brokenline Namespace Reference

Classes
struct	PreparedBrokenLineData
	data needed for the Broken Line fit procedure. More...

Typedefs
using	karimaki_circle_fit = riemannFit::CircleFit
	< Karimäki's parameters: (phi, d, k=1/R) More...

Functions
template<typename M3xN , typename M6xN , typename V4 , int n>
__host__ __device__ void	circleFit (const M3xN &hits, const M6xN &hits_ge, const V4 &fast_fit, const double bField, PreparedBrokenLineData< n > &data, karimaki_circle_fit &circle_results)
	Performs the Broken Line fit in the curved track case (that is, the fit parameters are the interceptions u and the curvature correction ). More...
template<typename M3xN , typename V4 >
__host__ __device__ void	fastFit (const M3xN &hits, V4 &result)
	A very fast helix fit. More...
template<int n>
riemannFit::HelixFit	helixFit (const riemannFit::Matrix3xNd< n > &hits, const Eigen::Matrix< float, 6, 4 > &hits_ge, const double bField)
	Helix fit by three step: -fast pre-fit (see Fast_fit() for further info); -circle fit of the hits projected in the transverse plane by Broken Line algorithm (see BL_Circle_fit() for further info); -line fit of the hits projected on the (pre-fitted) cilinder surface by Broken Line algorithm (see BL_Line_fit() for further info); Points must be passed ordered (from inner to outer layer). More...
template<typename V4 , typename M6xN , int n>
__host__ __device__ void	lineFit (const M6xN &hits_ge, const V4 &fast_fit, const double bField, const PreparedBrokenLineData< n > &data, riemannFit::LineFit &line_results)
	Performs the Broken Line fit in the straight track case (that is, the fit parameters are only the interceptions u). More...
template<int n>
__host__ __device__ riemannFit::MatrixNd< n >	matrixC_u (const riemannFit::VectorNd< n > &weights, const riemannFit::VectorNd< n > &sTotal, const riemannFit::VectorNd< n > &varBeta)
	Computes the n-by-n band matrix obtained minimizing the Broken Line's cost function w.r.t u. This is the whole matrix in the case of the line fit and the main n-by-n block in the case of the circle fit. More...
__host__ __device__ double	multScatt (const double &length, const double bField, const double radius, int layer, double slope)
	Computes the Coulomb multiple scattering variance of the planar angle. More...
template<typename M3xN , typename V4 , int n>
__host__ __device__ void	prepareBrokenLineData (const M3xN &hits, const V4 &fast_fit, const double bField, PreparedBrokenLineData< n > &results)
	Computes the data needed for the Broken Line fit procedure that are mainly common for the circle and the line fit. More...
__host__ __device__ riemannFit::Matrix2d	rotationMatrix (double slope)
	Computes the 2D rotation matrix that transforms the line y=slope*x into the line y=0. More...
__host__ __device__ void	translateKarimaki (karimaki_circle_fit &circle, double x0, double y0, riemannFit::Matrix3d &jacobian)
	Changes the Karimäki parameters (and consequently their covariance matrix) under a translation of the coordinate system, such that the old origin has coordinates (x0,y0) in the new coordinate system. The formulas are taken from Karimäki V., 1990, Effective circle fitting for particle trajectories, Nucl. Instr. and Meth. A305 (1991) 187. More...

Typedef Documentation

using brokenline::karimaki_circle_fit = typedef riemannFit::CircleFit

< Karimäki's parameters: (phi, d, k=1/R)

< covariance matrix:
|cov(phi,phi)|cov( d ,phi)|cov( k ,phi)|
|cov(phi, d )|cov( d , d )|cov( k , d )|
|cov(phi, k )|cov( d , k )|cov( k , k )|
as defined in Karimäki V., 1990, Effective circle fitting for particle trajectories, Nucl. Instr. and Meth. A305 (1991) 187.

Definition at line 18 of file BrokenLine.h.

Function Documentation

template<typename M3xN , typename M6xN , typename V4 , int n>

__host__ __device__ void brokenline::circleFit ( const M3xN &  hits, const M6xN &  hits_ge, const V4 &  fast_fit, const double  bField, PreparedBrokenLineData< n > &  data, karimaki_circle_fit &  circle_results  )

inline

Performs the Broken Line fit in the curved track case (that is, the fit parameters are the interceptions u and the curvature correction ).

Parameters

hits	hits coordinates.
hits_cov	hits covariance matrix.
fast_fit	pre-fit result in the form (X0,Y0,R,tan(theta)).
bField	magnetic field in Gev/cm/c.
data	PreparedBrokenLineData.
circle_results	struct to be filled with the results in this form: -par parameter of the line in this form: (phi, d, k); -cov covariance matrix of the fitted parameter; -chi2 value of the cost function in the minimum.

The function implements the steps 2 and 3 of the Broken Line fit with the curvature correction.
The step 2 is the least square fit, done by imposing the minimum constraint on the cost function and solving the consequent linear system. It determines the fitted parameters u and and their covariance matrix. The step 3 is the correction of the fast pre-fitted parameters for the innermost part of the track. It is first done in a comfortable coordinate system (the one in which the first hit is the origin) and then the parameters and their covariance matrix are transformed to the original coordinate system.

