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Merge branch 'develop' into fix/LP_QP_stype

release/4.3a0
Frank Dellaert 2020-10-01 21:37:45 -04:00
parent dc1057f314 323fb85cff
commit 52518c9b0b
11 changed files with 159 additions and 92 deletions
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40
gtsam/geometry/Cal3Bundler.cpp
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@ -25,13 +25,13 @@ namespace gtsam {
/* ************************************************************************* */
Cal3Bundler::Cal3Bundler() :
f_(1), k1_(0), k2_(0), u0_(0), v0_(0) {
f_(1), k1_(0), k2_(0), u0_(0), v0_(0), tol_(1e-5) {
}
/* ************************************************************************* */
Cal3Bundler::Cal3Bundler(double f, double k1, double k2, double u0, double v0) :
f_(f), k1_(k1), k2_(k2), u0_(u0), v0_(v0) {
}
Cal3Bundler::Cal3Bundler(double f, double k1, double k2, double u0, double v0,
double tol)
: f_(f), k1_(k1), k2_(k2), u0_(u0), v0_(v0), tol_(tol) {}
/* ************************************************************************* */
Matrix3 Cal3Bundler::K() const {
@ -94,21 +94,24 @@ Point2 Cal3Bundler::uncalibrate(const Point2& p, //
}
/* ************************************************************************* */
Point2 Cal3Bundler::calibrate(const Point2& pi, const double tol) const {
Point2 Cal3Bundler::calibrate(const Point2& pi,
OptionalJacobian<2, 3> Dcal,
OptionalJacobian<2, 2> Dp) const {
// Copied from Cal3DS2 :-(
// but specialized with k1,k2 non-zero only and fx=fy and s=0
const Point2 invKPi((pi.x() - u0_)/f_, (pi.y() - v0_)/f_);
double x = (pi.x() - u0_)/f_, y = (pi.y() - v0_)/f_;
const Point2 invKPi(x, y);
// initialize by ignoring the distortion at all, might be problematic for pixels around boundary
Point2 pn = invKPi;
Point2 pn(x, y);
// iterate until the uncalibrate is close to the actual pixel coordinate
const int maxIterations = 10;
int iteration;
for (iteration = 0; iteration < maxIterations; ++iteration) {
if (distance2(uncalibrate(pn), pi) <= tol)
if (distance2(uncalibrate(pn), pi) <= tol_)
break;
const double x = pn.x(), y = pn.y(), xx = x * x, yy = y * y;
const double px = pn.x(), py = pn.y(), xx = px * px, yy = py * py;
const double rr = xx + yy;
const double g = (1 + k1_ * rr + k2_ * rr * rr);
pn = invKPi / g;
@ -118,6 +121,25 @@ Point2 Cal3Bundler::calibrate(const Point2& pi, const double tol) const {
throw std::runtime_error(
"Cal3Bundler::calibrate fails to converge. need a better initialization");
// We make use of the Implicit Function Theorem to compute the Jacobians from uncalibrate
// Given f(pi, pn) = uncalibrate(pn) - pi, and g(pi) = calibrate, we can easily compute the Jacobians
// df/pi = -I (pn and pi are independent args)
// Dcal = -inv(H_uncal_pn) * df/pi = -inv(H_uncal_pn) * (-I) = inv(H_uncal_pn)
// Dp = -inv(H_uncal_pn) * df/K = -inv(H_uncal_pn) * H_uncal_K
Matrix23 H_uncal_K;
Matrix22 H_uncal_pn, H_uncal_pn_inv;
if (Dcal || Dp) {
// Compute uncalibrate Jacobians
uncalibrate(pn, Dcal ? &H_uncal_K : nullptr, H_uncal_pn);
H_uncal_pn_inv = H_uncal_pn.inverse();
if (Dp) *Dp = H_uncal_pn_inv;
if (Dcal) *Dcal = -H_uncal_pn_inv * H_uncal_K;
}
return pn;
}

