MuonLayerHough.cxx

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/*
  Copyright (C) 2002-2021 CERN for the benefit of the ATLAS collaboration
*/
 
#include "MuonLayerHough/MuonLayerHough.h"
 
#include <TH1.h>
#include <TString.h>  // for Form
#include <memory.h>
 
#include <bitset>
#include <cmath>
#include <cstdlib>
#include <iostream>
 
#include "AthenaKernel/getMessageSvc.h"
#include "GaudiKernel/MsgStream.h"
#include "GaudiKernel/SystemOfUnits.h"
namespace {
    // Avoid floating point exceptions arising from cases of x = 0 or x = PI
    // by extending the inverse tan function towards a large number
    // Avoid FPEs occuring in clang 10 , e.g.,
    // FPEAuditor  2   1 WARNING FPE OVERFLOW in [Execute] of [MuGirlStauAlg] on event 257948896 0 0
    // by large number
    float cot(const float x) {
        const float arg = M_PI_2 - x;
        // Tan becomes singular at -pi/2 and pi/2
        if (std::abs(arg - M_PI_2) <= FLT_EPSILON || std::abs(arg + M_PI_2) <= FLT_EPSILON) return 1.e8;
        return std::tan(arg);
    }   
}  // namespace
namespace MuonHough {
 
    MuonLayerHough::MuonLayerHough(const RegionDescriptor& descriptor) :
         m_descriptor(descriptor) {
        // calculate the number of bins
        m_nbins = (m_descriptor.yMaxRange - m_descriptor.yMinRange) / m_descriptor.yBinSize;  // the same for all the cycles
        m_binsize = (m_descriptor.yMaxRange - m_descriptor.yMinRange) / m_nbins;
        m_invbinsize = 1. / m_binsize;  // binsize in the floating plane
 
        // setup the histograms; determines the number of theta slices
        m_histos.reserve(m_descriptor.nthetaSamples);  // it contains all the theta layers
        for (unsigned int i = 0; i < m_descriptor.nthetaSamples; ++i) m_histos.emplace_back(new unsigned int[m_nbins]);
        reset();
    }
 
    void MuonLayerHough::reset() {
        for (unsigned int i = 0; i < m_histos.size(); ++i) memset(m_histos[i].get(), 0, sizeof(unsigned int) * m_nbins);
        max = 0;
        maxhist = -1;
        maxbin = -1;
    }
 
    void MuonLayerHough::fill(float x, float y, float weight) {
        int cycles = m_histos.size();
        for (int ci = 0; ci < cycles; ++ci) {
            float dtheta = m_descriptor.thetaStep;
            float dthetaOffset = 2 * m_descriptor.thetaStep * (ci - (cycles - 1) / 2.);
            float theta = std::atan2(x, y);
            float zref = (m_descriptor.referencePosition - x) * cot(theta - dthetaOffset) + y;
            float z0 = (m_descriptor.referencePosition - x) * cot(theta - dthetaOffset + dtheta) + y;
            float z1 = (m_descriptor.referencePosition - x) * cot(theta - dthetaOffset - dtheta) + y;
 
            float zmin = z0 < z1 ? z0 : z1;
            float zmax = z0 < z1 ? z1 : z0;
            int bincenter = (zref - m_descriptor.yMinRange) * m_invbinsize;
 
            int binmin = (zmin - m_descriptor.yMinRange) * m_invbinsize;
            int binmax = (zmax - m_descriptor.yMinRange) * m_invbinsize;
            if (binmin - bincenter < -3) binmin = bincenter - 3;
            if (binmin == bincenter) binmin -= 1;
            if (binmax == bincenter) binmax += 1;
            if (binmin >= m_nbins) continue;
            if (binmax - bincenter > 3) binmax = bincenter + 3;
            if (binmax < 0) continue;
 