Definition at line 314 of file BrokenLine.h.

References ecalTB2006H4_GenSimDigiReco_cfg::bField, riemannFit::CircleFit::chi2, gather_cfg::cout, riemannFit::CircleFit::cov, mps_fire::i, l1tstage2_dqm_sourceclient-live_cfg::invert, matrixC_u(), multScatt(), dqmiodumpmetadata::n, riemannFit::CircleFit::par, brokenline::PreparedBrokenLineData< n >::qCharge, riemannFit::CircleFit::qCharge, brokenline::PreparedBrokenLineData< n >::radii, rotationMatrix(), slope, riemannFit::sqr(), mathSSE::sqrt(), brokenline::PreparedBrokenLineData< n >::sTotal, brokenline::PreparedBrokenLineData< n >::sTransverse, translateKarimaki(), brokenline::PreparedBrokenLineData< n >::varBeta, and brokenline::PreparedBrokenLineData< n >::zInSZplane.

Referenced by helixFit().

                                                                                 {
    circle_results.qCharge = data.qCharge;
    auto& radii = data.radii;
    const auto& sTransverse = data.sTransverse;
    const auto& sTotal = data.sTotal;
    auto& zInSZplane = data.zInSZplane;
    auto& varBeta = data.varBeta;
    const double slope = -circle_results.qCharge / fast_fit(3);
    varBeta *= 1. + riemannFit::sqr(slope);  // the kink angles are projected!

    for (u_int i = 0; i < n; i++) {
      zInSZplane(i) = radii.block(0, i, 2, 1).norm() - fast_fit(2);
    }

    riemannFit::Matrix2d vMat;           // covariance matrix
    riemannFit::VectorNd<n> weightsVec;  // weights
    riemannFit::Matrix2d rotMat;         // rotation matrix point by point
    for (u_int i = 0; i < n; i++) {
      vMat(0, 0) = hits_ge.col(i)[0];               // x errors
      vMat(0, 1) = vMat(1, 0) = hits_ge.col(i)[1];  // cov_xy
      vMat(1, 1) = hits_ge.col(i)[2];               // y errors
      rotMat = rotationMatrix(-radii(0, i) / radii(1, i));
      weightsVec(i) =
          1. / ((rotMat * vMat * rotMat.transpose())(1, 1));  // compute the orthogonal weight point by point
    }

    riemannFit::VectorNplusONEd<n> r_uVec;
    r_uVec(n) = 0;
    for (u_int i = 0; i < n; i++) {
      r_uVec(i) = weightsVec(i) * zInSZplane(i);
    }

    riemannFit::MatrixNplusONEd<n> c_uMat;
    c_uMat.block(0, 0, n, n) = matrixC_u(weightsVec, sTransverse, varBeta);
    c_uMat(n, n) = 0;
    //add the border to the c_uMat matrix
    for (u_int i = 0; i < n; i++) {
      c_uMat(i, n) = 0;
      if (i > 0 && i < n - 1) {
        c_uMat(i, n) +=
            -(sTransverse(i + 1) - sTransverse(i - 1)) * (sTransverse(i + 1) - sTransverse(i - 1)) /
            (2. * varBeta(i) * (sTransverse(i + 1) - sTransverse(i)) * (sTransverse(i) - sTransverse(i - 1)));
      }
      if (i > 1) {
        c_uMat(i, n) +=
            (sTransverse(i) - sTransverse(i - 2)) / (2. * varBeta(i - 1) * (sTransverse(i) - sTransverse(i - 1)));
      }
      if (i < n - 2) {
        c_uMat(i, n) +=
            (sTransverse(i + 2) - sTransverse(i)) / (2. * varBeta(i + 1) * (sTransverse(i + 1) - sTransverse(i)));
      }
      c_uMat(n, i) = c_uMat(i, n);
      if (i > 0 && i < n - 1)
        c_uMat(n, n) += riemannFit::sqr(sTransverse(i + 1) - sTransverse(i - 1)) / (4. * varBeta(i));
    }

#ifdef CPP_DUMP
    std::cout << "CU5\n" << c_uMat << std::endl;
#endif
    riemannFit::MatrixNplusONEd<n> iMat;
    math::cholesky::invert(c_uMat, iMat);
#ifdef CPP_DUMP
    std::cout << "I5\n" << iMat << std::endl;
#endif

    riemannFit::VectorNplusONEd<n> uVec = iMat * r_uVec;  // obtain the fitted parameters by solving the linear system

    // compute (phi, d_ca, k) in the system in which the midpoint of the first two corrected hits is the origin...

