Table Of ContentGeneral Broken Lines as advanced track fitting method
ClausKleinwort
DESYDeutschesElektronen-Synchrotron,Notkestraße85,22607Hamburg,Germany
Abstract
2
InHEPexperimentsthedescriptionofthetrajectoryofachargedparticleisobtainedfromafittomeasurementsin
1
trackingdetectors. Theparametrizationofthetrajectoryhastoaccountforbendinginthemagneticfield,energyloss
0
2 andmultiplescatteringinthedetectormaterial.Generalbrokenlinesimplementatrackmodelwithproperdescription
ofmultiplescatteringleadingtolinearequationswithaspecialstructureofthecorrespondingmatrixallowingfora
n
fast solution with the computing time depending linearly on the number of measurements. The calculation of the
a
J full covariance matrix along the trajectory enables the application to track based alignment and calibration of large
0 detectorswithglobalmethods.
2
Keywords: Trackfitting,Multiplescattering,Globalleast-squaresfit,Millepede,Brokenlines
]
t
e
d 1. Introduction computingtimeO(n3). Progressivemethods[2]likethe
- Ka´lma´nfilter[3]updatingthetrackparametersforeach
s
Thetrajectoryofachargedparticleinahomogeneous
n additionalmeasurementandscattererwithacomputing
i magnetic field neglecting the interactions with the de- time O(n) have become the standard method for track
.
s tector material is described by a helix. In a global co-
fitting.
c
ordinate system (x,y,z) with the magnetic field in z-
i
s directionitcanbeparametrizedbytheinversemomen- The broken lines method is a fast global track refit
y tum(timescharge)q/p,anangleφ atandthedistance adding the description of multiple scattering to an ini-
h 0
p d0 to the point of closest approach in the (x,y)-plane, tial trajectory and able to determine the complete co-
[ thedipangleλtothatplaneandtheoffsetz atthepoint variance matrix of all track parameters. This allows
0
of closest approach. Energy loss in the detector mate- theusageastrackmodelfortrackbasedalignmentand
1
rial due to ionization or radiation (for electrons) leads calibration with the global Millepede-II [4] algorithm.
v
0 to a reduction of the momentum. Multiple scattering, Correctionsandcovariancematricesforthelocaltrack
2 mainly due to Coulomb interaction with the electrons parameters (at single points) can be determined with a
3 intheatoms,resultsinrandomchangesindirectionand computingtimeO(n)exploitingthesparsityofthema-
4 spatialpositionwithexpectationvaluesofzeroandvari- trixofthecorrespondinglinearequationssystem.
.
1 ancedependingonthetraversedmaterialandtheparti- The original formulation [5] describes the case of
0
cle momentum. In addition the magnetic field is usu- a tracking detector with a solenoidal magnetic field
2
1 allyhomogeneousonlyinanapproximation. Therefore andindependenttwo-dimensionaltrackinginthebend-
: moreadvancedtrackmodelsarerequired. ing and perpendicular to the bending plane. It con-
v Theeffectofmultiplescatteringcanbetakenintoac- structs the planar trajectories from the measurements
i
X count in different ways [1]. For global methods it can including the material around those as thick scatterers.