            if (binmin < 0) binmin = 0;
            if (binmax >= m_nbins) binmax = m_nbins - 1;
            if (m_debug)
                std::cout << " filling " << x << " y " << binmin << " " << binmax << " w " << weight << " center " << bincenter << " range "
                          << (zmin - m_descriptor.yMinRange) * m_invbinsize << " " << (zmax - m_descriptor.yMinRange) * m_invbinsize
                          << std::endl;
            for (int n = binmin; n <= binmax; ++n) {
                unsigned int& val = m_histos[ci][n];
                int w = 1000 * weight;
                if (w < 0 && (int)val < -w)
                    val = 0;
                else
                    val += weight;
                if (val > max) {
                    max = val;
                    maxhist = ci;
                    maxbin = n;
                }
            }
        }
    }
 
    void MuonLayerHough::fillLayer(const HitVec& hits, bool subtract) {
        if (hits.empty()) return;
 
        // outer loop over cycles
        int cycles = m_histos.size();
        for (int ci = 0; ci < cycles; ++ci) {
            // float dtheta = m_descriptor.thetaStep;
            // float dthetaOffset = 2*m_descriptor.thetaStep*(ci-(cycles-1)/2.);
 
            int prevlayer = -1;
            int prevbinmin = 10000;
            int prevbinmax = -1;
            // inner loop over hits
            HitVec::const_iterator it = hits.begin();
            HitVec::const_iterator it_end = hits.end();
            for (; it != it_end; ++it) {
                float x = (*it)->x;
                float y1 = (*it)->ymin;
                float y2 = (*it)->ymax;
                std::pair<int, int> minMax = range((*it)->x, (*it)->ymin, (*it)->ymax, ci);
                int binmin = std::max(minMax.first,0);
                int binmax = std::min(minMax.second, m_nbins - 1);
                if (binmin >= m_nbins || binmax < 0) continue;
                
                if (m_debug) {
                    std::cout << " filling hit " << x << " refpos " << m_descriptor.referencePosition << " ymin " << y1 << " ymax " << y2
                              << " layer " << (*it)->layer << " binmin " << binmin << " max " << binmax;
                    if ((*it)->debugInfo()) {
                        const HitDebugInfo* db1 = (*it)->debugInfo();
                        std::cout << " sec " << db1->sector << " r " <<Muon::MuonStationIndex::regionName(db1->region) << " type " 
                                  << db1->type << " lay " << Muon::MuonStationIndex::layerName(db1->layer)
                                  << " slay " << db1->sublayer << std::endl;
                    } else
                        std::cout << std::endl;
                }
                // first hit within range
                if (prevbinmax == -1) {
                    if (m_debug) std::cout << " first range " << binmin << "  " << binmax << std::endl;
                    prevbinmin = binmin;
                    prevbinmax = binmax;
                    prevlayer = (*it)->layer;
                    continue;
                }
 
                if (binmin < prevbinmin && prevlayer == (*it)->layer) {
                    std::cout << "Error hits are out of order: min " << binmin << " max " << binmax << " lay " << (*it)->layer << std::endl;
                }
                // if the max value of the previous hit is smaller than the current minvalue fill the histogram of the previous hit
                // do the same when reached last hit
                if (prevbinmax < binmin || prevlayer != (*it)->layer) {
                    if (m_debug)
                        std::cout << " filling range " << prevbinmin << " " << prevbinmax << " new min " << binmin << "  " << binmax
                                  << " weight " << (*it)->w << std::endl;
                    for (int n = prevbinmin; n <= prevbinmax; ++n) {
                        unsigned int& val = m_histos[ci][n];
                        int w = 1000 * (*it)->w;
                        if (subtract) w *= -1;
                        if (w < 0 && (int)val < -w)
                            val = 0;
                        else
                            val += w;
                        if (val > max) {
                            max = val;
                            maxhist = ci;
                            maxbin = n;
                        }
                    }
                    prevbinmin = binmin;
                    prevbinmax = binmax;
                    prevlayer = (*it)->layer;
 
