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 # Tracking in a nutshell
 
 Track reconstruction is the process to recover the properties of a charged
 particle from a set of measurements caused by the interaction with some form of
 sensitive detector. The goal is to find which measurements are likely to have
 been caused by which particle, to group them accordingly, and to estimate the
 associated trajectory. Such charged particle trajectories form the basic input
 to the majority of higher-level reconstruction procedures in many cases.
 
 :::{figure} /figures/tracking/tracking.svg
 :width: 400px
 :align: center
 Illustration of a track reconstruction chain starting from spacepoints to fully
 formed tracks.
 :::
 
 This section provides a high-level view of a track reconstruction chain, and is
 largely based on [^phd:gessinger:2021]. It gives an overview of the basic
 building blocks conceptually, and also connects these building blocks with the
 concrete implementations in the core ACTS library, where available.
 
 ## Charged particle detection
 
 The first step in the chain to reconstruct charged particles is their
 detection using sensitive elements. Charged particle detection can be
 achieved in a variety of ways with very different technologies. It can be
 achieved by measuring the interaction of charged particles with matter. When
 this occurs, the interacting particle typically ionizes the surrounding
 material. Particle detectors make use of this fact by converting the
 resulting charge into a measurable signal in various ways.
 
 (segmentation)=
 :::{figure} /figures/tracking/segmentation.svg
 :align: center
 :width: 400px
 
 Illustration of a one-dimensional (a) and a two-dimensional segmentation (b) of a silicon sensor.
 :::
 
 A very common electronic detection approach is the use of semiconducting
 particle detectors, often made of silicon. When a charged particle traverses
 such a sensor, it ionizes the material in the depletion zone, caused by the
 interface of two different semiconducting materials. The result are pairs of
 opposite charges. These charge pairs are separated by an electric field and
 drift toward the electrodes. At this point, an electric signal is created
 which can be amplified and read out. By means of segmentation, the measured
 signal can be associated with a location on the sensor. Silicon sensors are
 usually segmented in one dimension (*strips*) or in two dimensions
 (*pixels*) (see {numref}`segmentation`).
 
 (track_parametrization)=
 ## Track parametrization
 
 To express the properties of a particle's trajectory, a choice of parameters
 has to be made. The parameters need to express all the relevant quantities of
 interest. In the presence of a magnetic field, which affects charged
 trajectories, the global position and momentum, as well as the charge are
 needed to fully specify the particle properties. In addition, a time parameter
 can be included. Apart from the global reference frame, track quantities often
 need to be represented with respect to a surface. This can be achieved with a
 parametrization like
 
 \begin{equation*}
   \vec x = \left(l_0, l_1, \phi, \theta, q/p, t\right)^T
 \end{equation*}
 
 although other parameter conventions exist as well.
 {numref}`parameters` illustrates this choice of parameters. $l_0$, $l_1$ are
 the local coordinates of the corresponding surface, $\phi \in [-\pi,\pi)$ and
 $\theta \in [0,\pi]$ are the angles in the transverse and longitudinal
 direction of the global frame, expressed with respect to the current location
 along the trajectory, as indicated in {numref}`parameters` (b). $\theta$ is
 the polar angle relative to the positive $z$-axis, and $\phi$ is the azimuth
 angle in the transverse plane. Finally, $q/p$ combines the charge of the
 particle with the inverse momentum. In {numref}`parameters` (a), the global
 momentum vector $\vec p$ is shown, which can be recovered from the parameters
 $\vec x$ using $\phi$, $\theta$ and $q/p$.
 
 (parameters)=
 :::{figure} /figures/tracking/parameters.svg
 :width: 500px
 :align: center
 Illustration of the parametrization of a particle track with respect to a
 two-dimensional surface. (a) shows the local position, global momentum and
 their corresponding uncertainties. (b) displays the angles $\phi$ and $\theta$
 in the transverse and longitudinal planes.
 :::
 
 (perigee)=
 :::{figure} /figures/tracking/perigee.svg
 :width: 200px
 :align: center
 Illustration of the perigee parametrization which uses the point of closest
 approach relative to a reference point. The impact parameter $d_0$, the
 position $l$ and the momentum vector $\vec p$ are shown.
 :::
 
 Aside from the nominal quantities captured in $\vec x$, the related
 uncertainties and correlations need to be taken into account as well. They
 can be expressed as a $6\times 6$ covariance matrix like
 
 \begin{equation*}
   C =
   \begin{bmatrix}
    \sigma^2(l_0)& \text{cov}(l_0,l_1) & \text{cov}(l_0, \phi) & \text{cov}(l_0, \theta) & \text{cov}(l_0, q/p) & \text{cov}(l_0, t) \\
    . & \sigma^2(l_1) & \text{cov}(l_1, \phi) & \text{cov}(l_1, \theta) & \text{cov}(l_1, q/p) & \text{cov}(l_1, t) \\
    . & . &  \sigma^2(\phi) & \text{cov}(\phi,\theta) & \text{cov}(\phi, q/p) & \text{cov}(\phi, t) \\
    . & . & . & \sigma^2(\theta) & \text{cov}(\theta, q/p) & \text{cov}(\theta, t) \\
    . & . & . & . & \sigma^2(q/p) & \text{cov}(q/p, t) \\
    . & . & . & . & . & \sigma^2(t)
   \end{bmatrix}
 \end{equation*}
 
 Here, $\text{cov}(X,Y)$ is the covariance of variables $X$ and $Y$, while
 $\sigma^2(X)$ are the regular variances. As the covariance matrix $C$ is
 symmetric, only the upper right half is shown in the matrix above. The
 uncertainties associated with the local position, as well as the momentum
 direction are indicated in {numref}`parameters` (a) as an ellipse and a cone
 around the momentum vector $\vec p$, respectively.
 