r beaddedtothemeasurementerrorsleadingtoanondi- In the presence of measurements with components in
a
agonalcovariancematrixorexplicitlyfittedbyscatter- both planes a single trajectory in three dimensions is
inganglesasadditionaltrackparameters. Inbothcases required. General broken lines are describing trajec-
this requires the inversion of a large matrix of order n tories in space in an arbitrary magnetic field for ar-
(numberofmeasurementsornumberofscatterers)with bitrary measurements and material distribution. The
three-dimensional broken lines trajectories from one-
or two- dimensional independent measurements dedi-
Emailaddress:[email protected](Claus
Kleinwort) cated to the track-based alignment of the CMS silicon
PreprintsubmittedtoNuclearInstrumentsandMethodsA January23,2012
tracker [6] with Millepede-II [4] are an intermediate momentum q/p. The general broken lines fit deter-
step. They are constructed from the material between minesfromthefitparametersx = (∆q/p,u ,..,u )at
1 nscat
the measurement planes by equivalent thin scatterers. eachthinscattereritrackparametercorrections∆p =
loc
Measurementsareparametrizedbytheinterpolationof (∆q/p,u(cid:48),u). Theslopebeforethescatteringisdefined
i i
(small) offsets perpendicular to the track direction de- by (∆q/p,ui−1,ui) and afterwards by (∆q/p,ui,ui+1)
finedatthosescatterersandacommoncorrection∆q/p (eqn (6) in [6]). At the measurement planes the track
oftheinversemomentum. Multiplescatteringkinksare parameter corrections ∆p = (∆q/p,u(cid:48) ,u ) can be
loc int int
definedbytripletsofthinscatterers. Thecorresponding obtained from interpolation of the enclosing scatterers
linear equation system Ax = b with a bordered band (eqn (8) and inserting u for u in eqn (6) in [6]). In
int 0
matrixAwithasizendependingonthenumberofscat- anycase∆p dependsonlyon∆q/pandtwoadjacent
loc
terers n , band width m (usually 5) and border size offsets. Therefore for the covariance matrices of the
scat
b = 1 is built from the Jacobians of the propagation track parameter corrections the bordered band part of
of the offsets between measurements and scatterers on the covariance matrix A−1 of the fit parameters is suf-
the initial trajectory and solved by root-free Cholesky ficient. Forasparsematrixtheelementsoftheinverse
decomposition. Inthefollowingthefurtherfeaturesof matrix inside the sparsity pattern can by calculated by
generalbrokenlinesarepresented,theprocedureisdis- special methods without those outside that pattern [9].
cussedandacomparisonwiththeKa´lma´nfilterisper- ThisallowstoobtaintheborderedbandpartofA−1with
formed. O(n(m+b)2)operationslinearinthenumberofscatter-
ers.
2. Towardsgeneralbrokenlines
2.3. Measurements
2.1. Localcoordinatesystem
The measurements m are (the residuals with respect
Ateachmeasurementplaneandeachthinscatterera
to the initial trajectory of) arbitrary observables with
local(orthonormal)coordinatesystem(u ,u ,w)isde-
1 2 predictionsdefinedbythelocaltrackparametersinthe
fined. Thenaturalchoiceforthew-directionisperpen-
measurement plane. This can be one or two dimen-
dicular to the measurement plane for a measurement
sional position measurements or track segments con-
and parallel to the track direction for a scatterer. A
taining slope information. The linearized prediction
local system moving with the track is the curvilinear is m(x) = H x. In case the corresponding covari-
m
frame (x ,y ,z ) with z in track direction and x in
⊥ ⊥ ⊥ ⊥ ⊥ ance matrices V are not diagonal they should be di-
m
the global (x,y)-plane. At each thin scatterer the two-
agonalizedandthemeasurementsandpredictionsmod-
dimensionaloffsetu = (u ,u )isdefinedasfitparame-
1 2 ified accordingly. This allows the usage of univariate
ter.
M-estimatorsforoutlierdown-weightingandtheinter-
The variance V of the multiple scattering kinks k
k facingtoMillepede-II[4]expectingindependentscalar
inthelocalsystemcanbecalculatedfromthevariance
measurements.
(cid:0) (cid:1)
θ2 10 [7]inthecurvilinearsystembyparametertrans-
0 01
formation [8] from the curvilinear to the local slopes
2.4. Iterations
whichdependsontheprojectionsc = e ·e ofthe
i track ui
offsetdirectionse ofthelocalframeontothetrackdi- Incasetheresultingfitparametersarenotsmallcor-
ui
rectione : rections as assumed for the linearization of the track
track
model (eqn (3) in [6]) the trajectory has to be iterated.
Vk = ∂∂((xu(cid:48)(cid:48)1,,uy(cid:48)(cid:48)2))(cid:32)θ002 θ02(cid:33)(cid:34)∂∂((xu(cid:48)(cid:48)1,,uy(cid:48)(cid:48)2))(cid:35)T= Tofhitsheremquuilrteipslienistciaalttneorinn-gzekrionkvsa:luke(sxk)0=inHthkxep+rekd0i.ctTiohne
⊥ ⊥ 0θ2 ⊥(cid:32)1⊥−c2 c c (cid:33) expectationvalueremains(cid:104)k(x)(cid:105)=0.