                } else {
                    // update the maximum value of the window
                    if (m_debug)
                        std::cout << " updating range " << prevbinmin << " " << prevbinmax << " hit " << binmin << "  " << binmax
                                  << "  new " << prevbinmin << " " << binmax << std::endl;
                    prevbinmax = binmax;
                }
            }
            if (prevbinmax != -1) {
                if (m_debug)
                    std::cout << " filling last range " << prevbinmin << " " << prevbinmax << " weight " << hits.back()->w << std::endl;
                for (int n = prevbinmin; n <= prevbinmax; ++n) {
                    unsigned int& val = m_histos[ci][n];
                    int w = 1000 * hits.back()->w;
                    if (subtract) w *= -1;
                    if (w < 0 && (int)val < -w)
                        val = 0;
                    else
                        val += w;
                    if (val > max) {
                        max = val;
                        maxhist = ci;
                        maxbin = n;
                    }
                }
            }
        }
    }
 
    void MuonLayerHough::fillLayer2(const HitVec& hits, bool subtract) {
        if (hits.empty()) return;
 
        std::vector<int> layerCounts(m_nbins, 0);
        int sign = subtract ? -1000 : 1000;
        // outer loop over cycles
        int cycles = m_histos.size();
        for (int ci = 0; ci < cycles; ++ci) {
            // keep track of the previous layer
            int prevlayer = hits.front()->layer;
 
            // inner loop over hits
            HitVec::const_iterator it = hits.begin();
            HitVec::const_iterator it_end = hits.end();
            for (; it != it_end; ++it) {
                // if we get to the next layer process the current one and fill the Hough space
                if (prevlayer != (*it)->layer) {
                    for (int i = 0; i < m_nbins; ++i) {
                        if (subtract && -layerCounts[i] >= static_cast<int>(m_histos[ci][i]))
                            m_histos[ci][i] = 0;
                        else
                            m_histos[ci][i] += layerCounts[i];
                        layerCounts[i] = 0;  // reset bin
                    }
                    prevlayer = (*it)->layer;
                }
 
                // get bin range
                std::pair<int, int> minMax = range((*it)->x, (*it)->ymin, (*it)->ymax, ci);
                int binmin = minMax.first;
                int binmax = minMax.second;
 
                // check wether we are within the Hough space
                if (binmin >= m_nbins) continue;
                if (binmax < 0) continue;
 
                // adjust boundaries if needed
                if (binmin < 0) binmin = 0;
                if (binmax >= m_nbins) binmax = m_nbins - 1;
 
                // output hit for debug purposes
                if (m_debug) {
                    std::cout << " cycle(theta layers) " << ci << " filling hit " << (*it)->x << " refpos "
                              << m_descriptor.referencePosition << " ymin " << (*it)->ymin << " ymax " << (*it)->ymax << " layer "
                              << (*it)->layer << " weight " << (*it)->w << " binmin " << binmin << " max " << binmax;
                    std::cout << " zmin " << binmin * m_binsize + m_descriptor.yMinRange << " zmax "
                              << binmax * m_binsize + m_descriptor.yMinRange;
                    if ((*it)->debugInfo()) {
                        const HitDebugInfo* db1 = (*it)->debugInfo();
                        std::cout << " sec " << db1->sector << " r " << Muon::MuonStationIndex::regionName(db1->region)
                         << " type " << db1->type << " lay " << Muon::MuonStationIndex::layerName(db1->layer)
                         << " bc " << db1->uniqueID << std::endl;
                    } else
                        std::cout << std::endl;
                }
                int weight = sign * (*it)->w;  // set the hit weight
                // set bits to true
                for (; binmin <= binmax; ++binmin) layerCounts[binmin] = weight;
            }  // end of loopin gover hits
            // if the last set of hits was not filled, fill them now; just for the last layer
            for (int i = 0; i < m_nbins; ++i) {
                if (subtract && -layerCounts[i] >= static_cast<int>(m_histos[ci][i]))
                    m_histos[ci][i] = 0;
                else
                    m_histos[ci][i] += layerCounts[i];
                // if( m_debug && layerCounts[i] != 0 ) std::cout << " filling layer " << prevlayer << " bin " << i << " val " <<
                // layerCounts[i] << " tot " << m_histos[ci][i] << std::endl;
                layerCounts[i] = 0;  // reset bin
            }
        }
    }
 