19
gtsam/geometry/Cal3Bundler.h
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@ -33,6 +33,7 @@ private:
double f_; ///< focal length
double k1_, k2_; ///< radial distortion
double u0_, v0_; ///< image center, not a parameter to be optimized but a constant
double tol_; ///< tolerance value when calibrating
public:
@ -51,8 +52,10 @@ public:
* @param k2 second radial distortion coefficient (quartic)
* @param u0 optional image center (default 0), considered a constant
* @param v0 optional image center (default 0), considered a constant
* @param tol optional calibration tolerance value
*/
Cal3Bundler(double f, double k1, double k2, double u0 = 0, double v0 = 0);
Cal3Bundler(double f, double k1, double k2, double u0 = 0, double v0 = 0,
double tol = 1e-5);
virtual ~Cal3Bundler() {}
@ -117,8 +120,17 @@ public:
Point2 uncalibrate(const Point2& p, OptionalJacobian<2, 3> Dcal = boost::none,
OptionalJacobian<2, 2> Dp = boost::none) const;
/// Convert a pixel coordinate to ideal coordinate
Point2 calibrate(const Point2& pi, const double tol = 1e-5) const;
/**
* Convert a pixel coordinate to ideal coordinate xy
* @param p point in image coordinates
* @param tol optional tolerance threshold value for iterative minimization
* @param Dcal optional 2*3 Jacobian wrpt Cal3Bundler parameters
* @param Dp optional 2*2 Jacobian wrpt intrinsic coordinates
* @return point in intrinsic coordinates
*/
Point2 calibrate(const Point2& pi,
OptionalJacobian<2, 3> Dcal = boost::none,
OptionalJacobian<2, 2> Dp = boost::none) const;
/// @deprecated might be removed in next release, use uncalibrate
Matrix2 D2d_intrinsic(const Point2& p) const;
@ -164,6 +176,7 @@ private:
ar & BOOST_SERIALIZATION_NVP(k2_);
ar & BOOST_SERIALIZATION_NVP(u0_);
ar & BOOST_SERIALIZATION_NVP(v0_);
ar & BOOST_SERIALIZATION_NVP(tol_);
}
/// @}

13
gtsam/geometry/Point3.cpp
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@ -75,17 +75,16 @@ double dot(const Point3 &p, const Point3 &q, OptionalJacobian<1, 3> H1,
return p.x() * q.x() + p.y() * q.y() + p.z() * q.z();
}
Point3Pair mean(const std::vector<Point3Pair> &abPointPairs) {
Point3Pair means(const std::vector<Point3Pair> &abPointPairs) {
const size_t n = abPointPairs.size();
Point3 aCentroid(0, 0, 0), bCentroid(0, 0, 0);
if (n == 0) throw std::invalid_argument("Point3::mean input Point3Pair vector is empty");
Point3 aSum(0, 0, 0), bSum(0, 0, 0);
for (const Point3Pair &abPair : abPointPairs) {
aCentroid += abPair.first;
bCentroid += abPair.second;
aSum += abPair.first;
bSum += abPair.second;
}
const double f = 1.0 / n;
aCentroid *= f;
bCentroid *= f;
return make_pair(aCentroid, bCentroid);
return {aSum * f, bSum * f};
}
/* ************************************************************************* */

5
gtsam/geometry/Point3.h
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@ -62,13 +62,14 @@ GTSAM_EXPORT double dot(const Point3& p, const Point3& q,
/// mean
template <class CONTAINER>
GTSAM_EXPORT Point3 mean(const CONTAINER& points) {
if (points.size() == 0) throw std::invalid_argument("Point3::mean input container is empty");
Point3 sum(0, 0, 0);
sum = std::accumulate(points.begin(), points.end(), sum);
return sum / points.size();
}
/// mean of Point3 pair
GTSAM_EXPORT Point3Pair mean(const std::vector<Point3Pair>& abPointPairs);
/// Calculate the two means of a set of Point3 pairs
GTSAM_EXPORT Point3Pair means(const std::vector<Point3Pair> &abPointPairs);
template <typename A1, typename A2>
struct Range;

