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 // This file is part of the ACTS project.
 //
 // Copyright (C) 2016 CERN for the benefit of the ACTS project
 //
 // This Source Code Form is subject to the terms of the Mozilla Public
 // License, v. 2.0. If a copy of the MPL was not distributed with this
 // file, You can obtain one at https://mozilla.org/MPL/2.0/.
 
 #pragma once
 
 #include "Acts/Definitions/Algebra.hpp"
 #include "Acts/Surfaces/PerigeeSurface.hpp"
 #include "Acts/Vertexing/LinearizedTrack.hpp"
 #include "Acts/Vertexing/TrackAtVertex.hpp"
 #include "Acts/Vertexing/Vertex.hpp"
 
 namespace Acts::KalmanVertexUpdater::detail {
 
 /// Cache object, the comments indicate the names of the variables in Ref. (1)
 /// @tparam nDimVertex number of dimensions of the vertex. Can be 3 (if we only
 /// fit its spatial coordinates) or 4 (if we also fit time).
 template <unsigned int nDimVertex>
 struct Cache {
   using VertexVector = Vector<nDimVertex>;
   using VertexMatrix = SquareMatrix<nDimVertex>;
   // \tilde{x_k}
   VertexVector newVertexPos = VertexVector::Zero();
   // C_k
   VertexMatrix newVertexCov = VertexMatrix::Zero();
   // C_k^-1
   VertexMatrix newVertexWeight = VertexMatrix::Zero();
   // C_{k-1}^-1
   VertexMatrix oldVertexWeight = VertexMatrix::Zero();
   // W_k
   SquareMatrix3 wMat = SquareMatrix3::Zero();
 };
 
 /// @brief Calculates updated vertex position and covariance as well as the
 /// updated track momentum when adding/removing linTrack. Saves the result in
 /// cache.
 ///
 /// @tparam nDimVertex number of dimensions of the vertex. Can be 3 (if we only
 /// fit its spatial coordinates) or 4 (if we also fit time).
 ///
 /// @param vtx Vertex
 /// @param linTrack Linearized track to be added or removed
 /// @param trackWeight Track weight
 /// @param sign +1 (add track) or -1 (remove track)
 /// @note Tracks are removed during the smoothing procedure to compute
 /// the chi2 of the track wrt the updated vertex position
 /// @param[out] cache A cache to store the results of this function
 template <unsigned int nDimVertex>
 void calculateUpdate(const Vertex& vtx, const Acts::LinearizedTrack& linTrack,
                      const double trackWeight, const int sign,
                      Cache<nDimVertex>& cache) {
   constexpr unsigned int nBoundParams = nDimVertex + 2;
   using ParameterVector = Vector<nBoundParams>;
   using ParameterMatrix = SquareMatrix<nBoundParams>;
   // Retrieve variables from the track linearization. The comments indicate the
   // corresponding symbol used in Ref. (1).
   // A_k
   const Matrix<nBoundParams, nDimVertex> posJac =
       linTrack.positionJacobian.block<nBoundParams, nDimVertex>(0, 0);
   // B_k
   const Matrix<nBoundParams, 3> momJac =
       linTrack.momentumJacobian.block<nBoundParams, 3>(0, 0);
   // p_k
   const ParameterVector trkParams =
       linTrack.parametersAtPCA.head<nBoundParams>();
   // c_k
   const ParameterVector constTerm = linTrack.constantTerm.head<nBoundParams>();
   // TODO we could use `linTrack.weightAtPCA` but only if we would always fit
   // time.
   // G_k
   // Note that, when removing a track, G_k -> - G_k, see Ref. (1).
   // Further note that, as we use the weighted formalism, the track weight
   // matrix (i.e., the inverse track covariance matrix) should be multiplied
   // with the track weight from the AMVF formalism. Here, we choose to
   // consider these two multiplicative factors directly in the updates of
   // newVertexWeight and newVertexPos.
   const ParameterMatrix trkParamWeight =
       linTrack.covarianceAtPCA.block<nBoundParams, nBoundParams>(0, 0)
           .inverse();
 
   // Retrieve current position of the vertex and its current weight matrix
   const Vector<nDimVertex> oldVtxPos =
       vtx.fullPosition().template head<nDimVertex>();
   // C_{k-1}^-1
   cache.oldVertexWeight =
       (vtx.fullCovariance().template block<nDimVertex, nDimVertex>(0, 0))
           .inverse();
 
   // W_k
   cache.wMat = (momJac.transpose() * (trkParamWeight * momJac)).inverse();
 
   // G_k^B = G_k - G_k*B_k*W_k*B_k^(T)*G_k
   ParameterMatrix gBMat = trkParamWeight - trkParamWeight * momJac *
                                                cache.wMat * momJac.transpose() *
                                                trkParamWeight;
 