    std::vector<TH1*> MuonLayerHough::rootHistos(const std::string& prefix, const float* rmi, const float* rma) const {
        std::vector<TH1*> hists;
        hists.reserve(m_histos.size());
 
        float rmin = rmi ? *rmi : m_descriptor.yMinRange;
        float rmax = rma ? *rma : m_descriptor.yMaxRange;
 
        int cycles = m_histos.size();
        for (int ci = 0; ci < cycles; ++ci) {
            TString hname = prefix + "_hist";
            hname += ci;
            TH1F* h = new TH1F(hname, hname, m_nbins, rmin, rmax);  // register the new histograms here
            for (int n = 0; n < m_nbins; ++n) h->SetBinContent(n + 1, m_histos[ci][n] * 0.001);
            hists.push_back(h);
        }
        return hists;
    }
 
    bool MuonLayerHough::findMaximum(Maximum& maximum, const MuonLayerHoughSelector& selector) const {
        const float preMaxCut = selector.getMinCutValue();  // in this case it is likely to be 1.9
        maximum.max = 0;
        maximum.pos = 0;
        maximum.theta = 0;
        maximum.refpos = 0;
        maximum.refregion = DetRegIdx::DetectorRegionUnknown;
        maximum.refchIndex = ChIdx::ChUnknown;
        maximum.binpos = -1;
        maximum.binposmin = -1;
        maximum.binposmax = -1;
        maximum.binsize = -1;
        maximum.bintheta = -1;
        maximum.triggerConfirmed = 0;
        maximum.hits.clear();
        maximum.hough = this;
 
        if (preMaxCut < 0) return false;  // this is where there is no cut
 
        float tmax = 0;
        int posb = -1;
        int thetab = -1;
        int imaxval = preMaxCut * 1000;
        const int cycles = m_histos.size();
        // loop over histograms and find maximum
        for (int ci = 0; ci < cycles; ++ci) {
            const float scale =
                1. - 0.01 * std::abs(ci - cycles / 2.0);  // small deweighting of non pointing bins; weighting more on the central part
            for (int n = 0; n < m_nbins; ++n) {
                const int val = m_histos[ci][n];
                if (val < imaxval) continue;
 
                if (scale * val > tmax) {
                    tmax = scale * val;
                    posb = n;
                    thetab = ci;
                }
            }
        }
        if (posb == -1) return false;  // didn't find a max
 
        const float candidatePos = m_descriptor.yMinRange + m_binsize * posb;
        if (tmax < selector.getCutValue(candidatePos)) return false;
 
        maximum.max = tmax * 0.001;
        maximum.pos = candidatePos;
        maximum.theta =
            -m_descriptor.thetaStep * (thetab - (m_histos.size() - 1) * 0.5) + std::atan2(m_descriptor.referencePosition, maximum.pos);
        maximum.refpos = m_descriptor.referencePosition;
        maximum.refregion = m_descriptor.region;
        maximum.refchIndex = m_descriptor.chIndex;
        maximum.binpos = posb;
        maximum.binposmin = posb;
        maximum.binposmax = posb;
        maximum.binsize = m_binsize;
        maximum.bintheta = thetab;
 
        // determin width of maximum; now to be 70% of the original height
        unsigned int imax = m_histos[thetab][posb];
        unsigned int sidemax = 0.7 * imax;
        // loop down, catch case the maximum is in the first bin
        for (int n = posb != 0 ? posb - 1 : posb; n >= 0; --n) {
            if (m_histos[thetab][n] > sidemax) {
                maximum.binposmin = n;
            } else {
                break;
            }
        }
        for (int n = posb + 1; n < m_nbins; ++n) {
            if (m_histos[thetab][n] > sidemax) {
                maximum.binposmax = n;
            } else {
                break;
            }
        }
        return true;
    }
 
