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Nuclear Instruments and Methods in Physics Research A 595 (2008) 228–232

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

E-m

journal homepage: www.elsevier.com/locate/nima

In search of the rings: Approaches to Cherenkov ring finding andreconstruction in high energy physics

Guy Wilkinson

Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK

a r t i c l e i n f o

Available online 19 July 2008

Keywords:

RICH pattern recognition

Likelihood techniques

Hough Transforms

02/$ - see front matter & 2008 Elsevier B.V. A

016/j.nima.2008.07.066

ail address: guy.wilkinson@physics.ox.ac.uk

a b s t r a c t

After considering the challenges of Cherenkov ring finding in High Energy Physics (HEP) experiments we

review two common approaches to this problem: likelihood methods and Hough Transform algorithms.

We conclude by considering some more exotic techniques.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

In designing a successful RICH system, the adoption anddevelopment of a suitable reconstruction and pattern recognitionalgorithm is as important for the detector’s success as considera-tions such as choice of radiator, mirror optics, photodetector, etc.Here we survey the problems encountered when ring finding inparticle physics RICH detectors, and discuss the two mostcommon categories of algorithm: likelihood and Hough Transformtechniques. We finish by very briefly considering other possibleapproaches.

When listing examples we consciously avoid quantifying theoverall particle identification performance of each experiment, asit is not possible in the space available to separate thecontributions to this figure of merit coming from the intrinsicdetector quality and from the reconstruction algorithm.

1.1. Challenges of Cherenkov ring finding

Many factors complicate the problem of ring-finding in particlephysics experiments. The most important are listed below.

�

High multiplicity environment: Any RICH system exposed to ahigh particle flux records many overlapping rings. A primaryproblem in finding signal rings is in understanding thebackground arising from the rings produced by other particles. � Rings without associated tracks: Experiments which are reliant

on high efficiency tracking will generally employ patternrecognition algorithms for the RICH which make use oftracking information. Inevitably some fraction of ringswill have no associated track. Generally the particles giving

ll rights reserved.

rise to these trackless rings are of no physics interest inthemselves, but they lead to a particularly dangerous back-ground category.

� Split, partial and distorted rings: Mirror layout and detector

configuration may lead to rings which are far from beingcircular (an extreme case is the example of the BaBar DIRC),and sometimes divided between separated detector planes.Limited geometrical efficiency may also give rise to significantgaps in rings.

� Sparsely populated rings: Below a certain density of hits, rings

become challenging to identify, for example the LHCb aerogel(hits �5 per ring).

� Non-ring backgrounds: Processes other than Cherenkov rings

can produce hits in the photodetectors. These include machinebackground, light from particles radiating in the photodetectorwindow, and electronic noise.

� Finite resolution: Any degradation in the ring sharpness leads to

a corresponding falloff in signal to background discrimination.

Many of these challenges are apparent from the display of asimulated LHCb RICH 1 event, shown in Fig. 1.

1.2. Reconstructing the Cherenkov angle

A very large class of RICH pattern recognition algorithms areperformed in Cherenkov angle (yC) space. Unless the rings are self-evident circles, the yC-space approach is generally preferred toring finding in the coordinates of the photodetector plane, as inyC-space the rings manifest themselves as spikes, albeit smearedby the reconstruction resolution, rather than distorted, partial oreven disjoint circles. Transforming the hit points from detectorspace to yC-space is therefore a vital first step in many RICHreconstruction procedures. For each hit point, however, thistransformation needs to know the trajectory of the parent particle.

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-60

-40

-20

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-60x local (cm)

y lo

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Fig. 1. A simulated LHCb RICH 1 event display, with aerogel (large) and C4F10

(small) rings. The solid red circles are the hits from primary particles, the open

blue squares are from secondaries and background processes. The solid rings are

the result of the pattern recognition with physics quality tracks, the broken rings

those from lower quality tracks. Some rings have no associated tracks.

-250 -125 0 125 2502 ln ( Lπ/LK)

0

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Num

ber o

f tra

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10Fig. 2. CLEO-III/-c RICH: distribution of 2ðlnLp=LKÞ for a sample of true kaons

(solid) and pions (open) accumulated through kinematically selected D� decays.

G. Wilkinson / Nuclear Instruments and Methods in Physics Research A 595 (2008) 228–232 229

Therefore, an association between hit and particle must beassumed, and tracking information be accessed. The hit–particleassociation is of course not a priori known, and so it is a commonpractice to perform the transformation for all possible associa-tions, and give the subsequent pattern-recognition step thetask of deciding the best solution. The problem of yC reconstruc-tion has been discussed many times in the literature [1]. Ananalytic solution suitable for a mirror focusing RICH may be foundin Ref. [2].