65
gtsam/geometry/Pose3.cpp
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@ -24,10 +24,11 @@
#include <limits>
#include <string>
using namespace std;
namespace gtsam {
using std::vector;
using Point3Pairs = vector<Point3Pair>;
/** instantiate concept checks */
GTSAM_CONCEPT_POSE_INST(Pose3);
@ -212,18 +213,20 @@ Matrix3 Pose3::ComputeQforExpmapDerivative(const Vector6& xi, double nearZeroThr
#else
// The closed-form formula in Barfoot14tro eq. (102)
double phi = w.norm();
if (std::abs(phi)>nearZeroThreshold) {
const double sinPhi = sin(phi), cosPhi = cos(phi);
const double phi2 = phi * phi, phi3 = phi2 * phi, phi4 = phi3 * phi, phi5 = phi4 * phi;
const Matrix3 WVW = W * V * W;
if (std::abs(phi) > nearZeroThreshold) {
const double s = sin(phi), c = cos(phi);
const double phi2 = phi * phi, phi3 = phi2 * phi, phi4 = phi3 * phi,
phi5 = phi4 * phi;
// Invert the sign of odd-order terms to have the right Jacobian
Q = -0.5*V + (phi-sinPhi)/phi3*(W*V + V*W - W*V*W)
+ (1-phi2/2-cosPhi)/phi4*(W*W*V + V*W*W - 3*W*V*W)
- 0.5*((1-phi2/2-cosPhi)/phi4 - 3*(phi-sinPhi-phi3/6.)/phi5)*(W*V*W*W + W*W*V*W);
}
else {
Q = -0.5*V + 1./6.*(W*V + V*W - W*V*W)
- 1./24.*(W*W*V + V*W*W - 3*W*V*W)
+ 1./120.*(W*V*W*W + W*W*V*W);
Q = -0.5 * V + (phi - s) / phi3 * (W * V + V * W - WVW) +
(1 - phi2 / 2 - c) / phi4 * (W * W * V + V * W * W - 3 * WVW) -
0.5 * ((1 - phi2 / 2 - c) / phi4 - 3 * (phi - s - phi3 / 6.) / phi5) *
(WVW * W + W * WVW);
} else {
Q = -0.5 * V + 1. / 6. * (W * V + V * W - WVW) -
1. / 24. * (W * W * V + V * W * W - 3 * WVW) +
1. / 120. * (WVW * W + W * WVW);
}
#endif
@ -381,39 +384,33 @@ Unit3 Pose3::bearing(const Pose3& pose, OptionalJacobian<2, 6> Hself,
}
/* ************************************************************************* */
boost::optional<Pose3> Pose3::Align(const std::vector<Point3Pair>& abPointPairs) {
boost::optional<Pose3> Pose3::Align(const Point3Pairs &abPointPairs) {
const size_t n = abPointPairs.size();
if (n < 3)
return boost::none; // we need at least three pairs
if (n < 3) {
return boost::none; // we need at least three pairs
}
// calculate centroids
Point3 aCentroid(0,0,0), bCentroid(0,0,0);
for(const Point3Pair& abPair: abPointPairs) {
aCentroid += abPair.first;
bCentroid += abPair.second;
}
double f = 1.0 / n;
aCentroid *= f;
bCentroid *= f;
const auto centroids = means(abPointPairs);
// Add to form H matrix
Matrix3 H = Z_3x3;
for(const Point3Pair& abPair: abPointPairs) {
Point3 da = abPair.first - aCentroid;
Point3 db = abPair.second - bCentroid;
for (const Point3Pair &abPair : abPointPairs) {
const Point3 da = abPair.first - centroids.first;
const Point3 db = abPair.second - centroids.second;
H += da * db.transpose();
}
}
// ClosestTo finds rotation matrix closest to H in Frobenius sense
Rot3 aRb = Rot3::ClosestTo(H);
Point3 aTb = Point3(aCentroid) - aRb * Point3(bCentroid);
const Rot3 aRb = Rot3::ClosestTo(H);
const Point3 aTb = centroids.first - aRb * centroids.second;
return Pose3(aRb, aTb);
}
boost::optional<Pose3> align(const vector<Point3Pair>& baPointPairs) {
vector<Point3Pair> abPointPairs;
for (const Point3Pair& baPair: baPointPairs) {
abPointPairs.push_back(make_pair(baPair.second, baPair.first));
boost::optional<Pose3> align(const Point3Pairs &baPointPairs) {
Point3Pairs abPointPairs;
for (const Point3Pair &baPair : baPointPairs) {
abPointPairs.emplace_back(baPair.second, baPair.first);
}
return Pose3::Align(abPointPairs);
}