    void MuonLayerHough::associateHitsToMaximum(MuonLayerHough::Maximum& maximum, const HitVec& hits) const {
        if (maximum.bintheta == -1 || maximum.binposmax == -1 || maximum.binposmin == -1) return;
        // loop over hits and find those that are compatible with the maximum
        HitVec::const_iterator it = hits.begin();
        HitVec::const_iterator it_end = hits.end();
        for (; it != it_end; ++it) {
            // calculate the bins associated with the hit and check whether any of they are part of the maximum
            std::pair<int, int> minMax = range((*it)->x, (*it)->ymin, (*it)->ymax, maximum.bintheta);
            if (m_debug) {
                std::cout << " associating hit: x " << (*it)->x << " ymin " << (*it)->ymin << " ymax " << (*it)->ymax << " layer "
                          << (*it)->layer << " range " << minMax.first << "  " << minMax.second << "  max range " << maximum.binposmin
                          << "  " << maximum.binposmax << " prd " << (*it)->prd << " tgc " << (*it)->tgc;
                if ((*it)->debugInfo()) std::cout << " trigC " << (*it)->debugInfo()->trigConfirm;
                std::cout << std::endl;
            }
            if (minMax.first > maximum.binposmax) continue;   // minimum bin large than the maximum, drop
            if (minMax.second < maximum.binposmin) continue;  // maximum bin smaller than the minimum, drop
            // keep everything else
            maximum.hits.push_back(*it);
            if ((*it)->debugInfo() && (*it)->debugInfo()->trigConfirm > 0) ++maximum.triggerConfirmed;
        }
    }
 
    std::pair<float, float> MuonLayerHough::maximum(float x, float y, int& posbin, int& thetabin) const {
        unsigned int max = 0;
        thetabin = -1;
        posbin = -1;
 
        int cycles = m_histos.size();
        for (int ci = 0; ci < cycles; ++ci) {
            int relbin = ci - (cycles - 1) / 2;
            float dtheta = m_descriptor.thetaStep;
            float dthetaOffset = 2 * m_descriptor.thetaStep * (relbin);  // if bintheta = cycles, this is the same as dtheta * (cycles + 1 )
            float theta = std::atan2(x, y);
            float z0 = (m_descriptor.referencePosition - x) * cot(theta - dthetaOffset + dtheta) +
                       y;  // move the angle by a step, recalculate the new y value
            float z1 = (m_descriptor.referencePosition - x) * cot(theta - dthetaOffset - dtheta) + y;
 
            float zmin = z0 < z1 ? z0 : z1;
            float zmax = z0 < z1 ? z1 : z0;
            int binmin = (zmin - m_descriptor.yMinRange) / m_binsize - 1;
            int binmax = (zmax - m_descriptor.yMinRange) / m_binsize + 1;
            if (binmin >= m_nbins) continue;
            if (binmax < 0) continue;
 
            if (binmin < 0) binmin = 0;
            if (binmax >= m_nbins) binmax = m_nbins - 1;
            for (int n = binmin; n < binmax; ++n) {
                if (max < m_histos[ci][n]) {
                    max = m_histos[ci][n];
                    posbin = n;
                    thetabin = ci;
                }
            }
        }
        float dthetaOffset = 2 * m_descriptor.thetaStep * (thetabin - (cycles - 1) / 2.);
        return std::make_pair(0.001 * max, -dthetaOffset);
    }
 
    float MuonLayerHough::layerConfirmation(float x, float y, float range) const {
        unsigned int max = 0;
        if (x == 0) return 0;
 
        float yloc = m_descriptor.referencePosition * y / x;
        int bincenter = (yloc - m_descriptor.yMinRange) / m_binsize;
        int scanRange = range / m_binsize;
        int binmin = bincenter - scanRange;
        int binmax = bincenter + scanRange;
        if (binmin >= m_nbins) return 0;
        if (binmax < 0) return 0;
 