    radii.block(0, 0, 2, 1) /= radii.block(0, 0, 2, 1).norm();
    radii.block(0, 1, 2, 1) /= radii.block(0, 1, 2, 1).norm();

    riemannFit::Vector2d dVec = hits.block(0, 0, 2, 1) + (-zInSZplane(0) + uVec(0)) * radii.block(0, 0, 2, 1);
    riemannFit::Vector2d eVec = hits.block(0, 1, 2, 1) + (-zInSZplane(1) + uVec(1)) * radii.block(0, 1, 2, 1);
    auto eMinusd = eVec - dVec;
    auto eMinusd2 = eMinusd.squaredNorm();
    auto tmp1 = 1. / eMinusd2;
    auto tmp2 = sqrt(riemannFit::sqr(fast_fit(2)) - 0.25 * eMinusd2);

    circle_results.par << atan2(eMinusd(1), eMinusd(0)), circle_results.qCharge * (tmp2 - fast_fit(2)),
        circle_results.qCharge * (1. / fast_fit(2) + uVec(n));

    tmp2 = 1. / tmp2;

    riemannFit::Matrix3d jacobian;
    jacobian << (radii(1, 0) * eMinusd(0) - eMinusd(1) * radii(0, 0)) * tmp1,
        (radii(1, 1) * eMinusd(0) - eMinusd(1) * radii(0, 1)) * tmp1, 0,
        circle_results.qCharge * (eMinusd(0) * radii(0, 0) + eMinusd(1) * radii(1, 0)) * tmp2,
        circle_results.qCharge * (eMinusd(0) * radii(0, 1) + eMinusd(1) * radii(1, 1)) * tmp2, 0, 0, 0,
        circle_results.qCharge;

    circle_results.cov << iMat(0, 0), iMat(0, 1), iMat(0, n), iMat(1, 0), iMat(1, 1), iMat(1, n), iMat(n, 0),
        iMat(n, 1), iMat(n, n);

    circle_results.cov = jacobian * circle_results.cov * jacobian.transpose();

    //...Translate in the system in which the first corrected hit is the origin, adding the m.s. correction...

    translateKarimaki(circle_results, 0.5 * eMinusd(0), 0.5 * eMinusd(1), jacobian);
    circle_results.cov(0, 0) +=
        (1 + riemannFit::sqr(slope)) * multScatt(sTotal(1) - sTotal(0), bField, fast_fit(2), 2, slope);

    //...And translate back to the original system

    translateKarimaki(circle_results, dVec(0), dVec(1), jacobian);

    // compute chi2
    circle_results.chi2 = 0;
    for (u_int i = 0; i < n; i++) {
      circle_results.chi2 += weightsVec(i) * riemannFit::sqr(zInSZplane(i) - uVec(i));
      if (i > 0 && i < n - 1)
        circle_results.chi2 +=
            riemannFit::sqr(uVec(i - 1) / (sTransverse(i) - sTransverse(i - 1)) -
                            uVec(i) * (sTransverse(i + 1) - sTransverse(i - 1)) /
                                ((sTransverse(i + 1) - sTransverse(i)) * (sTransverse(i) - sTransverse(i - 1))) +
                            uVec(i + 1) / (sTransverse(i + 1) - sTransverse(i)) +
                            (sTransverse(i + 1) - sTransverse(i - 1)) * uVec(n) / 2) /
            varBeta(i);
    }
  }

template<typename M3xN , typename V4 >

__host__ __device__ void brokenline::fastFit ( const M3xN &  hits, V4 &  result  )

inline

A very fast helix fit.

Parameters

hits	the measured hits.

Returns

(X0,Y0,R,tan(theta)).

Warning

sign of theta is (intentionally, for now) mistaken for negative charges.

Definition at line 258 of file BrokenLine.h.

References a, funct::abs(), b, c, riemannFit::cross2D(), ztail::d, d1, alignCSCRings::e, dqmiodumpmetadata::n, mps_fire::result, mathSSE::sqrt(), and createJobs::tmp.

Referenced by for(), and helixFit().

                                                                        {
    constexpr uint32_t n = M3xN::ColsAtCompileTime;

    int mId = 1;

    if constexpr (n > 3) {
      riemannFit::Vector2d middle = 0.5 * (hits.block(0, n - 1, 2, 1) + hits.block(0, 0, 2, 1));
      auto d1 = (hits.block(0, n / 2, 2, 1) - middle).squaredNorm();
      auto d2 = (hits.block(0, n / 2 - 1, 2, 1) - middle).squaredNorm();
      mId = d1 < d2 ? n / 2 : n / 2 - 1;
    }

    const riemannFit::Vector2d a = hits.block(0, mId, 2, 1) - hits.block(0, 0, 2, 1);
    const riemannFit::Vector2d b = hits.block(0, n - 1, 2, 1) - hits.block(0, mId, 2, 1);
    const riemannFit::Vector2d c = hits.block(0, 0, 2, 1) - hits.block(0, n - 1, 2, 1);

    auto tmp = 0.5 / riemannFit::cross2D(c, a);
    result(0) = hits(0, 0) - (a(1) * c.squaredNorm() + c(1) * a.squaredNorm()) * tmp;
    result(1) = hits(1, 0) + (a(0) * c.squaredNorm() + c(0) * a.squaredNorm()) * tmp;
    // check Wikipedia for these formulas

    result(2) = sqrt(a.squaredNorm() * b.squaredNorm() * c.squaredNorm()) / (2. * std::abs(riemannFit::cross2D(b, a)));
    // Using Math Olympiad's formula R=abc/(4A)

    const riemannFit::Vector2d d = hits.block(0, 0, 2, 1) - result.head(2);
    const riemannFit::Vector2d e = hits.block(0, n - 1, 2, 1) - result.head(2);

    result(3) = result(2) * atan2(riemannFit::cross2D(d, e), d.dot(e)) / (hits(2, n - 1) - hits(2, 0));
    // ds/dz slope between last and first point
  }

template<int n>

riemannFit::HelixFit brokenline::helixFit ( const riemannFit::Matrix3xNd< n > &  hits, const Eigen::Matrix< float, 6, 4 > &  hits_ge, const double  bField  )

inline

Helix fit by three step: -fast pre-fit (see Fast_fit() for further info);
-circle fit of the hits projected in the transverse plane by Broken Line algorithm (see BL_Circle_fit() for further info);
-line fit of the hits projected on the (pre-fitted) cilinder surface by Broken Line algorithm (see BL_Line_fit() for further info);
Points must be passed ordered (from inner to outer layer).