 (particle_propagation)=
 ## Particle propagation
 
 :::{tip}
 A dedicated description of the ACTS implementation of particle propagation can be found [here](propagation_impl).
 :::
 
 A central part of track reconstruction is the ability to calculate the trajectory
 of a charged particle, given its properties at a given point. This process,
 called *particle propagation* or *extrapolation*, is used to predict a
 particle's properties after it has travelled a certain distance. In many cases,
 the projected intersection with various types of surfaces is desired. The
 trajectory of a charged particle is governed by the [magnetic
 field](#magnetic-field-core) through which it travels, as well as any
 [material effects](#material-core). In case of a homogeneous magnetic field,
 and in the absence of material interaction, the particle follows a helical
 trajectory. Such a helix can be calculated purely analytically, although
 intersections require numerical methods nevertheless.
 
 Often, magnetic fields are not homogeneous, however. In the presence of such
 changing fields, the corresponding differential equations of motions need to be
 solved using numerical integration techniques.
 
 ### Numerical integration
 
 In ACTS, numerical integration is done using the *Runge-Kutta-Nyström* (RKN) method.
 Commonly used in the variant at fourth order, the RKN method describes how to calculate a
 solution to an initial value problem that can be formulated generically like
 
 $$
   \frac{dy}{dt} = f(t,y), \qquad y(t_0) = y_0,
 $$
 
  where $y_0$ refers to the initial value of $y$ at $t_0$, and
 $f(t,y)$ is the functional form describing the dynamics. The method then
 successively approximates the analytical solution $y(t)$ in a stepwise
 fashion. At each step $(t_n, y_n)$, the goal is effectively to approximate
 the next step $y(t_{n+1})$. Using a step size $h$, the algorithm evaluates
 the function $f$ at four points $k_{1-4}$:
 
 $$
   \begin{aligned}
     k_1 &= f(t_n, y_n) \\
     k_2 &= f\left( t_n + \frac h 2, y_n + h \frac{k_1} 2 \right) \\
     k_3 &= f\left( t_n + \frac h 2, y_n + h \frac{k_2} 2 \right)\\
     k_4 &= f\left( t_n + h, y_n + hk_3 \right).
   \end{aligned}
 $$
 
 
 (rk)=
 :::{figure} /figures/tracking/rk.svg
 :width: 400px
 :align: center
 Illustration of the RKN method approximating a first order
 differential equation. Shown is the calculation of an estimate $y_{n+1}$ at
 $t_{n+1} = t_n + h$, based on the current step $(t_n,y_n)$. Shown are the four
 distinct points at which function $y(t)$ is evaluated, and which are blended to
 form the estimate.
 :::
 
 
 Looking at {numref}`rk`, the meaning of these four points in relation
 to the step size $h$ can be understood. $k_1$ is the derivative at the
 current location, $k_{2,3}$ use $k_1$ and $k_2$ respectively to calculate two
 envelope derivatives at $h/2$ and $k_4$ uses $k_3$ to make an estimate of the
 derivative at $h$. Combining $k_{1-4}$, $(t_{n+1},y_{n+1})$ can be calculated
 as the approximation of $y(t_{n+1})$ like
 
 $$
   \begin{aligned}
     y_{n+1} &= y_n + \frac 1 6 h ( k_1 + 2 k_2 + 2 k_2 + k_4)\\
     t_{n+1} &= t_n + h
   \end{aligned}
 $$
 
 by effectively averaging the four derivatives. It is apparent that
 the step size crucially influences the accuracy of the approximation. A large
 step size weakens the approximation, especially if the magnetic field changes
 strongly. On the other hand, a too small step size will negatively affect the
 execution time of the algorithm.
 
 The Runge-Kutta-Nyström method from above can be adapted to handle second order
 differential equations, as is needed for the equations of motion in question,
 
 $$
   \frac{d^2 \vec r}{ds^2} = \frac q p \left( \frac{d\vec r}{ds} \times \vec B (\vec r) \right) = f(s, \vec r, \vec T), \qquad \vec T \equiv \frac{d \vec r}{ds},
 $$
 
 with the global position $\vec r$, the path element $s$, the
 normalized tangent vector $\vec T$ and the magnetic field $\vec B(\vec r)$ at
 the global position. A slight modification of $k_{1-4}$ is also required,
 incorporating the first derivative of $f(s, \vec r, \vec r')$, finally
 leading to
 
 $$
   \begin{aligned}
     \vec T_{n+1} &= \vec T_n + \frac h 6 (k_1 + 2k_2 + 2k_3 + k_4) \\
     \vec r_{n+1} &= \vec r_n + h \vec T_n + \frac{h^2}{6} (k_1 + k_2 + k_3).
   \end{aligned}
 $$
 
 A strategy exists to dynamically adapt the step size according to the magnetic
 field strength, with the definition of a target accuracy that the algorithm
 tries to achieve. Here, the step size $h$ will successively be decreased and
 the approximation recalculated until the accuracy goal is achieved. Even with
 these additional calculations, the approach is still preferable over a
 consistently low step size.
 
 ### Covariance transport
 
 Aside from the prediction of the track parameters at a given path length, a key
 ingredient to many dependent applications are the uncertainties in the form of
 the associated covariance matrix $C$. Conversions between covariance matrices
 $C^i\to C^f$ can generally be achieved like
 
 $$
   C^f = J \cdot C^i \cdot J^T,
 $$
 
 using the Jacobian matrix
 
 $$
   J = \begin{bmatrix}
     \frac{\partial l_0^f}{\partial l_0^i} & \cdots & \frac{\partial l_0^f}{\partial (q/p)^i} \\
     \vdots & \ddots & \vdots \\
     \frac{\partial (q/p)^f}{\partial l_0^i} & \cdots & \frac{\partial (q/p)^f}{\partial (q/p)^i}
   \end{bmatrix},
 $$
 
 between initial and final parameters $\vec x^i$ and $\vec x^f$. The
 task therefore becomes calculating the necessary Jacobians to achieve correct
 transformation.
 
 One part is the transformation between different coordinate systems, but at the
 same location along the trajectory. For this purpose, generic Jacobians can be
 calculated between each coordinate system type, and a common coordinate system.
 The common coordinate system used for this purpose is the curvilinear frame,
 which consists of the global direction angles, and a plane surface located at
 the track location, with the normal of the plane surface aligned with the track
 momentum. By using Jacobians to the curvilinear frame and the corresponding
 inverse matrices, conversions between any two coordinate systems can be
 performed.
 