        int maxbin = -1;
        if (binmin < 0) binmin = 0;
        if (binmax >= m_nbins) binmax = m_nbins - 1;
        int cyclesmin = 0;
        int cycles = m_histos.size();
        if (cycles > 3) {
            int c0 = cycles / 2;
            cyclesmin = c0 - 1;
            cycles = c0 + 1;
        }
        for (int n = binmin; n < binmax; ++n) {
            for (int ci = cyclesmin; ci < cycles; ++ci) {
                if (max < m_histos[ci][n]) {
                    max = m_histos[ci][n];
                    maxbin = n;
                }
            }
        }
        if (range == 900) std::cout << " layerConfirmation " << max << " bin " << maxbin << " entry " << bincenter << std::endl;
        return 0.001 * max;
    }
 
    std::pair<float, float> MuonLayerHough::layerConfirmation(unsigned int thetaBin, float x, float y, float range) const {
        unsigned int max = 0;
        if (x == 0) return std::make_pair(0., 0.);
        if (thetaBin >= m_histos.size()) return std::make_pair(0., 0.);
        float yloc = m_descriptor.referencePosition * y / x;
        int bincenter = (yloc - m_descriptor.yMinRange) / m_binsize;
        int scanRange = range / m_binsize;
        int binmin = bincenter - scanRange;
        int binmax = bincenter + scanRange;
        if (binmin >= m_nbins) return std::make_pair(0., 0.);
        if (binmax < 0) return std::make_pair(0., 0.);
 
        int maxbin = -1;
        if (binmin < 0) binmin = 0;
        if (binmax >= m_nbins) binmax = m_nbins - 1;
        for (int n = binmin; n < binmax; ++n) {
            if (max < m_histos[thetaBin][n]) {
                max = m_histos[thetaBin][n];
                maxbin = n;
            }
        }
        std::cout << " layerConfirmation " << max << " bin " << maxbin << " entry " << bincenter << " val " << yval(max) << std::endl;
        return std::make_pair(0.001 * max, yval(maxbin));
    }
 
    std::pair<int, int> MuonLayerHough::range(const float x, const float y1, const float y2, const int bintheta) const {
        int cycles = m_histos.size();
        float dx = m_descriptor.referencePosition - x;
        float dtheta = m_descriptor.thetaStep;
        if (dtheta <= 0)
            throw std::runtime_error(
                Form("File: %s, Line: %d\nMuonLayerHough::range() - dtheta is not positive (%.4f)", __FILE__, __LINE__, dtheta));
        float dthetaOffset = 2 * dtheta * (bintheta - (cycles - 1) / 2.);
 
        float theta1 = std::atan2(x, y1) - dthetaOffset;
        float z01 = dx * cot(theta1 + dtheta) + y1;
        float z11 = dx * cot(theta1 - dtheta) + y1;
        float zmin1 = std::min(z01, z11);
        float zmax1 = std::max(z01, z11);
 
        float theta2 = std::atan2(x, y2) - dthetaOffset;
        float z02 = dx * cot(theta2 + dtheta) + y2;
        float z12 = dx * cot(theta2 - dtheta) + y2;
        float zmin2 = std::min(z02, z12);
        float zmax2 = std::max(z02, z12);
 
        const float zmin = std::min(zmin1, zmin2);
        const float zmax = std::max(zmax1, zmax2);
        
        constexpr float cavern_size = 100.*Gaudi::Units::meter;     
        const float flt_lower_bin = std::max(-cavern_size, (zmin - m_descriptor.yMinRange) * m_invbinsize);
        const float flt_upper_bin = std::min(cavern_size,  (zmax - m_descriptor.yMinRange) * m_invbinsize);
        const int lower_bin = std::floor(flt_lower_bin);
        const int upper_bin = std::floor(flt_upper_bin);
       
        return std::make_pair(lower_bin, upper_bin);  // convert the output to bins
    }
 