2. Track-based likelihood algorithms

Track-based likelihood algorithms represent the most commonapproach to the problem of RICH pattern recognition. Examples ofdetectors which have followed, or intended to adopt this method,include SLD [3], DELPHI [4], HERA-B [5], HERMES [6], CLEO-III/c[7], BaBar [8] and LHCb [2].

The distribution of a set of Cherenkov hits around a projectedring centre may be compared, through a likelihood calculation,with that expected from a given hypothesis, taking into account ofthe number of hits and their position in yC-space (the expectedflat distribution of hits in the azimuthal coordinate may also beexploited). Important inputs to this calculation are the efficiencyof detection (and its geometrical dependency), the resolution ofthe Cherenkov angle, and both the level and distribution ofbackground. The likelihood can naturally account for featuressuch as ambiguities in the path the photon took in reaching thephotodetector, and can include each possible association of trackand hits. In the likelihood maximisation, each of the possibleparticle assignments are considered for the track. In addition toprocessing spatial information, other ingredients can be included.In the BaBar DIRC, for instance, the expected arrival time of eachhit provides important input.

Attractive attributes of the likelihood approach are that itperforms the steps of pattern recognition and particle identifica-tion in a single procedure. It also returns for each track astatistically meaningful quantity, that is the difference in log-likelihoods between difference hypotheses, which can be cut onaccording to the needs of the physics analysis. Fig. 2 shows anexample from CLEO.

Two distinct categories of likelihood approaches can bedistinguished: local likelihoods and global likelihoods. In the formerthe computation and particle identification is performed indivi-dually for each track. In the latter, a single likelihood is built up forthe entire event, which is then maximised by flipping the particlehypothesis of each track in turn. This strategy is convenient forthose detectors which have multiple radiators (e.g. SLD, DELPHI,HERMES and LHCb) by allowing all information to be treatedsimultaneously. The main motive, however, arises from theobservation that in experiments with a high charged trackmultiplicity the principal source of background for signal ringsare those rings from other particles in the event. The globallikelihood takes account of this cross-feed in the mostcorrect manner. For this reason, many experiments preferadopting the global likelihood approach for their default offlinereconstruction. Exceptions include low-occupancy experiments,such as CLEO-III/-c, and experiments where other factors makethe local approach more performant (e.g. HERA-B—see below).Finally, it can be noted that the global likelihood method willalways be slower than the local algorithm, which is intrinsicallyfaster and can be deployed only on those tracks of interest in thephysics analysis. For this reason the local method remains a verypromising candidate for use in high level trigger applications, forinstance in LHCb [10].

Despite the general success of the track-based likelihoodmethod (global or local), two weaknesses may be identified.

The first problem concerns the issue of background categor-isation. In the ideal case the global likelihood proceeds on theassumption that for every ring there is an associated track, theparticle hypothesis of which may be varied in order to achieve acomplete understanding of all observed hits. (This statementneglects those hits of non-ring origin, which in many examples

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G. Wilkinson / Nuclear Instruments and Methods in Physics Research A 595 (2008) 228–232230

are a small contribution.) Of course certain rings, the majorityarising from secondaries, have no associated track, and this leadsto difficulties for the algorithm. In the limit that there are manysuch trackless rings, the global approach may no longer bepractical, and instead it will be better to apply a local algorithmwith some parameterised expectation for the background. This iswhat is done, for example, in HERA-B [5]. Even at lower levels thedistribution of this background cannot be predicted event-by-event, and it will lead to tails in the likelihood distributions, whichwill in turn give rise to wrong particle associations in regions ofthe momentum spectrum where the performance would other-wise be very good. One way to limit the extent of these tails is torestrict the weight a single hit point may have in the likelihoodcalculation, so that outliers do not distort the result unduly. Thebenefit of such an approach as implemented at LHCb can be seenin Fig. 3.

A second difficulty enters through the problem of correlatederrors. Certain error contributions, most notably that associatedwith the track direction, are correlated between all the photo-electron hits on the ring. If we designate the contribution from thetracking to the Cherenkov angle error per hit by strk, the sum ofother contributions to be spe, and the total number of photo-electrons on the ring to be Npe, then the overall error on

ΔLL-100

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102

-80 -60 -40 -20 0 20 40 60 80 100

60 80 100Kaon ID Efficiency / %

Pio

n M

isID

Effi

cien

cy /

%

10

1

RICH Kaon ID

Normal00

2525

5050

Modified with10-3 limit

Fig. 3. LHCb PID performance before and after truncating the weight of outliers in

the likelihood fit. Top: the difference in log likelihood (DLL between the pion and

kaon hypotheses before (solid black squares) and after (solid red circles in interval

�20oDLLo80) the truncation. Note how after truncation the tail at low DLL is

suppressed. Bottom: the PID performance as the cut value on DLL (indicated on

the curve) is changed. Note the significant improvement for the truncated

(‘10�3 limit’) case.

the Cherenkov angle integrated over the ring is given bysring ¼

pðs2

pe=Npe þ s2trkÞ, in the case where the tracking error is

fully correlated between all hits. A track based reconstruction istherefore limited in resolution, and this may have importantconsequences, most significantly in the high momentumregime.