   // C_k^-1
   cache.newVertexWeight = cache.oldVertexWeight + sign * trackWeight *
                                                       posJac.transpose() *
                                                       gBMat * posJac;
   // C_k
   cache.newVertexCov = cache.newVertexWeight.inverse();
 
   // \tilde{x_k}
   cache.newVertexPos =
       cache.newVertexCov * (cache.oldVertexWeight * oldVtxPos +
                             sign * trackWeight * posJac.transpose() * gBMat *
                                 (trkParams - constTerm));
 }
 
 template <unsigned int nDimVertex>
 double vertexPositionChi2Update(const Vector4& oldVtxPos,
                                 const Cache<nDimVertex>& cache) {
   Vector<nDimVertex> posDiff =
       cache.newVertexPos - oldVtxPos.template head<nDimVertex>();
 
   // Calculate and return corresponding chi2
   return posDiff.transpose() * (cache.oldVertexWeight * posDiff);
 }
 
 template <unsigned int nDimVertex>
 double trackParametersChi2(const LinearizedTrack& linTrack,
                            const Cache<nDimVertex>& cache) {
   constexpr unsigned int nBoundParams = nDimVertex + 2;
   using ParameterVector = Vector<nBoundParams>;
   using ParameterMatrix = SquareMatrix<nBoundParams>;
   // A_k
   const Matrix<nBoundParams, nDimVertex> posJac =
       linTrack.positionJacobian.block<nBoundParams, nDimVertex>(0, 0);
   // B_k
   const Matrix<nBoundParams, 3> momJac =
       linTrack.momentumJacobian.block<nBoundParams, 3>(0, 0);
   // p_k
   const ParameterVector trkParams =
       linTrack.parametersAtPCA.head<nBoundParams>();
   // c_k
   const ParameterVector constTerm = linTrack.constantTerm.head<nBoundParams>();
   // TODO we could use `linTrack.weightAtPCA` but only if we would always fit
   // time.
   // G_k
   const ParameterMatrix trkParamWeight =
       linTrack.covarianceAtPCA.block<nBoundParams, nBoundParams>(0, 0)
           .inverse();
 
   // A_k * \tilde{x_k}
   const ParameterVector posJacVtxPos = posJac * cache.newVertexPos;
 
   // \tilde{q_k}
   Vector3 newTrkMom = cache.wMat * momJac.transpose() * trkParamWeight *
                       (trkParams - constTerm - posJacVtxPos);
 
   // \tilde{p_k}
   ParameterVector linearizedTrackParameters =
       constTerm + posJacVtxPos + momJac * newTrkMom;
 
   // r_k
   ParameterVector paramDiff = trkParams - linearizedTrackParameters;
 
   // Return chi2
   return paramDiff.transpose() * (trkParamWeight * paramDiff);
 }
 
 /// @brief Calculates a covariance matrix for the refitted track parameters
 ///
 /// @tparam nDimVertex number of dimensions of the vertex. Can be 3 (if we only
 /// fit its spatial coordinates) or 4 (if we also fit time).
 ///
 /// @param wMat W_k matrix from Ref. (1)
 /// @param crossCovVP Cross-covariance matrix between vertex position and track
 /// momentum
 /// @param vtxCov Vertex covariance matrix
 /// @param newTrkParams Refitted track parameters
 template <unsigned int nDimVertex>
 Acts::BoundMatrix calculateTrackCovariance(
     const SquareMatrix3& wMat, const Matrix<nDimVertex, 3>& crossCovVP,
     const SquareMatrix<nDimVertex>& vtxCov, const BoundVector& newTrkParams) {
   // D_k^n
   SquareMatrix<3> momCov =
       wMat + crossCovVP.transpose() * vtxCov.inverse() * crossCovVP;
 
   // Full x, y, z, phi, theta, q/p, and, optionally, t covariance matrix. Note
   // that we call this set of parameters "free" in the following even though
   // that is not quite correct (free parameters correspond to
   // x, y, z, t, px, py, pz)
   constexpr unsigned int nFreeParams = nDimVertex + 3;
   SquareMatrix<nFreeParams> freeTrkCov(SquareMatrix<nFreeParams>::Zero());
 
   freeTrkCov.template block<3, 3>(0, 0) = vtxCov.template block<3, 3>(0, 0);
   freeTrkCov.template block<3, 3>(0, 3) = crossCovVP.template block<3, 3>(0, 0);
   freeTrkCov.template block<3, 3>(3, 0) =
       crossCovVP.template block<3, 3>(0, 0).transpose();
   freeTrkCov.template block<3, 3>(3, 3) = momCov;
 