0 2 1 2 (1)
(1−c21−c22)2 c1c2 1−c21 2.5. Seeding
With at least one offset defined perpendicular to the Thegeneralbrokenlinesareseededbyaninitialtra-
trackdirectionthisisadiagonalmatrix. jectory. Alternatively to the result of a fit of the mea-
surements (internal seeding) this can be a prediction
2.2. Localtrackparameters fromanotherpartofthedetector(externalseeding).The
Inthelocalframeatrackcanbedescribedbyanoff- seeding track parameters at some reference point are
set (u ,u ), a slope (u(cid:48),u(cid:48)) = ∂(u1,u2) and the inverse usedforthepropagationalongthetrajectoryaccording
1 2 1 2 ∂w
2
tothemagneticfield(andaverageenergyloss)tocalcu- themeasurementandmultiplescatteringerrorsfromdi-
lateresidualsforthemeasurementsandparametertrans- agonalcovariancematricesVaretheingredientsforthe
formationmatrices. Ifthetrackfithastobeiteratedto localfitinMillepede-II[4]. Afterwardsthelinearequa-
accountfornonlineareffectstheinitialtrajectorycould tion system Ax = b is constructed from the individual
bebasedongeneralbrokenlinesitself. Onaverageno contributions(HTV−1H,HTV−1morHTV−1k )ofmea-
0
changeofthetrackparametersatthereferencepointis surements, kinksandoptionallytheexternalseed. The
expectedfromrefitting: (cid:104)∆p (cid:105)=(cid:104)H x(cid:105)=0. bandpartA ofthematrixisdecomposedintoadiago-
seed s u
nalmatrixDandatriagonalbandmatrixLwithunitdi-
2.6. Minimization agonal(Au = LDLT)andtheoffsetsxu aredetermined
fromD,Landb bysolvingLz=b andLTx =D−1z
Thefitparametersaredeterminedbyminimizing: u u u
[5]. ThebandpartofA−1iscalculatedaccordingto[9].
u
Usingblockmatrixalgebrathecompletesolutionxand
χ2(x)=n(cid:88)meas(cid:0)H x−m(cid:1)TV−1(cid:0)H x−m(cid:1) the bordered band part of A−1 are obtained. From this
m,i i m,i m,i i
i=1 corrections and covariance matrices for local track pa-
(frommeasurements) rameters at any point (measurement, scatterer or seed)
canbederived.
+n(cid:88)scat−1(cid:0)H x+k (cid:1)TV−1(cid:0)H x+k (cid:1) (fromkinks) For all this steps (propagation, prediction, construc-
k,i 0,i k,i k,i 0,i tion, decomposition, solution and inversion) the num-
i=2
ber of operations is linear in the number of points or
+(H x)TV−1(H x) (fromexternalseed) (2)
s s s parameters and a sizable contribution to the total com-
puting time. As all predictions Hx depend at most on
The minimization leads to a linear equation system
Ax=bwithxofsizen=2n +1: threeconsecutiveoffsetsthematrixAuhasabandwidth
scat m = 3·2−1 = 5. Incasethetransformationmatrices
n(cid:88)meas n(cid:88)scat−1 JandSarediagonalthetwooffsetcomponentsarede-
A= HTm,iV−m1,iHm,i+ HTk,iV−k,1iHk,i+HTsV−s1Hs coupledandthebandwidthreducestom=(3−1)·2=4
i=1 i=2 speedingupthebandmatrixoperations(O(n(m+b)2)).
(3a) In additionthe construction of A is fasterdue to fewer
n(cid:88)meas n(cid:88)scat−1 non-zeroelementsinH.