The optimum resolution can in principle be recovered througha second-stage ring refit. The likelihood step is used to define a setof hits potentially arising from the track-in-question, and then thering is refit with the constraint of the ring centre from the trackremoved, thereby suppressing the strk term in the ring resolution.Such a procedure is straightforward to implement when the ringsare good approximations to circles, but not when they areelliptical. However, a transformation exists, the so-called stereo-

graphical projection, which enables a circular representation to beobtained [11]. The dimensions of the resulting circle only dependon the track direction at second order. Refitting the hits on thiscircle, and rejecting outliers, allows both for an improvement inresolution, and a suppression of tails from poorly understoodbackground [12].

3. Hough transforms: spatial and angular

The method of Hough Transforms is an established techniquefor pattern recognition, with several applications in High EnergyPhysics (HEP). The approach is very suitable for RICH reconstruc-tion, being unaffected by topological gaps in curves, split imagesand in general rather robust against noise.

The principle of Hough Transforms as applied in circular ringsearching can be understood as follows. Each point in detectorspace can be associated with an infinite number of circles, ordifferent radii and centres. Thus each point may be turned into ahypersurface in three-dimensional Hough Transform space. Alldetected points may be transformed in this manner and theproblem of pattern recognition then becomes that of searching foran intersection of hypersurfaces in Hough Transform space. Inpractical applications, however, it is usual to use externalconstraints to fix one or more parameters and thereby reducethe search to a one- or two-dimensional problem.

The most common strategy for using Hough Transforms inRICH detectors is to use tracking information to determine theexpected ring centre. In practice, this is equivalent to looking for apeak in yC-space. A good example of a detector which uses thisapproach is the ALICE HMPID. Here, the challenge lies in the highbackground levels expected in the LHC heavy ion environment,particularly in Pb–Pb collisions. Although, in contrast to track-based global likelihood techniques, no attempt is made to predictthe exact background contribution event-by-event, the knownparticle density allows the general distribution to be modelledwith confidence. This calculation permits a weight to be applied toeach entry in the yC-histogram, in effect subtracting the back-ground, as is indicated in Fig. 4. A sliding window, of width 3–4times the single photoelectron resolution, is moved over thedistribution and a weighted average taken of the contents. Moredetails on the baseline ALICE strategy for pattern recognition, andpossible extensions, can be found in Ref. [13].

Hough Transforms may also be used in a manner which doesnot require tracking information. A frequent goal in heavy ionexperiments is to look for low mass electron–positron pairs.Therefore, it is of interest to search for saturated rings: the HoughTransform space in this case is that of the coordinates of the ringcentre. An excess above threshold in a cluster of bins in this spacewill signify that a ring has been found and reveal its position. Thismethod has been used with success by CERES [14] at the CERNSPS. An essentially identical procedure is used in large volume

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Fig. 4. ALICE HMPID: distribution of Cherenkov angle Zc at different particle densities for a given particle type integrated over many tracks and events. The background

distribution is well modelled by the broken line. The solid histogram shows the signal after the weighting procedure.

G. Wilkinson / Nuclear Instruments and Methods in Physics Research A 595 (2008) 228–232 231

water detectors, such as Super-Kamiokande [15], to find andidentify the products of neutrino interactions.

4. Other approaches

Other methods have been explored for RICH ring finding.Many of these approaches arise from the desire to have apattern recognition which is largely independent of trackinginformation. They include fuzzy clustering and deformabletemplate algorithms. None of these have shown sufficiently goodperformance to be considered a serious alternative for real RICHdata analysis.

A novel approach which has attracted attention more recentlyis the use of Markov Chain Monte Carlos (MCMC) as a tool forRICH pattern recognition [16]. Here one uses the MCMC to samplefrom a proposal distribution of Cherenkov rings which gave

rise to the observed distribution of photodetector hits. The MCMCallows for an efficient sampling of parameter space. The methodworks well and is being further evaluated in realistic detectorsimulations [10].

5. Summary

Attention to the strategy and details of the reconstructionand pattern recognition are vital for the success of a RICH system.The majority of experiments have achieved excellent performanceby using methods based on likelihood fits or Hough Transforms.Almost always tracking information plays a central role in thering search. It is no doubt due to the success of these approachesthat alternative methods have not yet been adopted as a baselinestrategy in any prominent experiment, although such methodsexist.

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Acknowledgements

I wish to thank the organisers for a stimulating and enjoyableconference. I am grateful to Domenico Di Bari, Chris Jones, RalucaMuresan and Mitesh Patel for their assistance in assembling thetalk and these proceedings.

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