   // Fill time (cross-)covariances
   if constexpr (nFreeParams == 7) {
     freeTrkCov.template block<3, 1>(0, 6) = vtxCov.template block<3, 1>(0, 3);
     freeTrkCov.template block<3, 1>(3, 6) =
         crossCovVP.template block<1, 3>(3, 0).transpose();
     freeTrkCov.template block<1, 3>(6, 0) = vtxCov.template block<1, 3>(3, 0);
     freeTrkCov.template block<1, 3>(6, 3) =
         crossCovVP.template block<1, 3>(3, 0);
     freeTrkCov(6, 6) = vtxCov(3, 3);
   }
 
   // Jacobian relating "free" and bound track parameters
   constexpr unsigned int nBoundParams = nDimVertex + 2;
   Matrix<nBoundParams, nFreeParams> freeToBoundJac(
       Matrix<nBoundParams, nFreeParams>::Zero());
 
   // TODO: Jacobian is not quite correct
   // First row
   freeToBoundJac(0, 0) = -std::sin(newTrkParams[2]);
   freeToBoundJac(0, 1) = std::cos(newTrkParams[2]);
 
   double tanTheta = std::tan(newTrkParams[3]);
 
   // Second row
   freeToBoundJac(1, 0) = -freeToBoundJac(0, 1) / tanTheta;
   freeToBoundJac(1, 1) = freeToBoundJac(0, 0) / tanTheta;
 
   // Dimension of the part of the Jacobian that is an identity matrix
   constexpr unsigned int nDimIdentity = nFreeParams - 2;
   freeToBoundJac.template block<nDimIdentity, nDimIdentity>(1, 2) =
       Matrix<nDimIdentity, nDimIdentity>::Identity();
 
   // Full perigee track covariance
   BoundMatrix boundTrackCov(BoundMatrix::Identity());
   boundTrackCov.block<nBoundParams, nBoundParams>(0, 0) =
       (freeToBoundJac * (freeTrkCov * freeToBoundJac.transpose()));
 
   return boundTrackCov;
 }
 
 template <unsigned int nDimVertex>
 void updateVertexWithTrackImpl(Vertex& vtx, TrackAtVertex& trk, int sign) {
   if constexpr (nDimVertex != 3 && nDimVertex != 4) {
     throw std::invalid_argument(
         "The vertex dimension must either be 3 (when fitting the spatial "
         "coordinates) or 4 (when fitting the spatial coordinates + time).");
   }
 
   double trackWeight = trk.trackWeight;
 
   // Set up cache where entire content is set to 0
   Cache<nDimVertex> cache;
 
   // Calculate update and save result in cache
   calculateUpdate(vtx, trk.linearizedState, trackWeight, sign, cache);
 
   // Get fit quality parameters wrt to old vertex
   auto [chi2, ndf] = vtx.fitQuality();
 
   // Chi2 of the track parameters
   double trkChi2 = trackParametersChi2(trk.linearizedState, cache);
 
   // Update of the chi2 of the vertex position
   double vtxPosChi2Update = vertexPositionChi2Update(vtx.fullPosition(), cache);
 
   // Calculate new chi2
   chi2 += sign * (vtxPosChi2Update + trackWeight * trkChi2);
 
   // Calculate ndf
   ndf += sign * trackWeight * 2.;
 
   // Updating the vertex
   if constexpr (nDimVertex == 3) {
     vtx.fullPosition().head<3>() = cache.newVertexPos.template head<3>();
     vtx.fullCovariance().template topLeftCorner<3, 3>() =
         cache.newVertexCov.template topLeftCorner<3, 3>();
   } else if constexpr (nDimVertex == 4) {
     vtx.fullPosition() = cache.newVertexPos;
     vtx.fullCovariance() = cache.newVertexCov;
   }
   vtx.setFitQuality(chi2, ndf);
 
   if (sign == 1) {
     // Update track
     trk.chi2Track = trkChi2;
     trk.ndf = 2 * trackWeight;
   }
   // Remove trk from current vertex by setting its weight to 0
   else if (sign == -1) {
     trk.trackWeight = 0.;
   } else {
     throw std::invalid_argument(
         "Sign for adding/removing track must be +1 (add) or -1 (remove).");
   }
 }
 
 template <unsigned int nDimVertex>
 void updateTrackWithVertexImpl(TrackAtVertex& track, const Vertex& vtx) {
   if constexpr (nDimVertex != 3 && nDimVertex != 4) {
     throw std::invalid_argument(
         "The vertex dimension must either be 3 (when fitting the spatial "
         "coordinates) or 4 (when fitting the spatial coordinates + time).");
   }
 
   using VertexVector = Vector<nDimVertex>;
   using VertexMatrix = SquareMatrix<nDimVertex>;
   constexpr unsigned int nBoundParams = nDimVertex + 2;
   using ParameterVector = Vector<nBoundParams>;
   using ParameterMatrix = SquareMatrix<nBoundParams>;
   // Check if linearized state exists
   if (!track.isLinearized) {
     throw std::invalid_argument("TrackAtVertex object must be linearized.");
   }
 