b= HT V−1m − HT V−1k (3b)
m,i m,i i k,i k,i 0,i
i=1 i=2
4. ComparisionwithKa´lma´nfilter
3. Procedure 4.1. Mathematicalequivalence
An externally seeded general broken lines trajectory
Theinitialtrajectoryprovidesinthelocalcoordinate with one measurement has only one offset defined and
system for the measurements the residuals m with (di- the slope has to be used directly as fit parameter: x =
agonal)covariancematrixVm,forthescattererstheini- ∆p = (∆q/p,u(cid:48),u ). The solution of the linear
tial (usually zero) kinks k with covariance matrix V , loc,1 1 1
0 k equationsystemis:
optionallytheexternalseedandfortheparametertrans-
(cid:104) (cid:105)
formations from one point (measurement, scatterer or x=A−1 HT V−1m (4a)
seeTdh)etolintheearniezxedtotrnaectkhmeJoadceolboifatnhseTgie=ne∂rapllobcr,io+k1/e∂nplilnoce,is. A−1 =(cid:104)V−1+mH,1T mV,1−11H (cid:105)−1 (4b)
s m,1 m,1 m,1
is defined at each point by the offset part (d,S,J) =
±
∂u/∂(q/p,u(cid:48),u) oftheJacobiansforthetransformtion ThisisequivalenttothefilteringstepofaKa´lma´nfilter
±
to the next and the previous scatterer. For the forward intheweightedmeanformalism(equation(8b)in[3]):
pprooinpta-gtoa-tipooninthteJyacaorbeicaanlscuTla,tefodrfrthoembthacekpwroadrducctassoefathde- xk =Ck(cid:20)(cid:16)Ckk−1(cid:17)−1xkk−1+HtkV−k1mk(cid:21) (5a)
dfoitrimoneadl.pIanrttihael innevxetrssitoenpsth(feornothne-zoeffroseetlepmaretn)tasreofptehre- C =(cid:20)(cid:16)Ck−1(cid:17)−1+HtV−1H (cid:21)−1 (5b)
k k k k k
matricesHforthelinearizedpredictionsarecalculated
fromthe2×2matricesandvectorsJ ,W =±S−1and Astheinitialtrajectoryhasbeenbasedonthetrackpa-
± ± ±
d (seeeqn(7)and(8)in[6]).Themeasurementsm,the rameters from the external seed the prediction xk−1 is
± k
kinksk,thederivativesfromthesparsematricesHand zero,C correspondstoA−1,Ck−1toV andm tom .
k k s k 1
3
Ingeneralonlythecovariancematricesforthetrack 5 5
parametersatsinglepointsarecalculatedbytheKa´lma´n 4.5 4.5
4 4
filter.Fortheglobalcovariancematrixfromalltrackpa-
3.5 3.5
rametersofthetrajectoryrequiredbyglobalalignment 3 3
andcalibrationmethodsanextensionoftheKa´lma´nfil- 2.5 2.5
2 2
terisdescribedin[10]. 1.5 1.5
1 1
4.2. Computationaldifference 0.5 0.5
0 0
TrackfittingwiththeKa´lma´nfilteralgorithm[3]isa GBL KWM KGM GBL KWM+S KGM+S
sequentialprocedureaddingmeasurementsandscatter-
5 5
ers(processnoise)tothetrajectoryoneatthetime. The 4.5 4.5
4 4
propagation of track parameters and covariance matri-
3.5 3.5
cesfrompointtopointinthefilteringstepinvolvesmul- 3 3
tiplicationsof5×5matriceswithO(53)operations.The 2.5 2.5
2 2
filtering in the weighted mean formalism is dominated
1.5 1.5
bytheinversionof5×5matrices(O(53)operations,C 1 1
0.5 0.5
in equation (5b)) for the addition of a single measure-
0 0
ment. Usingthegainmatrixformalismrequiresthein- GBL KWM KGM GBL KWM+S KGM+S
versionofmatricesofthesizeofthedimensiondofthe
measurements (O(d3) operations) and several multipli-
Figure 1: Computing time for track refitting using different algo-
cationsof5×dmatrices(O(52·d),O(5·d2)operations).
rithms,propagationJacobiansandsetupsofscatterersnormalizedto
The smoothing step to get optimal local track parame- thefastestcase. Thetopplotsarewiththedetailedhelixpropagator,
tersatallandnotonlythelastpointinvolvesinversion thebottomoneswithasimplifiedversion,theleftonesforfiltering
andmultiplicationsof5×5matrices(O(53)operations). (onetrackend)onlyandtherightonesforfilteringandsmoothing
(resultsatbothends). Themethodsshownaregeneralbrokenlines
Forsmalldthegainmatrixformalismshouldbefastest.