23
gtsam/geometry/tests/testCal3Bundler.cpp
View File

@ -52,12 +52,14 @@ TEST( Cal3Bundler, calibrate )
Point2 pn(0.5, 0.5);
Point2 pi = K.uncalibrate(pn);
Point2 pn_hat = K.calibrate(pi);
CHECK( traits<Point2>::Equals(pn, pn_hat, 1e-5));
CHECK(traits<Point2>::Equals(pn, pn_hat, 1e-5));
}
/* ************************************************************************* */
Point2 uncalibrate_(const Cal3Bundler& k, const Point2& pt) { return k.uncalibrate(pt); }
Point2 calibrate_(const Cal3Bundler& k, const Point2& pt) { return k.calibrate(pt); }
/* ************************************************************************* */
TEST( Cal3Bundler, Duncalibrate)
{
@ -69,11 +71,20 @@ TEST( Cal3Bundler, Duncalibrate)
Matrix numerical2 = numericalDerivative22(uncalibrate_, K, p);
CHECK(assert_equal(numerical1,Dcal,1e-7));
CHECK(assert_equal(numerical2,Dp,1e-7));
CHECK(assert_equal(numerical1,K.D2d_calibration(p),1e-7));
CHECK(assert_equal(numerical2,K.D2d_intrinsic(p),1e-7));
Matrix Dcombined(2,5);
Dcombined << Dp, Dcal;
CHECK(assert_equal(Dcombined,K.D2d_intrinsic_calibration(p),1e-7));
}
/* ************************************************************************* */
TEST( Cal3Bundler, Dcalibrate)
{
Matrix Dcal, Dp;
Point2 pn(0.5, 0.5);
Point2 pi = K.uncalibrate(pn);
Point2 actual = K.calibrate(pi, Dcal, Dp);
CHECK(assert_equal(pn, actual, 1e-7));
Matrix numerical1 = numericalDerivative21(calibrate_, K, pi);
Matrix numerical2 = numericalDerivative22(calibrate_, K, pi);
CHECK(assert_equal(numerical1,Dcal,1e-5));
CHECK(assert_equal(numerical2,Dp,1e-5));
}
/* ************************************************************************* */

9
gtsam/geometry/tests/testPinholeCamera.cpp
View File

@ -336,6 +336,15 @@ TEST( PinholeCamera, range3) {
EXPECT(assert_equal(Hexpected2, D2, 1e-7));
}
/* ************************************************************************* */
TEST( PinholeCamera, Cal3Bundler) {
Cal3Bundler calibration;
Pose3 wTc;
PinholeCamera<Cal3Bundler> camera(wTc, calibration);
Point2 p(50, 100);
camera.backproject(p, 10);
}
/* ************************************************************************* */
int main() { TestResult tr; return TestRegistry::runAllTests(tr); }
/* ************************************************************************* */

16
gtsam/geometry/tests/testPoint3.cpp
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@ -166,22 +166,22 @@ TEST (Point3, normalize) {
//*************************************************************************
TEST(Point3, mean) {
Point3 expected_a_mean(2, 2, 2);
Point3 expected(2, 2, 2);
Point3 a1(0, 0, 0), a2(1, 2, 3), a3(5, 4, 3);
std::vector<Point3> a_points{a1, a2, a3};
Point3 actual_a_mean = mean(a_points);
EXPECT(assert_equal(expected_a_mean, actual_a_mean));
Point3 actual = mean(a_points);
EXPECT(assert_equal(expected, actual));
}
TEST(Point3, mean_pair) {
Point3 a_mean(2, 2, 2), b_mean(-1, 1, 0);
Point3Pair expected_mean = std::make_pair(a_mean, b_mean);
Point3Pair expected = std::make_pair(a_mean, b_mean);
Point3 a1(0, 0, 0), a2(1, 2, 3), a3(5, 4, 3);
Point3 b1(-1, 0, 0), b2(-2, 4, 0), b3(0, -1, 0);
std::vector<Point3Pair> point_pairs{{a1,b1},{a2,b2},{a3,b3}};
Point3Pair actual_mean = mean(point_pairs);
EXPECT(assert_equal(expected_mean.first, actual_mean.first));
EXPECT(assert_equal(expected_mean.second, actual_mean.second));
std::vector<Point3Pair> point_pairs{{a1, b1}, {a2, b2}, {a3, b3}};
Point3Pair actual = means(point_pairs);
EXPECT(assert_equal(expected.first, actual.first));
EXPECT(assert_equal(expected.second, actual.second));
}
//*************************************************************************