   // Extract vertex position and covariance
   // \tilde{x_n}
   const VertexVector vtxPos = vtx.fullPosition().template head<nDimVertex>();
   // C_n
   const VertexMatrix vtxCov =
       vtx.fullCovariance().template block<nDimVertex, nDimVertex>(0, 0);
 
   // Get the linearized track
   const LinearizedTrack& linTrack = track.linearizedState;
   // Retrieve variables from the track linearization. The comments indicate the
   // corresponding symbol used in Ref. (1).
   // A_k
   const Matrix<nBoundParams, nDimVertex> posJac =
       linTrack.positionJacobian.block<nBoundParams, nDimVertex>(0, 0);
   // B_k
   const Matrix<nBoundParams, 3> momJac =
       linTrack.momentumJacobian.block<nBoundParams, 3>(0, 0);
   // p_k
   const ParameterVector trkParams =
       linTrack.parametersAtPCA.head<nBoundParams>();
   // c_k
   const ParameterVector constTerm = linTrack.constantTerm.head<nBoundParams>();
   // TODO we could use `linTrack.weightAtPCA` but only if we would always fit
   // time.
   // G_k
   const ParameterMatrix trkParamWeight =
       linTrack.covarianceAtPCA.block<nBoundParams, nBoundParams>(0, 0)
           .inverse();
 
   // Cache object filled with zeros
   Cache<nDimVertex> cache;
 
   // Calculate the update of the vertex position when the track is removed. This
   // might be unintuitive, but it is needed to compute a symmetric chi2.
   calculateUpdate(vtx, linTrack, track.trackWeight, -1, cache);
 
   // Refit track momentum with the final vertex position
   Vector3 newTrkMomentum = cache.wMat * momJac.transpose() * trkParamWeight *
                            (trkParams - constTerm - posJac * vtxPos);
 
   // Track parameters, impact parameters are set to 0 and momentum corresponds
   // to newTrkMomentum. TODO: Make transition fitterParams -> fittedMomentum.
   BoundVector newTrkParams(BoundVector::Zero());
 
   // Get phi and theta and correct for possible periodicity changes
   const auto correctedPhiTheta =
       Acts::detail::normalizePhiTheta(newTrkMomentum(0), newTrkMomentum(1));
   newTrkParams(BoundIndices::eBoundPhi) = correctedPhiTheta.first;     // phi
   newTrkParams(BoundIndices::eBoundTheta) = correctedPhiTheta.second;  // theta
   newTrkParams(BoundIndices::eBoundQOverP) = newTrkMomentum(2);        // qOverP
 
   // E_k^n
   const Matrix<nDimVertex, 3> crossCovVP =
       -vtxCov * posJac.transpose() * trkParamWeight * momJac * cache.wMat;
 
   // Difference in positions. cache.newVertexPos corresponds to \tilde{x_k^{n*}} in Ref. (1).
   VertexVector posDiff =
       vtxPos - cache.newVertexPos.template head<nDimVertex>();
 
   // r_k^n
   ParameterVector paramDiff =
       trkParams - (constTerm + posJac * vtxPos + momJac * newTrkMomentum);
 
   // New chi2 to be set later
   double chi2 =
       posDiff.dot(
           cache.newVertexWeight.template block<nDimVertex, nDimVertex>(0, 0) *
           posDiff) +
       paramDiff.dot(trkParamWeight * paramDiff);
 
   Acts::BoundMatrix newTrackCov = calculateTrackCovariance<nDimVertex>(
       cache.wMat, crossCovVP, vtxCov, newTrkParams);
 
   // Create new refitted parameters
   std::shared_ptr<PerigeeSurface> perigeeSurface =
       Surface::makeShared<PerigeeSurface>(vtxPos.template head<3>());
 
   BoundTrackParameters refittedPerigee =
       BoundTrackParameters(perigeeSurface, newTrkParams, std::move(newTrackCov),
                            track.fittedParams.particleHypothesis());
 
   // Set new properties
   track.fittedParams = refittedPerigee;
   track.chi2Track = chi2;
   track.ndf = 2 * track.trackWeight;
 
   return;
 }
 
 }  // namespace Acts::KalmanVertexUpdater::detail

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