Parameters

hits	Matrix3xNd hits coordinates in this form: |x1|x2|x3|...|xn| |y1|y2|y3|...|yn| |z1|z2|z3|...|zn|
hits_cov	Matrix3Nd covariance matrix in this form (()->cov()): |(x1,x1)|(x2,x1)|(x3,x1)|(x4,x1)|.|(y1,x1)|(y2,x1)|(y3,x1)|(y4,x1)|.|(z1,x1)|(z2,x1)|(z3,x1)|(z4,x1)| |(x1,x2)|(x2,x2)|(x3,x2)|(x4,x2)|.|(y1,x2)|(y2,x2)|(y3,x2)|(y4,x2)|.|(z1,x2)|(z2,x2)|(z3,x2)|(z4,x2)| |(x1,x3)|(x2,x3)|(x3,x3)|(x4,x3)|.|(y1,x3)|(y2,x3)|(y3,x3)|(y4,x3)|.|(z1,x3)|(z2,x3)|(z3,x3)|(z4,x3)| |(x1,x4)|(x2,x4)|(x3,x4)|(x4,x4)|.|(y1,x4)|(y2,x4)|(y3,x4)|(y4,x4)|.|(z1,x4)|(z2,x4)|(z3,x4)|(z4,x4)| . . . . . . . . . . . . . . . |(x1,y1)|(x2,y1)|(x3,y1)|(x4,y1)|.|(y1,y1)|(y2,y1)|(y3,x1)|(y4,y1)|.|(z1,y1)|(z2,y1)|(z3,y1)|(z4,y1)| |(x1,y2)|(x2,y2)|(x3,y2)|(x4,y2)|.|(y1,y2)|(y2,y2)|(y3,x2)|(y4,y2)|.|(z1,y2)|(z2,y2)|(z3,y2)|(z4,y2)| |(x1,y3)|(x2,y3)|(x3,y3)|(x4,y3)|.|(y1,y3)|(y2,y3)|(y3,x3)|(y4,y3)|.|(z1,y3)|(z2,y3)|(z3,y3)|(z4,y3)| |(x1,y4)|(x2,y4)|(x3,y4)|(x4,y4)|.|(y1,y4)|(y2,y4)|(y3,x4)|(y4,y4)|.|(z1,y4)|(z2,y4)|(z3,y4)|(z4,y4)| . . . . . . . . . . . . . . . |(x1,z1)|(x2,z1)|(x3,z1)|(x4,z1)|.|(y1,z1)|(y2,z1)|(y3,z1)|(y4,z1)|.|(z1,z1)|(z2,z1)|(z3,z1)|(z4,z1)| |(x1,z2)|(x2,z2)|(x3,z2)|(x4,z2)|.|(y1,z2)|(y2,z2)|(y3,z2)|(y4,z2)|.|(z1,z2)|(z2,z2)|(z3,z2)|(z4,z2)| |(x1,z3)|(x2,z3)|(x3,z3)|(x4,z3)|.|(y1,z3)|(y2,z3)|(y3,z3)|(y4,z3)|.|(z1,z3)|(z2,z3)|(z3,z3)|(z4,z3)| |(x1,z4)|(x2,z4)|(x3,z4)|(x4,z4)|.|(y1,z4)|(y2,z4)|(y3,z4)|(y4,z4)|.|(z1,z4)|(z2,z4)|(z3,z4)|(z4,z4)|
bField	magnetic field in the center of the detector in Gev/cm/c, in order to perform the p_t calculation.

Warning

see BL_Circle_fit(), BL_Line_fit() and Fast_fit() warnings.

Returns

(phi,Tip,p_t,cot(theta)),Zip), their covariance matrix and the chi2's of the circle and line fits.

Definition at line 584 of file BrokenLine.h.

References funct::abs(), riemannFit::CircleFit::chi2, riemannFit::LineFit::chi2, riemannFit::HelixFit::chi2_circle, riemannFit::HelixFit::chi2_line, circleFit(), riemannFit::CircleFit::cov, riemannFit::LineFit::cov, riemannFit::HelixFit::cov, data, fastFit(), geometryCSVtoXML::line, lineFit(), riemannFit::CircleFit::par, riemannFit::LineFit::par, riemannFit::HelixFit::par, prepareBrokenLineData(), riemannFit::CircleFit::qCharge, riemannFit::HelixFit::qCharge, and riemannFit::sqr().

Referenced by PixelNtupletsFitter::run().