 The second part is the calculation of the evolution of the covariance matrix
 during the propagation between surfaces. To this end, a semi-analytical
 method which calculates the effective derivatives between two consecutive
 RKN steps can be used. By accumulating the Jacobian
 matrices calculated for each step, the effective Jacobian between the
 starting point and the destination can be obtained.
 
 ## Material effects
 
 Charged particles interact with matter as they pass through it. Since
 particle detectors inevitably consist of some form or material, this effect
 cannot be completely avoided. By building tracking detectors as light as
 possible, and arranging passive components, such as services and support
 structures carefully, the material a particle encounters before being
 measured can be reduced. Charged particles traversing any form of matter
 undergo elastic and inelastic interactions with the atomic structure of the
 material, depending on the particle properties.
 
 (multiple_scattering)=
 :::{figure} /figures/tracking/multiple_scattering.svg
 :width: 200px
 :align: center
 Illustration of the effect of multiple scattering on the
 trajectory of a charged particle passing through a block of material.
 Entering from the left, it undergoes a series of scattering events,
 deflecting the trajectory statistically, before exiting on the right.
 :::
 
 In elastic interactions, the particle does not lose a significant amount of
 energy, while its trajectory is affected. {numref}`multiple_scattering` shows a
 sketch of the way multiple Coulomb scattering affects the direction of a
 particle trajectory. In addition, a shift in the transverse plane relative to
 the incident direction can occur. As the scattering events occur in
 statistically independent directions, the means of both the deflection and
 offset tends toward zero as the number of scatters increases. Therefore, in the
 numerical particle propagation, this can be accounted for by simply increasing
 the uncertainties associated with the direction, depending on the amount of
 material encountered.
 
 On the other hand, there are interactions during which the particle loses some
 of its energy. Relevant processes here are ionization, as well as
 bremsstrahlung for light particles like electrons. For hadronic particles,
 hadronic interactions with the nuclei of surrounding material is another
 process of interest. In such hadronic interactions, the incoming particle often
 disintegrates, and does not propagate further. Since the size of ionization
 losses only fluctuates to a small degree for thin layers of material, they can
 usually be accounted for by reducing the trajectory energy correspondingly. For
 bremsstrahlung, where fluctuations are much larger and the effect cannot be
 modelled adequately with a Gaussian distribution, dedicated techniques are
 needed (see [](gsf_core)).
 
 Two main approaches are implemented in ACTS. The first approximates the
 material interaction by using a description that averages the real material
 onto thin surfaces across the detector (more on this in
 [](#geometry-and-material-modelling)). When the propagation encounters such a
 surface, it retrieves the material properties, and executes parametrized
 modifications to the particle properties and uncertainties. In the second
 approach, material effects are continuously incorporated during propagation,
 rather than at discrete locations. The latter approach is especially suited for
 propagation through volumes of dense material, where the discretization of the
 material distribution will not work as well.
 
 ## Geometry and material modelling
 
 :::{tip}
 A dedicated description of the ACTS implementation of the tracking geometry model can be found [here](geometry_impl).
 :::
 
 A detailed model of the geometry of an experiment is required for tracking. In
 many cases, external information is needed to associate a sensitive element
 with a position and rotation in the laboratory frame. In case of silicon
 sensors, the intrinsic information captured by the sensor is restricted to the
 measurement plane. Using a transformation matrix, this local measurement can be
 turned into a global one.
 
 Full simulation using tools like Geant4 are frequently used in HEP.
 It includes its own geometry
 description framework. For the precise simulation of particle interactions
 with the detector, this geometry modelling
 is highly detailed. Even very small details of the
 physical hardware can be crucial, and are often included in the geometry
 description. An example for this are readout chips on silicon sensors, or cooling
 elements. {numref}`geometry_detail` (a) shows a sketch of such a detailed
 geometry description. Shown as an example is a *layer* of silicon
 sensors in a barrel configuration. The green rectangles represent the actual
 sensitive surfaces, while other elements include cooling, readout and other components.
 
 (geometry_detail)=
 :::{figure} /figures/tracking/geometry_detail.svg
 :align: center
 
 Sketch of the way a fully detailed simulation geometry (a) models passive
 elements, in addition to the sensitive elements shown in green. (b) shows a
 simplified version, where all non-sensitive elements are approximated.
 :::
 
 In the majority of cases in track reconstruction, this detailed
 geometry is unnecessary. During track reconstruction, the aforementioned
 associated information needs to be accessible for measurements, so all
 sensitive elements need to be included in some form. Passive elements, on the
 other hand, are only required to factor in material interaction effects (see
 [](#particle-propagation)). Moreover, the fully detailed geometry comes
 at the disadvantage of introducing significant overhead during navigation. In
 this process, an algorithm attempts to figure out which elements the particle
 propagation needs to target, as the trajectory is likely to intersect them.
 With a geometry description this precise, the navigation process becomes a
 significant performance bottleneck.
 
 (layer_barrel)=
 :::{figure} /figures/tracking/layer_barrel.svg
 :width: 300px
 :align: center
 Sketch of the way sensitive elements are grouped into layers.
 Shown is an $xy$-view of a number of sensors, arranged as in e.g. the ATLAS
 silicon detector barrels. The grouping is based on their mounting radius.
 The layers are indicated in different colors.
 :::
 
 As a compromise between modelling accuracy and performance, ACTS
 uses a simplified geometry model. It
 focusses on the sensitive elements, which are strictly needed, while passive
 elements are discarded from the explicit description and approximated.
 {numref}`geometry_detail` (b) shows such a simplified geometry. Here, the
 sensitive elements are still shown in green, and other elements are greyed
 out, indicating that they are discarded. The sensitive elements are then
 grouped into layers, as sketched in {numref}`layer_barrel`. How exactly the
 grouping occurs depends on the concrete experiment geometry. In some cases, the layers have the shape
 of cylinder surfaces with increasing radii. This example is shown in the
 figure in the transverse plane at radii $r_{1,2,3}$. In the endcaps, where
 modules are often arranged on disks, these are used as the layer shape. An
 illustration of endcap disk layers can be found in {numref}`layer_ec`, where
 six disks are located at six distinct positions in $\pm z_{1,2,3}$, and
 shown in different colors.
 
