethan at
  // www.ethaneade.org/latex2html/lie/node29.html
  const Vector3 w = Rot3::Logmap(T.R_);
  const double lambda = log(T.s_);
  Vector7 result;
  result << w, GetV(w, lambda).inverse() * T.t_, lambda;
  if (Hm) {
    throw std::runtime_error("Similarity3::Logmap: derivative not implemented");
  }
  return result;
}

Similarity3 Similarity3::Expmap(const Vector7& v, OptionalJacobian<7, 7> Hm) {
  const auto w = v.head<3>();
  const auto u = v.segment<3>(3);
  const double lambda = v[6];
  if (Hm) {
    throw std::runtime_error("Similarity3::Expmap: derivative not implemented");
  }
  const Matrix3 V = GetV(w, lambda);
  return Similarity3(Rot3::Expmap(w), Point3(V * u), exp(lambda));
}

std::ostream &operator<<(std::ostream &os, const Similarity3& p) {
  os << "[" << p.rotation().xyz().transpose() << " "
      << p.translation().transpose() << " " << p.scale() << "]\';";
  return os;
}

Matrix4 Similarity3::matrix() const {
  Matrix4 T;
  T.topRows<3>() << R_.matrix(), t_;
  T.bottomRows<1>() << 0, 0, 0, 1.0 / s_;
  return T;
}

} // namespace gtsam