                                                            {
    riemannFit::HelixFit helix;
    riemannFit::Vector4d fast_fit;
    fastFit(hits, fast_fit);

    PreparedBrokenLineData<n> data;
    karimaki_circle_fit circle;
    riemannFit::LineFit line;
    riemannFit::Matrix3d jacobian;

    prepareBrokenLineData(hits, fast_fit, bField, data);
    lineFit(hits_ge, fast_fit, bField, data, line);
    circleFit(hits, hits_ge, fast_fit, bField, data, circle);

    // the circle fit gives k, but here we want p_t, so let's change the parameter and the covariance matrix
    jacobian << 1., 0, 0, 0, 1., 0, 0, 0,
        -std::abs(circle.par(2)) * bField / (riemannFit::sqr(circle.par(2)) * circle.par(2));
    circle.par(2) = bField / std::abs(circle.par(2));
    circle.cov = jacobian * circle.cov * jacobian.transpose();

    helix.par << circle.par, line.par;
    helix.cov = riemannFit::MatrixXd::Zero(5, 5);
    helix.cov.block(0, 0, 3, 3) = circle.cov;
    helix.cov.block(3, 3, 2, 2) = line.cov;
    helix.qCharge = circle.qCharge;
    helix.chi2_circle = circle.chi2;
    helix.chi2_line = line.chi2;

    return helix;
  }

template<typename V4 , typename M6xN , int n>

__host__ __device__ void brokenline::lineFit ( const M6xN &  hits_ge, const V4 &  fast_fit, const double  bField, const PreparedBrokenLineData< n > &  data, riemannFit::LineFit &  line_results  )

inline

Performs the Broken Line fit in the straight track case (that is, the fit parameters are only the interceptions u).

Parameters

hits	hits coordinates.
fast_fit	pre-fit result in the form (X0,Y0,R,tan(theta)).
bField	magnetic field in Gev/cm/c.
data	PreparedBrokenLineData.
line_results	struct to be filled with the results in this form: -par parameter of the line in this form: (cot(theta), Zip); -cov covariance matrix of the fitted parameter; -chi2 value of the cost function in the minimum.

The function implements the steps 2 and 3 of the Broken Line fit without the curvature correction.
The step 2 is the least square fit, done by imposing the minimum constraint on the cost function and solving the consequent linear system. It determines the fitted parameters u and their covariance matrix. The step 3 is the correction of the fast pre-fitted parameters for the innermost part of the track. It is first done in a comfortable coordinate system (the one in which the first hit is the origin) and then the parameters and their covariance matrix are transformed to the original coordinate system.

Definition at line 463 of file BrokenLine.h.

References riemannFit::LineFit::chi2, gather_cfg::cout, riemannFit::LineFit::cov, mps_fire::i, l1tstage2_dqm_sourceclient-live_cfg::invert, matrixC_u(), multScatt(), dqmiodumpmetadata::n, riemannFit::LineFit::par, brokenline::PreparedBrokenLineData< n >::qCharge, brokenline::PreparedBrokenLineData< n >::radii, rotationMatrix(), slope, riemannFit::sqr(), brokenline::PreparedBrokenLineData< n >::sTotal, createJobs::tmp, brokenline::PreparedBrokenLineData< n >::varBeta, w(), HLT_FULL_cff::weights, and brokenline::PreparedBrokenLineData< n >::zInSZplane.

Referenced by helixFit().

                                                                           {
    const auto& radii = data.radii;
    const auto& sTotal = data.sTotal;
    const auto& zInSZplane = data.zInSZplane;
    const auto& varBeta = data.varBeta;

    const double slope = -data.qCharge / fast_fit(3);
    riemannFit::Matrix2d rotMat = rotationMatrix(slope);

    riemannFit::Matrix3d vMat = riemannFit::Matrix3d::Zero();  // covariance matrix XYZ
    riemannFit::Matrix2x3d jacobXYZtosZ =
        riemannFit::Matrix2x3d::Zero();  // jacobian for computation of the error on s (xyz -> sz)
    riemannFit::VectorNd<n> weights = riemannFit::VectorNd<n>::Zero();
    for (u_int i = 0; i < n; i++) {
      vMat(0, 0) = hits_ge.col(i)[0];               // x errors
      vMat(0, 1) = vMat(1, 0) = hits_ge.col(i)[1];  // cov_xy
      vMat(0, 2) = vMat(2, 0) = hits_ge.col(i)[3];  // cov_xz
      vMat(1, 1) = hits_ge.col(i)[2];               // y errors
      vMat(2, 1) = vMat(1, 2) = hits_ge.col(i)[4];  // cov_yz
      vMat(2, 2) = hits_ge.col(i)[5];               // z errors
      auto tmp = 1. / radii.block(0, i, 2, 1).norm();
      jacobXYZtosZ(0, 0) = radii(1, i) * tmp;
      jacobXYZtosZ(0, 1) = -radii(0, i) * tmp;
      jacobXYZtosZ(1, 2) = 1.;
      weights(i) = 1. / ((rotMat * jacobXYZtosZ * vMat * jacobXYZtosZ.transpose() * rotMat.transpose())(
                            1, 1));  // compute the orthogonal weight point by point
    }

    riemannFit::VectorNd<n> r_u;
    for (u_int i = 0; i < n; i++) {
      r_u(i) = weights(i) * zInSZplane(i);
    }
#ifdef CPP_DUMP
    std::cout << "CU4\n" << matrixC_u(w, sTotal, varBeta) << std::endl;
#endif
    riemannFit::MatrixNd<n> iMat;
    math::cholesky::invert(matrixC_u(weights, sTotal, varBeta), iMat);
#ifdef CPP_DUMP
    std::cout << "I4\n" << iMat << std::endl;
#endif

    riemannFit::VectorNd<n> uVec = iMat * r_u;  // obtain the fitted parameters by solving the linear system