 (layer_ec)=
 :::{figure} /figures/tracking/layer_ec.svg
 :width: 400px
 :align: center
 Sketch of the way sensitive elements are grouped into layers.
 Shown is a view of a number of sensors, arranged as in e.g. the ATLAS
 silicon detector endcaps. They are grouped into disks based on their
 mounting position in $z$. The layers are indicated in different colors.
 :::
 
 During particle propagation, the navigation makes use of this layer
 system. Each layer contains a binned structure, which maps a bin to a set
 of sensitive surfaces that overlap with the bin area. This is illustrated in
 {numref}`geo_binning`, where the left picture shows the sensitive surface
 structure of an exemplary endcap disk. The picture on the right overlays the
 binning structure that can be used to enable fast retrieval of compatible
 sensitive surfaces. By performing a simple bin lookup, the navigation can
 ascertain which sensors it needs to attempt propagation to.
 
 
 (geo_binning)=
 :::{figure} /figures/tracking/surface_array.svg
 :width: 400px
 :align: center
 Illustration of the binning structure that is used to subdivide
 layer surfaces. (a) shows two sensor rings of
 different radii grouped into one disk layer. (b)
 overlays the binning structure that the navigation queries for compatible
 surfaces.
 :::
 
 Furthermore, layers are grouped into volumes. Each volume loosely corresponds
 to a region of the detector.
 Volumes are set up such that their boundary surfaces always touch another
 volume. An exception to this is the outermost volume. Each volume's boundary
 surfaces store which volume is located on their other side, essentially
 forming portals between the volumes. This glueing enables the geometry
 navigation between volumes. When the propagation has finished processing a
 set of layers, it attempts to target the boundary surfaces. Once a boundary
 surface is reached, the active volume is switched, and the next set of layers
 is processed.
 
 Care has to be taken to correctly model the passive material, that is
 initially discarded with non-sensitive elements. For the material effects to
 be correctly taken into account during particle propagation, the material is
 projected onto dedicated material surfaces. These material surfaces are
 spread across the detector geometry. Each layer is created with two
 *approach surfaces* on either side. Their distance can be interpreted as
 the thickness of the layer in question. Examples of these approach surfaces
 can be found in {numref}`geometry_detail`, at the inner and outer radius.
 Approach surfaces, and the boundary surfaces between volumes mentioned before,
 are candidates to receive a projection of the surrounding material.
 Additional artificial material layers can also be inserted to receive
 projected material.
 
 The projection procedure (see [](#material-core) and [](#material_mapping_howto_core)) works
 by extrapolating test particles using the fully detailed simulation geometry.
 During the extrapolation, the material properties of the geometry are sampled
 in small intervals. Subsequently, the same test particle is extrapolated
 through the tracking geometry. All material samples are then assigned and
 projected onto the closest material surface. Finally, the projection is
 averaged. The exact number and placement of the material surfaces has to be
 optimized to yield a sufficiently accurate representation of the inactive
 material in the detector.
 
 The numerical integration uses these projected material surfaces. Whenever
 such a surface is encountered in the propagation, the material properties are
 retrieved, and the corresponding modifications to the trajectory are
 executed. In case material is supposed to be integrated in a continuous way
 (as mentioned in [](#particle-propagation)), volumes can also store an
 effective volumetric material composition, which is queried by the numerical
 integration when needed. As the actual physical location of the detection
 hardware can vary over time, possible misalignment of the sensors needs to be
 handled correctly.
 
 ## Clustering
 
 :::{tip}
 See [](clustering_core) for information of the implementation of clustering in
 the core library.
 :::
 
 The actual track reconstruction procedure itself starts with the conversion of
 raw inputs that have been read out from the detector. In case of silicon
 detectors, the readout can either be performed in a binary way, only recording
 which segments fired, or the amount of charges measured in the segment can be
 recorded, e.g. via *time-over-threshold* readout. In all cases, the readout is
 attached to an identifier uniquely locating the segment on the corresponding
 sensor.
 
 As a next step, these raw readouts need to be *clustered*, in order to
 extract an estimate of where particles intersect with the sensor. The general
 strategy of clustering algorithms follows the Connected Component Analysis (CCA)
 approach, where subsets of segments are successively grouped into clusters.
 In case of the Pixel detector, this clustering occurs in two dimensions,
 corresponding to the segmentation of its sensors. Here, the CCA can
 either consider all eight surrounding pixels as neighboring a central one, or
 only consider the four non-diagonal ones, as shown in
 {numref}`clustering_cca`. The figure only shows the simplest possible
 cluster starting from the central pixel. In reality, the CCA will iteratively
 continue from the pixels on the cluster edges.
 
 (clustering_cca)=
 :::{figure} /figures/tracking/cca.svg
 :align: center
 :width: 400px
 Illustration of both eight and four cell connectivity.
 :::
 
 Subsequently, the effective cluster position needs to be estimated. Multiple
 factors play a role here. First of all, the average position of the cluster
 can be calculated either using only the geometry position of the segments,
 
 $$
   \vec r = \frac{1}{N} \sum_{i=1}^N \vec l_i,
 $$
 
 or be weighted by the charge collected in each segment:
 
 $$
   \vec r = \frac{1}{\sum_{i=1}^N q_i} \sum_{i=1}^N q_i \vec l_i.
 $$
 
 Here, $\vec l_i$ is the local position of the $i$-th segment while
 $q_i$ is its charge.
 
