New solution to circle fitting problem in analysis of RICH detector data

Abstract

The problem of approximating by a circle of measured data points, registered by RICH detector is considered. The situation is formulated in terms of the Bayesian estimation problem, the solution is based on the MetropolisHastings algorithm. IntroductionTile problem of recognition of circles in observed noisy planar data on RICH (Ring Imaging CHerenkov)detector is considered. RICH having the discrete cell structure register the energy dissipation produced by a passing elementary particle not in a single point, but in several adjacent cells where all this energy is distributed. The method which is used up to now consists of two steps: in the first one the centres of photon hits are estimated with the aid of a clustering procedure, the second step fits rings to these centres (Agakichiev et al., 1996). Both steps were combined to one robust procedure in Chernov et al. (1995). The simplest (and not too reliable) method uses weighted averaging, with the advantage that it can be employed also for on-line computations. As regards an off-line analysis, a number of different techniques (including methods of pattern recognition) can be considered.We have explored an application of the MCMC method. The problem is is formulated as a problem of Bayesian estimation of multivariate parameter in the framework of a probabilistic model of the physical phenomenon. The distribution of observed data is rather complicated, for instance it includes a distribution of points of hits and this distribution should be integrated out.The paper is organized as follows: First, properties of Metropolis-Hastings MCMC algorithm are recalled. Then the performance of Cherenkov photons detector RICH is described. Finally, the method for off-line analysis of Cherenkov rings is presented.

Trackless ring identification and pattern recognition in Ring Imaging Cherenkov (RICH) detectors

“…Trackless ring-finding algorithms have been proposed in the past, however. Reference [3] only came to the attention of the authors two years after the algorithm defined herein was implemented. The algorithm of [3], though independently concieved, has much in common with the one described here, and has much to commend it.…”

“…Reference [3] only came to the attention of the authors two years after the algorithm defined herein was implemented. The algorithm of [3], though independently concieved, has much in common with the one described here, and has much to commend it. Both methods use a Bayesian approach, implement a similar detector model 4 and explore the space of possible ring-configurations with a Markov chain Monte Carlo.…”

“…Note that we do not make any attempt to "maximise" p(R|H). We are not interested in where this function is a maximum, 3 though in some sense the sampling is likely to be localised near the maximum.…”

“…The performance of the algorithm is demonstrated in the context of the LHbC RICH simulation described in Section 10. (Not the LHCb experiment [1,2]) There are many examples of applications for ring finding pattern recognition both within high energy particle physics [3,4,5,6] and without [7,8].…”

“…The method chosen is a Metropolis-Hastings Monte Carlo (MHMC) sampling of a particular posterior distribution defined in the first half of the article. MHMC samplings in ring finders are not new [3] but their performance is strongly dependent on the choice of the so-called proposal distribution(s) they make use of (defined later). Indeed, the only freedom that one has in implementing the MHMC sampler, is in the definition of the proposal distribution(s) to be used -and so almost all of the second half of the article is devoted to describing the one used here.…”