    float extrapolate(const MuonLayerHough::Maximum& ref, const MuonLayerHough::Maximum& ex, bool doparabolic) {
        // z is always the precision plane. r is the reference plane, simple and robust
        double ref_z = ref.getGlobalZ();
        double ref_r = ref.getGlobalR();
        double ex_z = ex.getGlobalZ();
        double ex_r = ex.getGlobalR();
        float theta_ref = ref.getGlobalTheta();
        if (!doparabolic || ref_z == 0 || theta_ref == 0) {  // do linear extrapolation
            if (!ex.isEndcap()) {                            // if extrapolate to barell
                return ex_z - ex_r / ref_r * ref_z;
            } else {  // if extrapolate to endcap
                return ex_r - ex_z * ref_r / ref_z;
            }
        } else {  // do parabolic
            float expected = 0;
            float extrapolated_diff = 9999;
            const float tan_theta_ref = std::tan(theta_ref);
            const float invtan_theta_ref = 1. / tan_theta_ref;
            float r_start = static_cast<int>(ref.hough->m_descriptor.chIndex) % 2 > 0
                                ? 4900.
                                : 5200.;  // start of barrel B field; values could be further optimized; 5500.:6500.
            float z_start = 8500.;        // start of endcap B field; used to be 6500; should start at 8500
            float z_end = 12500.;         // end of endcap B field; used to be 12000; should end at 12500
            float r_SL = ref_r + (ex_z - ref_z) * tan_theta_ref;
            float z_SL = ref_z + (ex_r - ref_r) * invtan_theta_ref;
            // start extrapolation
            if (!ref.isEndcap()) {  // this is starting with barrel chamber; BEE is 6
                float rgeo = ref_r * ref_r - r_start * r_start;
                float rhoInv = (invtan_theta_ref * ref_r - ref_z) / rgeo;
                if (!ex.isEndcap()) {  // this ends with barrel chamber; BEE is 6
                    expected = ref_z + (ex_r - ref_r) * invtan_theta_ref + (ex_r - ref_r) * (ex_r - ref_r) * rhoInv;
                    extrapolated_diff = ex_z - expected;
                }       // end with barrel to barrel extrapolation
                else {  // this is a barrel to endcap extrapolation, mostly in the transition region
                    // recalculate z start
                    z_start = ref_z + (r_start - ref_r) * invtan_theta_ref + (r_start - ref_r) * (r_start - ref_r) * rhoInv;
                    float zgeo = ref_z * ref_z - z_start * z_start;
                    float rho = (tan_theta_ref * ref_z - ref_r) / zgeo;
                    expected = ref_r + (ex_z - ref_z) * tan_theta_ref + (ex_z - ref_z) * (ex_z - ref_z) * rho;
                    extrapolated_diff = ex_r - expected;
                }
            } else {  // this starts with endcap chamber;
                // swap the starting position if on the other side
                if (tan_theta_ref < 0) {
                    z_start = -z_start;
                    z_end = -z_end;
                }
                if (ex.isEndcap()) {                          // extrapolate to endcap
                    if (std::abs(ref_z) < std::abs(z_end)) {  // extrapolate from EI or EE, have to use linear
                        expected = r_SL;
                    } else {                                       // from EM or EO or EE
                        if (std::abs(ex_z) > std::abs(z_start)) {  // extrapolate to EM or EO
                            // extrapolate to outer layer, just using theta of the middle measurement; only works if the theta measurement
                            // is correct can extrapolate with either the outside theta or middle theta; outside theta is better; farther
                            // away from the B field
                            expected = r_SL;
                        } else {  // to EI
                            float r_end = ref_r + (z_end - ref_z) * tan_theta_ref;
                            float zgeo = z_start * z_start - z_end * z_end;
                            float rhoInv = (r_end - z_end * tan_theta_ref) / zgeo;
                            float tantheta = tan_theta_ref - 2 * (z_end - z_start) * rhoInv;
                            expected = ex_z * tantheta + (ex_z - z_start) * (ex_z - z_start) * rhoInv;
                        }
                    }
                    extrapolated_diff = ex_r - expected;
                } else {  // exrapolate to barrel; again verly likely to be in transition region, complicated B field; just use linear
                    expected = z_SL;
                    extrapolated_diff = ex_z - expected;
                }
            }
            return extrapolated_diff;
        }  // end of parabolic extrapolation
    }
}  // namespace MuonHough