10
gtsam/gtsam.i
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@ -959,6 +959,7 @@ class Cal3Bundler {
// Standard Constructors
Cal3Bundler();
Cal3Bundler(double fx, double k1, double k2, double u0, double v0);
Cal3Bundler(double fx, double k1, double k2, double u0, double v0, double tol);
// Testable
void print(string s) const;
@ -971,7 +972,7 @@ class Cal3Bundler {
Vector localCoordinates(const gtsam::Cal3Bundler& c) const;
// Action on Point2
gtsam::Point2 calibrate(const gtsam::Point2& p, double tol) const;
gtsam::Point2 calibrate(const gtsam::Point2& p) const;
gtsam::Point2 uncalibrate(const gtsam::Point2& p) const;
// Standard Interface
@ -1105,7 +1106,7 @@ typedef gtsam::PinholeCamera<gtsam::Cal3_S2> PinholeCameraCal3_S2;
//TODO (Issue 237) due to lack of jacobians of Cal3DS2_Base::calibrate, PinholeCamera does not apply to Cal3DS2/Unified
//typedef gtsam::PinholeCamera<gtsam::Cal3DS2> PinholeCameraCal3DS2;
//typedef gtsam::PinholeCamera<gtsam::Cal3Unified> PinholeCameraCal3Unified;
//typedef gtsam::PinholeCamera<gtsam::Cal3Bundler> PinholeCameraCal3Bundler;
typedef gtsam::PinholeCamera<gtsam::Cal3Bundler> PinholeCameraCal3Bundler;
#include <gtsam/geometry/StereoCamera.h>
class StereoCamera {
@ -2462,6 +2463,8 @@ class ISAM2Result {
size_t getVariablesRelinearized() const;
size_t getVariablesReeliminated() const;
size_t getCliques() const;
double getErrorBefore() const;
double getErrorAfter() const;
};
class ISAM2 {
@ -2766,8 +2769,7 @@ class SfmTrack {
class SfmData {
size_t number_cameras() const;
size_t number_tracks() const;
//TODO(Varun) Need to fix issue #237 first before this can work
// gtsam::PinholeCamera<gtsam::Cal3Bundler> camera(size_t idx) const;
gtsam::PinholeCamera<gtsam::Cal3Bundler> camera(size_t idx) const;
gtsam::SfmTrack track(size_t idx) const;
};

2
gtsam/nonlinear/ISAM2Result.h
View File

@ -176,6 +176,8 @@ struct ISAM2Result {
size_t getVariablesRelinearized() const { return variablesRelinearized; }
size_t getVariablesReeliminated() const { return variablesReeliminated; }
size_t getCliques() const { return cliques; }
double getErrorBefore() const { return errorBefore ? *errorBefore : std::nan(""); }
double getErrorAfter() const { return errorAfter ? *errorAfter : std::nan(""); }
};
} // namespace gtsam