(GBL)andKa´lma´nfilteringinweightedmean(KWM)orgainma-
Thegeneralbrokenlinescanaddseveral(n,between trix(KGM)formalism. Thecaseofthescatteringplanescoinciding
one and all) measurements with one fit . The effort is withthemeasurementplanesisindicatesbycircles,onethinscatterer
between measurements by squares and two thin scatterers between
defined by the construction of the linear equation sys-
measurementsbytriangles.
temfromthenon-zeroderivativesfromthepropagation
to the previous and next thin scatterer and its solution
usingO(n )·(m+1)2operationswithingeneralband faster for the simplified one (10–80%). This speedup
par
width m = 5 and n = 2n +1, depending on the is due to the reduced number of non-zero elements in
par scat
numberofthinscatterersn =2..2n. H for the matrix construction and reduced band width
scat
In a toy setup the performance of track refitting has m = 4. For the Ka´lma´n filter the gain matrix formal-
been compared. Tracks in a detector consisting of 10– ismisalittlefasterthantheweightedmeanformalism.
50planeswithtwoindependentmeasurements(d = 2) For fit results at both track ends Ka´lma´n filtering and
and thin scatterers coinciding with the measurement smoothing is compared with a general broken lines fit
planes(simplemodelforasilicontracker)oroneortwo includingtwotransformationsfromfittolocaltrackpa-
thin scatterers between adjacent planes have been sim- rameters. Inthisconfigurationthegeneralbrokenlines
ulated. ThemagneticfieldBisassumedtobeconstant arefasterinallcasestested(30–300%).
between adjacent points and the initial trajectory uses
adetailedhelixpropagator[8]orthesimplifiedversion
5. Summary
for weak deflection (limit |B|/p → 0) as in [6]. Start-
ingfromtheinitialtrajectorycorrectionsandcovariance A trajectory based on General Broken Lines (GBL)
matrixoftrackparametersatoneorbothtrackendshave is a track refit to add the description of multiple scat-
beencalculated.Theresultsofthistestareshowninfig- tering to an initial trajectory based on the propagation
ure1. ForfitresultsatonetrackendKa´lma´nfilteringis in a magnetic field (and average energy loss). The ini-
comparedwithageneralbrokenlinesfitincludingone tial trajectory can be the result of a fit of the measure-
transformation from fit to local track parameters. De- ments (internal seed) or a prediction from another de-
pending on the number of scatterers per measurement tector part (external seed). It is constructed from a se-
thefitwithgeneralbrokenlinesisalmostasfastasthe quence of (pairs of) thin scatterers describing the mul-
Ka´lma´n filter for the detailed propagator (±25%) and tiple scattering in the material between adjacent mea-
4
surementplanes. Predictionsforthemeasurementsare [10] W.D.Hulsbergen, Theglobalcovariancematrixoftracksfitted
obtainedbyinterpolationbetweentheenclosingscatter- withaKa´lma´nfilterandanapplicationindetectoralignment,
Nucl.Instr.andMethodsA,600(2009)471.
ersandtripletsofadjacentscatterersdefinekinkangles
with variance according to the material of the central
scatterer.Therequiredpropagation(ontheinitialtrajec-
tory)fromameasurementplaneorscatterertotheprevi-
ousandnextscattererisusinglocallyalinearizedtrack
model. This linearization may necessitate iterations of
thefittingprocedure. Thematrixoftheresultinglinear
equationssystemisaborderedbandmatrixallowingfor
a fast solution by a root-free Cholesky decomposition
inatimeproportionaltothenumberofmeasurements.
Atallscatterersandmeasurementplanescorrectionsfor
the local track parameters are determined. The corre-
spondingcovariancematricesandthepullsforthemea-
surementsandkinksrequireonlythecalculationofthe
borderedbandpartoftheinversematrix.
In a simulated detector, track refitting with GBL is
about as fast or a little faster than Ka´lma´n filtering,
depending on the multiple scattering setup and on the
complexityofthepropagationJacobian. Incomparison
withKa´lma´nfilteringandsmoothingitisuptoafactor
3faster.
A GBL fit with external seed and one (additional)
measurement is equivalent to the filtering step of the
trackfitwithaKa´lma´nfilter.
As the track refit can provide the complete covari-
ance matrix of all track parameters GBL are well
suitedastrackmodelforalignmentandcalibrationwith
Millepede-II[4].
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