 An illustration of the clusterization can be found in {numref}`clustering_image`,
 where a pixel sensor is shown to be intersected by a charged particle,
 entering on the lower left and exiting on the top right. Three cells shown
 with a red frame receive energy from the particle, but the amount is under
 the readout threshold. Four other cells receive energy above the threshold
 and are read out. The clustering will then group these four cells into a
 cluster, and subsequently estimate the cluster position based on the energy
 deposited in the cells. In case no charge information is not available
 for a given detector, the calculation is purely geometric.
 
 
 (clustering_image)=
 :::{figure} /figures/tracking/clustering.svg
 :align: center
 :width: 400px
 Illustration of the clustering of multiple pixels into a cluster,
 in a three-dimensional view on the left and a projection onto the
 $xy$-plane on the right. A particle enters the center in the lower left,
 crosses several segments before exiting the sensor on the top right. The
 cell colors indicate how far along the trajectory they are encountered.
 :::
 
 Another factor that needs to be accounted for is the drift direction of the
 created charges. In addition to the collection field of the sensor itself,
 the surrounding magnetic field modifies the drift direction by the
 *Lorentz-angle* $\theta_\text{L}$. Depending on the field strength, this
 additional angle can cause segments to be activated that would otherwise not
 be geometrically within reach of the charges. Other effects, such as the fact
 that the modules are not perfectly flat, as the geometry description assumes,
 or cross-talk between readout channels, also play a role at this stage.
 
 In the presence of high event activity, particle intersections on single
 sensors can be close enough to one another to result in clusters that are not
 clearly separated from each other. This circumstance can be somewhat
 mitigated by allowing tracks to share clusters with other particles, which
 comes at the price of allowing duplicated tracks to some extent.
 Additionally, merged clusters typically feature worse position resolution,
 which manifests itself since it negatively affects the final fit of the
 track.
 
 (tracking_sp_formation)=
 ## Spacepoint formation
 
 The basic input to most forms of pattern recognition algorithms for tracking
 are space points, which need to be assembled from the raw measurements. To this
 end, the raw measurements are combined with information provided by the
 geometry description, such as the location and rotation of the sensors. In this
 way, the locations, which are restricted to be local to the sensor surfaces
 intrinsically, can be converted into three dimensional points in space.  See
 [](spacepoint_formation_core) for a description of the implementation of
 spacepoint formation in the core library.
 
 {numref}`sensor` shows an illustration of the information that is consumed for
 a pixel measurement. Shown are three clusters on a sensor, which are caused by
 three tracks intersecting it. The corresponding cluster positions are indicated
 as well, and can be converted to global positions using the inverse of the
 global-to-local transformation matrix, that is provided by the geometry
 description.
 
 (sensor)=
 :::{figure} /figures/tracking/sp_l2g.svg
 :align: center
 :width: 400px
 Illustration of a pixel sensor and its local coordinate system in
 relation to the global laboratory frame. A transformation allows conversion
 between the two systems. Shown are three tracks intersecting the sensor,
 alongside clusters that they produce.
 :::
 
 In strip detectors, on the other hand, only a single
 dimension is segmented, and an individual measurement is therefore only
 constrained in one direction on the surface. However, usually the
 strip modules are mounted in pairs, with a stereo angle rotation
 between the pairs. To form global space points, measurements from both
 sensors of a pair need to be combined.
 Due to the stereo angle, a two dimensional
 location on the orthogonal projection plane relative to the two parallel
 pairs can be found. Using the global transformations of the pair, the
 combined measurement location can be converted to global coordinates. If
 multiple measurements are located on a stereo pair of strip sensors, there
 exists an ambiguity on how to combine strips to form space points, which has to be resolved.
 
 
 ## Seeding
 
 The next step after space point formation is pattern recognition, which be
 implemented in various ways.  Global methods exist which attempt to cluster
 space points, such as conformal mapping. In this approach, the space points are
 transformed into a feature parameter space that reveals patterns for hits
 belonging to the same track. In the specific example of a Hough transform, a
 parameter space $\left(\phi, q/p_\mathrm{T}\right)$ is used. As a result, each
 space point is effectively transformed into a line, as a series of combinations
 of these parameters would lead to the same space point. The lines from a set of
 space points of a single track will intersect in one common area. Such an
 intersection can be used to identify which space points originate from the same
 track. However, this task grows in complexity as detector activity increases
 and is susceptible to material effects. See [](seeding_core) for a description
 of the seeding implementation in the core library.
 
 Another group of approaches is the one of seeding and track following. These
 algorithms differ from the global ones in that they evaluate individual
 combinations of space points, and successively explore the events. One
 algorithm from this group is the cellular automaton that iteratively forms
 chains of space points going from one layer to the next.
 
 The main approach in ACTS is an algorithm that operates on coarse
 subdivisions of the detector is used. This seeding algorithm attempts to find
 triplets of space points from increasing radii which are likely to belong to
 the same track. It achieves this by iterating the combinatorial triplets and
 successively filtering them. Filtering is performed based on the momentum and
 impact parameters, which the algorithm attempts to estimate for each triplet.
 
 Under the assumption of a homogeneous magnetic field along the $z$-axis,
 charged particles should follow helical trajectories. In the transverse plane,
 the motion is circular, while it is a straight line in the $rz$-plane.  The
 transverse impact parameter and momentum can be estimated from the radius of
 the circle in the transverse plane like
 
 $$
   d_0 = \sqrt{c_x^2 + c_y^2} - \rho,
 $$
 
 with the circle center $(c_x, c_y)$ and radius $\rho$. The
 transverse momentum can be related to available quantities like
 
 $$
   p_\mathrm{T} \propto \cdot q B \rho
 $$
 
 with the charge $q$ and the magnetic field $B$. An intersection
 between the straight line in the $rz$-plane with the $z$-axis gives an
 estimate of the longitudinal impact parameter.
 An illustration of seeds in the transverse plane is found in
 {numref}`seeding_figure`. Note that seeds can incorporate hits spread across all of
 the layers shown, although this can be a configuration parameter.
 