    // line parameters in the system in which the first hit is the origin and with axis along SZ
    line_results.par << (uVec(1) - uVec(0)) / (sTotal(1) - sTotal(0)), uVec(0);
    auto idiff = 1. / (sTotal(1) - sTotal(0));
    line_results.cov << (iMat(0, 0) - 2 * iMat(0, 1) + iMat(1, 1)) * riemannFit::sqr(idiff) +
                            multScatt(sTotal(1) - sTotal(0), bField, fast_fit(2), 2, slope),
        (iMat(0, 1) - iMat(0, 0)) * idiff, (iMat(0, 1) - iMat(0, 0)) * idiff, iMat(0, 0);

    // translate to the original SZ system
    riemannFit::Matrix2d jacobian;
    jacobian(0, 0) = 1.;
    jacobian(0, 1) = 0;
    jacobian(1, 0) = -sTotal(0);
    jacobian(1, 1) = 1.;
    line_results.par(1) += -line_results.par(0) * sTotal(0);
    line_results.cov = jacobian * line_results.cov * jacobian.transpose();

    // rotate to the original sz system
    auto tmp = rotMat(0, 0) - line_results.par(0) * rotMat(0, 1);
    jacobian(1, 1) = 1. / tmp;
    jacobian(0, 0) = jacobian(1, 1) * jacobian(1, 1);
    jacobian(0, 1) = 0;
    jacobian(1, 0) = line_results.par(1) * rotMat(0, 1) * jacobian(0, 0);
    line_results.par(1) = line_results.par(1) * jacobian(1, 1);
    line_results.par(0) = (rotMat(0, 1) + line_results.par(0) * rotMat(0, 0)) * jacobian(1, 1);
    line_results.cov = jacobian * line_results.cov * jacobian.transpose();

    // compute chi2
    line_results.chi2 = 0;
    for (u_int i = 0; i < n; i++) {
      line_results.chi2 += weights(i) * riemannFit::sqr(zInSZplane(i) - uVec(i));
      if (i > 0 && i < n - 1)
        line_results.chi2 += riemannFit::sqr(uVec(i - 1) / (sTotal(i) - sTotal(i - 1)) -
                                             uVec(i) * (sTotal(i + 1) - sTotal(i - 1)) /
                                                 ((sTotal(i + 1) - sTotal(i)) * (sTotal(i) - sTotal(i - 1))) +
                                             uVec(i + 1) / (sTotal(i + 1) - sTotal(i))) /
                             varBeta(i);
    }
  }

template<int n>

__host__ __device__ riemannFit::MatrixNd<n> brokenline::matrixC_u ( const riemannFit::VectorNd< n > &  weights, const riemannFit::VectorNd< n > &  sTotal, const riemannFit::VectorNd< n > &  varBeta  )

inline

Computes the n-by-n band matrix obtained minimizing the Broken Line's cost function w.r.t u. This is the whole matrix in the case of the line fit and the main n-by-n block in the case of the circle fit.

Parameters

weights	weights of the first part of the cost function, the one with the measurements and not the angles ({i=1}^n w*(y_i-u_i)^2).
sTotal	total distance traveled by the particle from the pre-fitted closest approach.
varBeta	kink angles' variance.

Returns

the n-by-n matrix of the linear system

Definition at line 215 of file BrokenLine.h.

References mps_fire::i, dqmiodumpmetadata::n, riemannFit::sqr(), and HLT_FULL_cff::weights.

Referenced by circleFit(), and lineFit().

                                                                                                     {
    riemannFit::MatrixNd<n> c_uMat = riemannFit::MatrixNd<n>::Zero();
    for (u_int i = 0; i < n; i++) {
      c_uMat(i, i) = weights(i);
      if (i > 1)
        c_uMat(i, i) += 1. / (varBeta(i - 1) * riemannFit::sqr(sTotal(i) - sTotal(i - 1)));
      if (i > 0 && i < n - 1)
        c_uMat(i, i) +=
            (1. / varBeta(i)) * riemannFit::sqr((sTotal(i + 1) - sTotal(i - 1)) /
                                                ((sTotal(i + 1) - sTotal(i)) * (sTotal(i) - sTotal(i - 1))));
      if (i < n - 2)
        c_uMat(i, i) += 1. / (varBeta(i + 1) * riemannFit::sqr(sTotal(i + 1) - sTotal(i)));

      if (i > 0 && i < n - 1)
        c_uMat(i, i + 1) =
            1. / (varBeta(i) * (sTotal(i + 1) - sTotal(i))) *
            (-(sTotal(i + 1) - sTotal(i - 1)) / ((sTotal(i + 1) - sTotal(i)) * (sTotal(i) - sTotal(i - 1))));
      if (i < n - 2)
        c_uMat(i, i + 1) +=
            1. / (varBeta(i + 1) * (sTotal(i + 1) - sTotal(i))) *
            (-(sTotal(i + 2) - sTotal(i)) / ((sTotal(i + 2) - sTotal(i + 1)) * (sTotal(i + 1) - sTotal(i))));