49
gtsam_unstable/geometry/Similarity3.cpp
View File

@ -23,10 +23,14 @@
namespace gtsam {
using std::vector;
using PointPairs = vector<Point3Pair>;
namespace {
/// Subtract centroids from point pairs.
static std::vector<Point3Pair> subtractCentroids(const std::vector<Point3Pair>& abPointPairs, const Point3Pair& centroids) {
std::vector<Point3Pair> d_abPointPairs;
static PointPairs subtractCentroids(const PointPairs &abPointPairs,
const Point3Pair &centroids) {
PointPairs d_abPointPairs;
for (const Point3Pair& abPair : abPointPairs) {
Point3 da = abPair.first - centroids.first;
Point3 db = abPair.second - centroids.second;
@ -36,7 +40,8 @@ static std::vector<Point3Pair> subtractCentroids(const std::vector<Point3Pair>&
}
/// Form inner products x and y and calculate scale.
static const double calculateScale(const std::vector<Point3Pair>& d_abPointPairs, const Rot3& aRb) {
static const double calculateScale(const PointPairs &d_abPointPairs,
const Rot3 &aRb) {
double x = 0, y = 0;
Point3 da, db;
for (const Point3Pair& d_abPair : d_abPointPairs) {
@ -50,7 +55,7 @@ static const double calculateScale(const std::vector<Point3Pair>& d_abPointPairs
}
/// Form outer product H.
static Matrix3 calculateH(const std::vector<Point3Pair>& d_abPointPairs) {
static Matrix3 calculateH(const PointPairs &d_abPointPairs) {
Matrix3 H = Z_3x3;
for (const Point3Pair& d_abPair : d_abPointPairs) {
H += d_abPair.first * d_abPair.second.transpose();
@ -59,7 +64,8 @@ static Matrix3 calculateH(const std::vector<Point3Pair>& d_abPointPairs) {
}
/// This method estimates the similarity transform from differences point pairs, given a known or estimated rotation and point centroids.
static Similarity3 align(const std::vector<Point3Pair>& d_abPointPairs, const Rot3& aRb, const Point3Pair& centroids) {
static Similarity3 align(const PointPairs &d_abPointPairs, const Rot3 &aRb,
const Point3Pair &centroids) {
const double s = calculateScale(d_abPointPairs, aRb);
const Point3 aTb = (centroids.first - s * (aRb * centroids.second)) / s;
return Similarity3(aRb, aTb, s);
@ -67,8 +73,9 @@ static Similarity3 align(const std::vector<Point3Pair>& d_abPointPairs, const Ro
/// This method estimates the similarity transform from point pairs, given a known or estimated rotation.
// Refer to: http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2005/Zinsser05-PSR.pdf Chapter 3
static Similarity3 alignGivenR(const std::vector<Point3Pair>& abPointPairs, const Rot3& aRb) {
auto centroids = mean(abPointPairs);
static Similarity3 alignGivenR(const PointPairs &abPointPairs,
const Rot3 &aRb) {
auto centroids = means(abPointPairs);
auto d_abPointPairs = subtractCentroids(abPointPairs, centroids);
return align(d_abPointPairs, aRb, centroids);
}
@ -147,10 +154,12 @@ Point3 Similarity3::operator*(const Point3& p) const {
return transformFrom(p);
}
Similarity3 Similarity3::Align(const std::vector<Point3Pair>& abPointPairs) {
// Refer to: http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2005/Zinsser05-PSR.pdf Chapter 3
if (abPointPairs.size() < 3) throw std::runtime_error("input should have at least 3 pairs of points");
auto centroids = mean(abPointPairs);
Similarity3 Similarity3::Align(const PointPairs &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 = subtractCentroids(abPointPairs, centroids);
Matrix3 H = calculateH(d_abPointPairs);
// ClosestTo finds rotation matrix closest to H in Frobenius sense
@ -158,17 +167,18 @@ Similarity3 Similarity3::Align(const std::vector<Point3Pair>& abPointPairs) {
return align(d_abPointPairs, aRb, centroids);
}
Similarity3 Similarity3::Align(const std::vector<Pose3Pair>& abPosePairs) {
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");
if (n < 2)
throw std::runtime_error("input should have at least 2 pairs of poses");
// calculate rotation
vector<Rot3> rotations;
vector<Point3Pair> abPointPairs;
PointPairs abPointPairs;
rotations.reserve(n);
abPointPairs.reserve(n);
Pose3 wTa, wTb;
for (const Pose3Pair& abPair : abPosePairs) {
for (const Pose3Pair &abPair : abPosePairs) {
std::tie(wTa, wTb) = abPair;
rotations.emplace_back(wTa.rotation().compose(wTb.rotation().inverse()));
abPointPairs.emplace_back(wTa.translation(), wTb.translation());
@ -178,7 +188,7 @@ Similarity3 Similarity3::Align(const std::vector<Pose3Pair>& abPosePairs) {
return alignGivenR(abPointPairs, aRb);
}
Matrix4 Similarity3::wedge(const Vector7& xi) {
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);
@ -217,12 +227,13 @@ Matrix3 Similarity3::GetV(Vector3 w, double lambda) {
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 - exp(-lambda)) / lambda;
A = (1.0 - expMinLambda) / lambda;
alpha = 1.0 / (1.0 + theta2 / lambda2);
beta = (exp(-lambda) - 1 + lambda) / lambda2;
mu = (1 - lambda + (0.5 * lambda2) - exp(-lambda)) / lambda3;
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;