 
 (seeding_figure)=
 :::{figure} /figures/tracking/seeding.svg
 :width: 300px
 :align: center
 Sketch of seeds in the transverse plane for a number of tracks on
 four layers. Seeds can combine hits on any three of these layers. The shown
 seeds appear compatible with having originated in the center of the
 detector, which is also drawn.
 :::
 
 ## Track finding and track fitting
 
 In the track seeding and following approach, track candidates are built from
 the initial seeds. One method implemented in ACTS, the [Combinatorial Kalman
 Filter](#combinatorial-kalman-filter) (CKF), uses the *Kalman formalism*.
 Originally developed for monitoring and steering mechanical systems, it can
 also be used to iteratively calculate a track estimate. After a set of track
 candidates has been assembled and filtered (see [](#ambiguity-resolution)), an
 additional track fit is usually performed to extract the best estimate of the
 track. The Kalman formalism can also be used for this, with the addition of a
 smoothing step that has certain benefits.  Other fit strategies exist, such as
 a global $\chi^2$ fit that minimizes the distances between track-sensor
 intersections and measurements on all sensors at the same time. One drawback of
 this method is the necessity to invert very large matrices, which is
 computationally expensive.
 
 In a track fit, the Kalman formalism can be shown to yield optimal estimates
 for Gaussian uncertainties. This assumption breaks down when effects like
 bremsstrahlung come into play. An extension of the Kalman Filter (KF) exists
 that relies on the individual propagation of a set of trajectories, instead of
 a single one, to model these biased uncertainties by a sum of Gaussian
 components. This [Gaussian Sum Filter](gsf_core) (GSF) achieves better results when
 fitting particles such as electrons, likely to undergo bremsstrahlung, and is
 deployed in e.g. the ATLAS tracking chain.
 
 
 ### Kalman formalism and Kalman track fitter
 
 :::{tip}
 See [](kf_core) for a description of the implementation of the Kalman Filter in
 the core library.
 :::
 
 The basis of the Kalman formalism is a state vector, that can be identified
 with the set of track parameters $\vec x$. Note that the concrete
 parametrization plays a subordinate role in this context. Rather than building
 an estimate of the state of a system in real time, a Kalman track fit can be
 understood as estimating the parameters iteratively in steps. In the track
 fitting application, each step is defined by a measurement to be included.
 The evolution of the state vector is described by
 
 $$
   \vec x_k = \mathbf F_{k-1} \vec x_{k-1} + \vec w_{k-1},
 $$
 
 where the linear function $\mathbf F_{k-1}$ transports the state vector at
 step $k-1$ to step $k$. $\vec w_{k-1}$ is additional so-called process noise
 that affects the transport additively. Each step has an associated
 measurement, with the fixed relationship between the measurement and the state vector
 
 $$
   \vec m_k = \mathbf H_k \vec x_k + \epsilon_k.
 $$
 
 Here, $\mathbf H_k$ is the *measurement mapping function*, which
 transforms the state vector into the measurement space. In the ideal case,
 this purpose can be achieved by a simple projection matrix, which extracts a
 subspace of the state vector. Additionally, $\epsilon_k$ represents the
 measurement uncertainty.
 
 The Kalman fit process is divided into different phases:
 
 1. **Prediction** of the state vector at the next step $k+1$ based on the information at the current step $k$.
 2. **Filtering** of the prediction by incorporating the measurement associated to the step
 3. **Smoothing** of the state vector by walking back the steps and using information for the subsequent step $k+1$ to improve the estimate at the current step $k$.
 
 An illustration of these concepts is found in {numref}`kalman_filter`. Here,
 a series of three sensors is shown with measurements on them. The KF
 then predicts the track parameters at an intersection, shown in blue.
 Subsequently, a filtered set of parameters is calculated as a mixture between
 the measurement and the prediction. Not shown in this picture is the
 smoothing step.
 
 (kalman_filter)=
 :::{figure} /figures/tracking/kalman.svg
 :width: 400px
 :align: center
 Illustration of the KF. Two of the three stages, the prediction and the
 filtering are shown. The filtering updates the prediction with information from
 the measurement.
 :::
 
 
 In many cases, the first two phases run in tandem, with prediction and
 filtering happening alternatingly at each step. The smoothing phase is
 launched once the last measurement has been encountered.
 Starting from a state $k$, first, a prediction of the state vector at the
 next measurement location is obtained via
 
 (kf_pred)=
 $$
   \vec x_k^{k-1} = \mathbf F_{k-1} \vec x_{k-1},
 $$
 
 with the linear transport function from above. $\vec x_k^{k-1}$ is
 the prediction of the state vector at step $k$ based on step $k-1$. The next
 stage is the filtering. Here, the state vector is refined by taking into
 account the measurement at the current step. Following one of two variants of
 filtering from [^Fruhwirth:1987fm], the gain matrix formalism, the state
 vector is updated like
 
 $$
   \vec x_k = \vec x_k^{k-1} + \mathbf K_k \left( \vec m_k - \mathbf H_k \vec x_k^{k-1} \right),
 $$
 
 with the *Kalman gain matrix*
 
 $$
   \mathbf K_k = \mathbf C_k^{k-1} \mathbf H_k^\mathrm{T}
     \left(
       \mathbf V_k + \mathbf H_k \mathbf C_k^{k-1} \mathbf H_k^\mathrm{T}
     \right)^{-1}
     .
 $$
 
 Note that $\vec x_k$ is the filtered state vector at step $k$,
 based on information from previous steps and step $k$ itself. This is in
 contrast to $\vec x_k^{k-1}$, which is the prediction of the state vector at
 step $k$ based on $k-1$, and is used to calculate the filtered state vector.
 One input to these equations is the covariance matrix prediction $\mathbf
 C_k^{k-1}$ at step $k$ based on step $k-1$, which can be written as
 
 $$
   \mathbf C_k^{k-1}  = \mathbf F_{k-1} \mathbf C_{k-1} \mathbf F_{k-1}^\mathrm{T} + \mathbf Q_{k-1}
 $$
 
 in the linear version from [^Fruhwirth:1987fm], with the
 covariance $\mathbf C_{k-1}$ at step $k-1$, and the covariance $\mathbf
 Q_{k-1}$ associated with $\vec w_{k-1}$ from above. Also needed is $\mathbf
 V_k$, which is the covariance associated with $\epsilon_k$, effectively
 representing the measurement uncertainty.
 