      if (i < n - 2)
        c_uMat(i, i + 2) = 1. / (varBeta(i + 1) * (sTotal(i + 2) - sTotal(i + 1)) * (sTotal(i + 1) - sTotal(i)));

      c_uMat(i, i) *= 0.5;
    }
    return c_uMat + c_uMat.transpose();
  }

__host__ __device__ double brokenline::multScatt ( const double &  length, const double  bField, const double  radius, int  layer, double  slope  )

inline

Computes the Coulomb multiple scattering variance of the planar angle.

Parameters

length	length of the track in the material.
bField	magnetic field in Gev/cm/c.
radius	radius of curvature (needed to evaluate p).
layer	denotes which of the four layers of the detector is the endpoint of the multiple scattered track. For example, if Layer=3, then the particle has just gone through the material between the second and the third layer.

Warning

the formula used here assumes beta=1, and so neglects the dependence of theta_0 on the mass of the particle at fixed momentum.

Returns

the variance of the planar angle ((theta_0)^2 /3).

< inverse of radiation length of the material in cm

number between 1/3 (uniform material) and 1 (thin scatterer) to be manually tuned

Definition at line 53 of file BrokenLine.h.

References funct::abs(), fact, log, SiStripPI::min, and riemannFit::sqr().

Referenced by circleFit(), lineFit(), and prepareBrokenLineData().

                                                                                               {
    // limit R to 20GeV...
    auto pt2 = std::min(20., bField * radius);
    pt2 *= pt2;
    constexpr double inv_X0 = 0.06 / 16.;  
    //if(Layer==1) XXI_0=0.06/16.;
    // else XXI_0=0.06/16.;
    //XX_0*=1;

    constexpr double geometry_factor = 0.7;
    constexpr double fact = geometry_factor * riemannFit::sqr(13.6 / 1000.);
    return fact / (pt2 * (1. + riemannFit::sqr(slope))) * (std::abs(length) * inv_X0) *
           riemannFit::sqr(1. + 0.038 * log(std::abs(length) * inv_X0));
  }

template<typename M3xN , typename V4 , int n>

__host__ __device__ void brokenline::prepareBrokenLineData ( const M3xN &  hits, const V4 &  fast_fit, const double  bField, PreparedBrokenLineData< n > &  results  )

inline

Computes the data needed for the Broken Line fit procedure that are mainly common for the circle and the line fit.

Parameters

hits	hits coordinates.
fast_fit	pre-fit result in the form (X0,Y0,R,tan(theta)).
bField	magnetic field in Gev/cm/c.
results	PreparedBrokenLineData to be filled (see description of PreparedBrokenLineData).

Definition at line 150 of file BrokenLine.h.

References ecalTB2006H4_GenSimDigiReco_cfg::bField, riemannFit::cross2D(), d1, mps_fire::i, multScatt(), dqmiodumpmetadata::n, brokenline::PreparedBrokenLineData< n >::qCharge, brokenline::PreparedBrokenLineData< n >::radii, rotationMatrix(), slope, brokenline::PreparedBrokenLineData< n >::sTotal, brokenline::PreparedBrokenLineData< n >::sTransverse, brokenline::PreparedBrokenLineData< n >::varBeta, and brokenline::PreparedBrokenLineData< n >::zInSZplane.

Referenced by helixFit().

                                                                                            {
    riemannFit::Vector2d dVec;
    riemannFit::Vector2d eVec;

    int mId = 1;

    if constexpr (n > 3) {
      riemannFit::Vector2d middle = 0.5 * (hits.block(0, n - 1, 2, 1) + hits.block(0, 0, 2, 1));
      auto d1 = (hits.block(0, n / 2, 2, 1) - middle).squaredNorm();
      auto d2 = (hits.block(0, n / 2 - 1, 2, 1) - middle).squaredNorm();
      mId = d1 < d2 ? n / 2 : n / 2 - 1;
    }

    dVec = hits.block(0, mId, 2, 1) - hits.block(0, 0, 2, 1);
    eVec = hits.block(0, n - 1, 2, 1) - hits.block(0, mId, 2, 1);
    results.qCharge = riemannFit::cross2D(dVec, eVec) > 0 ? -1 : 1;

    const double slope = -results.qCharge / fast_fit(3);

    riemannFit::Matrix2d rotMat = rotationMatrix(slope);

    // calculate radii and s
    results.radii = hits.block(0, 0, 2, n) - fast_fit.head(2) * riemannFit::MatrixXd::Constant(1, n, 1);
    eVec = -fast_fit(2) * fast_fit.head(2) / fast_fit.head(2).norm();
    for (u_int i = 0; i < n; i++) {
      dVec = results.radii.block(0, i, 2, 1);
      results.sTransverse(i) = results.qCharge * fast_fit(2) *
                               atan2(riemannFit::cross2D(dVec, eVec), dVec.dot(eVec));  // calculates the arc length
    }
    riemannFit::VectorNd<n> zVec = hits.block(2, 0, 1, n).transpose();

    //calculate sTotal and zVec
    riemannFit::Matrix2xNd<n> pointsSZ = riemannFit::Matrix2xNd<n>::Zero();
    for (u_int i = 0; i < n; i++) {
      pointsSZ(0, i) = results.sTransverse(i);
      pointsSZ(1, i) = zVec(i);
      pointsSZ.block(0, i, 2, 1) = rotMat * pointsSZ.block(0, i, 2, 1);
    }
    results.sTotal = pointsSZ.block(0, 0, 1, n).transpose();
    results.zInSZplane = pointsSZ.block(1, 0, 1, n).transpose();

    //calculate varBeta
    results.varBeta(0) = results.varBeta(n - 1) = 0;
    for (u_int i = 1; i < n - 1; i++) {
      results.varBeta(i) = multScatt(results.sTotal(i + 1) - results.sTotal(i), bField, fast_fit(2), i + 2, slope) +
                           multScatt(results.sTotal(i) - results.sTotal(i - 1), bField, fast_fit(2), i + 1, slope);
    }
  }

__host__ __device__ riemannFit::Matrix2d brokenline::rotationMatrix ( double  slope )

inline

Computes the 2D rotation matrix that transforms the line y=slope*x into the line y=0.

Parameters

slope

tangent of the angle of rotation.

Returns

2D rotation matrix.

Definition at line 77 of file BrokenLine.h.

References makeMuonMisalignmentScenario::rot, riemannFit::sqr(), and mathSSE::sqrt().

Referenced by circleFit(), lineFit(), prepareBrokenLineData(), and reco::BeamSpot::rotatedCovariance3D().

                                                                             {
    riemannFit::Matrix2d rot;
    rot(0, 0) = 1. / sqrt(1. + riemannFit::sqr(slope));
    rot(0, 1) = slope * rot(0, 0);
    rot(1, 0) = -rot(0, 1);
    rot(1, 1) = rot(0, 0);
    return rot;
  }

__host__ __device__ void brokenline::translateKarimaki ( karimaki_circle_fit &  circle, double  x0, double  y0, riemannFit::Matrix3d &  jacobian  )

inline

Changes the Karimäki parameters (and consequently their covariance matrix) under a translation of the coordinate system, such that the old origin has coordinates (x0,y0) in the new coordinate system. The formulas are taken from Karimäki V., 1990, Effective circle fitting for particle trajectories, Nucl. Instr. and Meth. A305 (1991) 187.

Parameters

circle	circle fit in the old coordinate system. circle.par(0) is phi, circle.par(1) is d and circle.par(2) is rho.
x0	x coordinate of the translation vector.
y0	y coordinate of the translation vector.
jacobian	passed by reference in order to save stack.

Definition at line 97 of file BrokenLine.h.

References funct::cos(), Phase1L1TJetProducer_cfi::cosPhi, riemannFit::CircleFit::cov, RPCpg::mu, riemannFit::CircleFit::par, hitfit::scalar(), funct::sin(), Phase1L1TJetProducer_cfi::sinPhi, riemannFit::sqr(), and mathSSE::sqrt().

Referenced by circleFit().

                                                                                  {
    // Avoid multiple access to the circle.par vector.
    using scalar = std::remove_reference<decltype(circle.par(0))>::type;
    scalar phi = circle.par(0);
    scalar dee = circle.par(1);
    scalar rho = circle.par(2);

    // Avoid repeated trig. computations
    scalar sinPhi = sin(phi);
    scalar cosPhi = cos(phi);

    // Intermediate computations for the circle parameters
    scalar deltaPara = x0 * cosPhi + y0 * sinPhi;
    scalar deltaOrth = x0 * sinPhi - y0 * cosPhi + dee;
    scalar tempSmallU = 1 + rho * dee;
    scalar tempC = -rho * y0 + tempSmallU * cosPhi;
    scalar tempB = rho * x0 + tempSmallU * sinPhi;
    scalar tempA = 2. * deltaOrth + rho * (riemannFit::sqr(deltaOrth) + riemannFit::sqr(deltaPara));
    scalar tempU = sqrt(1. + rho * tempA);

    // Intermediate computations for the error matrix transform
    scalar xi = 1. / (riemannFit::sqr(tempB) + riemannFit::sqr(tempC));
    scalar tempV = 1. + rho * deltaOrth;
    scalar lambda = (0.5 * tempA) / (riemannFit::sqr(1. + tempU) * tempU);
    scalar mu = 1. / (tempU * (1. + tempU)) + rho * lambda;
    scalar zeta = riemannFit::sqr(deltaOrth) + riemannFit::sqr(deltaPara);
    jacobian << xi * tempSmallU * tempV, -xi * riemannFit::sqr(rho) * deltaOrth, xi * deltaPara,
        2. * mu * tempSmallU * deltaPara, 2. * mu * tempV, mu * zeta - lambda * tempA, 0, 0, 1.;

    // translated circle parameters
    // phi
    circle.par(0) = atan2(tempB, tempC);
    // d
    circle.par(1) = tempA / (1 + tempU);
    // rho after translation. It is invariant, so noop
    // circle.par(2)= rho;

    // translated error matrix
    circle.cov = jacobian * circle.cov * jacobian.transpose();
  }