 Similar to the state vector itself, the corresponding covariance matrix is
 also filtered using
 
 (kf_cov_pred)=
 $$
   \mathbf C_k = \left( \mathbb 1 - \mathbf K_k \mathbf H_k \right) \mathbf C_k^{k-1}.
 $$
 
 In the smoothing phase, the state vector at step $k$ is improved using the
 information from the subsequent step $k+1$ using
 
 $$
   \vec x_k^n = \vec x_k + \mathbf A_k \left( \vec x_{k+1}^n - \vec x_{k+1}^k \right).
 $$
 
 Here, $\vec x_{k+1}^n$ is the smoothed state vector and $\vec
 x_{k+1}^k$ the predicted state vector at the subsequent step $k+1$. Also
 needed is the *smoother gain matrix*
 
 $$
   \mathbf A_k = \mathbf C_k \mathbf F_k^\mathrm{T} \left( \mathbf C^k_{k+1} \right)^{-1},
 $$
 
 with the predicted covariance at step $k+1$, $\mathbf C_k^{k+1}$.
 Finally, the covariance at the current step $k$ can also be smoothed with
 
 $$
   \mathbf C_k^n = \mathbf C_k + \mathbf A_k \left(\mathbf C_{k+1}^n - \mathbf C_{k+1}^k \right) \mathbf A_k^\mathrm{T}.
 $$
 
 After smoothing, the parameter estimate at the first step contains information
 from all other measurement states. As mentioned above, in case the
 uncertainties entering the Kalman fit are Gaussian distributions without
 biases, the KF can be shown to be the optimal solution minimizing mean
 square estimation error. However, certain caveats exist. The KF assumes
 that a linear transport function $\mathbf F$ exists that can propagate the
 state vector. In the presence of inhomogeneous magnetic fields this is not
 the case. Instead of explicitly applying $\mathbf F$ to the state vector for
 the prediction, the ACTS KF turns to the numerical integration,
 discussed in [](#numerical-integration). With it, the prediction from
 [this equation](kf_pred) is simply the intersection of the extrapolated trajectory
 with the next sensitive surface. Aside from this, $\mathbf F$ is also used to
 transport the covariance between steps (see [here](kf_cov_pred)). Here, the
 semi-analytical method for covariance transport in the numerical integration
 can be used. $\mathbf F$ can then be identified with the transport
 Jacobian accumulated between surfaces.
 
 For smoothing, two possibilities exist to obtain the needed covariances from
 subsequent measurement steps. Either, the inverse transport Jacobian is used
 and applied, in a way similar to [this equation](kf_cov_pred), or the numerical
 integration is executed again in an inverse fashion, propagating from the
 subsequent step to the current one.
 
 ### Combinatorial Kalman Filter
 
 :::{tip}
 See [](ckf_core) for information on the CKF implementation found in the core
 library.
 :::
 
 As mentioned above, the Kalman formalism can be used for track finding. In this
 case, the smoothing step can be dropped, as the resulting track candidates are
 likely to be refit regardless, therefore saving some time. The CKF explores the
 event starting from an initial track seed. It does this by considering not only
 a single sequence of measurements, but allowing the branching of the fit at
 each sensitive surface that is encountered. To this end, all or a subset of
 measurements that are found on each surface are considered. Measurements are
 selected based on their compatibility with the current state estimate, by using
 their residuals. A predicted residual
 
 $$
   \vec r_k^{k-1} = \vec m_k - \mathbf H_k \vec x_k^{k-1},
 $$
 
 and a filtered residual
 
 $$
   \vec r_k = \vec m_k - \mathbf H_k \vec x_k
   ,
 $$
 
 can be defined, depending on which state estimate is compared with
 the measurement $\vec m_k$. Using the filtered residual, an effective
 $\chi^2$ increment
 
 $$
   \chi^2_+ = \vec r_k^\mathrm{T}
   \left[ \left( \mathbb 1 - \mathbf H_k  \mathbf K_k \right)  \mathbf V_k \right]^{-1}
   \vec r_k
 $$
 
 can be calculated. The global $\chi^2$ of the track candidate can
 be calculated as the sum of all $\chi^2_+$ across the steps. Measurements
 with a large $\chi^2_+$ are considered as outliers, which have low
 compatibility with the trajectory. By branching out for measurements below a
 certain $\chi^2_+$, and following the branches, a tree-like structure of
 compatible track candidates originating from a track seed is assembled. This
 feature is shown in {numref}`tracking_ckf`, which displays a circular
 trajectory, and a set of iteratively assembled track candidates. Basic
 quality criteria can be applied at this stage, to remove bad candidates. A
 dedicated [](#ambiguity-resolution).
 selects the candidates most likely to belong to real particle tracks.
 
 (tracking_ckf)=
 :::{figure} /figures/tracking/finding.svg
 :width: 300px
 :align: center
 Illustration of the way the CKF iteratively explores
 measurements from a seed outwards. Measurements are added successively, and
 can be shared between the resulting track candidates. Shown in green is a
 circular *real* trajectory.
 :::
 
 ## Ambiguity resolution
 
 Due to the combinatorial nature of track finding, and to achieve high
 efficiencies, this set of candidates is often large, and contains a
 non-negligible fraction of *fake* candidates. These fake candidates are either
 completely combinatorial, or arise from real particle measurements with
 combinatorial additions. Track candidates coming from a single seed necessarily
 share a common stem of measurements. Measurements can potentially also be
 shared between candidates from different seeds.
 
 One possibility to resolve this (as is done in e.g. ATLAS tracking) is an
 ambiguity resolution algorithm, that attempts to filter out as many undesirable
 tracks as possible. This is implemented by means of a scoring function, that
 combines properties of the track parameters. Higher scores are correlated with
 a larger probability to be a desirable track candidate. A larger number of hits
 results in an increase in the score, as longer compatible hit chains are less
 likely to be random combinations. On the other hand, missing hits in sensors
 where a hit was expected negatively impact the score.  Experiment specific
 scoring of hits from different subsystems is also implemented. The overall
 $\chi^2$ value computed for the track candidate also plays a role. Candidates
 that share hits with other candidates are penalized. Another quantity is the
 measured particle $p_\mathrm{Y}$, which enters the score, to give preference to
 tracks with large momenta. For tracks containing measurements with a
 substantial local $\chi^2_+$ at the start or end of the trajectory, the
 ambiguity resolution step can also attempt to remove these hits, and determine
 whether a refit without them yields a more favorable global $\chi^2$.
 
 Finally, the output of the ambiguity resolution step is a set of track candidates
 that contain an enhanced fraction of tracks from actual particles, while fake
 tracks are suppressed. They are passed into the final precision fit outlined
 in [](#track-finding-and-track-fitting), to extract the parameter estimate, and used
 for further aspects of reconstruction.
 
 ## Vertex reconstruction
 
 :::{tip}
 See [](vertexing_core) for a dedicated description of the vertexing as
 implemented in ACTS.
 :::
 
 A vertex is a point within the detector, where an interaction or a
 decay occurred. We distinguish between primary vertices (from
 collisions/interactions) and secondary vertices (from subsequent particle
 decays), see {numref}`vertexing_illust`. Primary vertices are further divided
 into hard-scatter and pile-up vertices. While primary vertices are located in
 the luminous region, secondary vertices are slightly displaced due to the finite
  life time of the decaying particle.
 
 (vertexing_illust)=
 :::{figure} /figures/tracking/vertexing.svg
 :width: 400px
 :align: center
 Illustration of a set of three vertices in a proton-proton
 collision. We distinguish between primary hard-scatter, primary pile-up, and
 secondary vertices.
 :::
 
 Vertices play an important role in higher-level reconstruction algorithms. For
 example, secondary vertices can help with the identification of particles:
 During *$b$-tagging*, a displaced vertex located inside a jet is a sign for the
 decay of a $b$-hadron.
 
 In analogy to track reconstruction, vertex reconstruction can be divided into
 two stages: vertex finding and vertex fitting. As a first step of vertex
 finding, we compute a rough estimate of the vertex position from a set of
 tracks. This first estimate can be calculated in many different ways, and is
 referred to as "vertex seed". Seeding algorithms differ for primary and
 secondary vertexing. For primary vertex seeding, one option is to use a
 histogram approach to cluster tracks on the $z$-axis[^phd:piacquadio:2010].
 This is based on the assumption that primary vertices will be close to the
 beamline. Other approaches model tracks as multivariate Gaussian distributions
 and identify regions of high track density as vertex seeds[^phd:schlag:2022].
 For secondary vertexing, seeds are formed from pairs of reconstructed tracks as
 the constraint to the beamline does not apply.
 
 Once a vertex seed is determined, tracks that are compatible with it are
 selected as part of the vertex finding.
 
 Before the vertex fit, we linearize tracks in the vicinity of the vertex seed
 under assuming that they follow a helical (for constant magnetic field) or
 straight (for no magnetic field) trajectory[^phd:piacquadio:2010]. The vertex
 fitter then uses this linearization to improve the position of the vertex seed.
 Furthermore, the track momenta are refitted under the assumption that the tracks
 originate at the vertex[^Fruhwirth:1987fm] [^billoirfitting:1992] .
 
 One issue with an approach like this is that the assignment of tracks to
 vertices is ambiguous. As an improvement, one can perform a multi-vertex fit,
 where vertices compete for tracks. This means that one track can be assigned to
 several vertices. Their contribution to each vertex fit is determined by a
 weight factor, which, in turn, depends on the tracks' compatibility with respect
 to all vertices[^fruwirth:amvfitting:2004].
 
 A flowchart of a multi-vertex reconstruction chain is shown in
 {numref}`vertexing_flowchart`.
 
 (vertexing_flowchart)=
 :::{figure} /figures/tracking/vertexing_flowchart.svg
 :width: 600px
 :align: center
 Simplified flowchart of multi-vertex reconstruction. From a set of seed tracks,
 we first compute a rough estimate of the vertex position, i.e., the vertex seed.
 Then, we evaluate the compatibility of all tracks with the the latter. If a
 track is deemed compatible, it is assigned a weight and attached to the vertex
 seed. Next, the vertex seed and all previously found vertices that share tracks
 with it are (re-)fitted. Finally, after convergence of the fit, we check whether
 the vertex candidate is merged with other vertices and discard it if that is the
 case. For the next iteration, all tracks that were assigned to the vertex seed
 and that have a weight above a certain threshold are removed from the seed
 tracks.
 :::
 
 [^phd:gessinger:2021]: Gessinger-Befurt, Paul, 30.04.2021. Development and improvement of track reconstruction software and search for disappearing tracks with the ATLAS experiment. [10.25358/openscience-5901](https://doi.org/10.25358/openscience-5901)
 [^Fruhwirth:1987fm]: R. Frühwirth, 1987, Application of Kalman filtering to track and vertex fitting, , [11.1016/0168-9002(87)90887-4](https://doi.org/10.1016/0168-9002(87)90887-4)
 [^phd:piacquadio:2010]: G. Piacquadio, 2010, Identification of b-jets and investigation of the discovery potential of a Higgs boson in the $W H \rightarrow l \nu \bar{b} b$ channel with the ATLAS experiment.
 [^phd:schlag:2022]: B. Schlag, 2022, Advanced Algorithms and Software for Primary Vertex Reconstruction and Search for Flavor-Violating Supersymmetry with the ATLAS Experiment.
 [^billoirfitting:1992]: P. Billoir et al., 2022, Fast vertex fitting with a local parametrization of tracks.
 [^fruwirth:amvfitting:2004]: R. Frühwirth et al., 2004, Adaptive Multi-Vertex fitting.

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