Left-Right Ambiguity Resolution

It is mentioned already, problems with the left-right ambiguity resolution occur naturally in the reconstruction of tracks in the OTR. In addition, the algorithm for the left-right ambiguity resolution allows to eliminate noise and wrongly associated hits.

In this step it is assumed that the track recognition is completed. Each track candidate is simply a sequence of fired wires $\{w_i\}$, $i=1,\dots,N$ for which two problems have to be solved:

1. Detection and removal of outliers, i.e. hits that most likely do not belong to the track.
2. Resolution of left-right (L/R) ambiguity for the other hits.

To solve both problems CATS uses a very promising approach that was proposed recently [#!CPC!#,#!NIM!#]. This approach is based on so-called elastic neural nets (ENN) [#!Durbin!#,#!Peterson!#] and generally employs the following heuristic idea:

In the presence of multiple scattering a track can be defined as the longest and the smoothest line which approaches the drift circles as close as possible.

Following this idea the problem of L/R ambiguity resolution and removal of outlying hits can be considered as an optimization problem in a form of either a variational problem or a problem of optimal trajectory control. Naturally, the optimization criterion is to be a sum of

1. penalty on non-smoothness of a track;
2. penalty on a sum of the minimal distances between the track and drift circles around the wires.

The latter term can also account for outlying hits: for large distances between the track and a given drift circle the penalty for the hit should be decreased in order to suppress the influence of outliers on the optimal solution.

Mathematically the ENN algorithm is an iterative numerical method to solve the given optimization problem. This method to solve L/R ambiguities has already successfully been applied to similar problems in [#!CPC!#,#!NIM!#]. The ENN in its elastic arm modification has also been studied by members of the HERA-B collaboration [#!Paus!#] where a full description and a detailed analysis of the method are presented.

Starting from the general ENN-based method to solve L/R ambiguities we describe a simplified version of the algorithm as implemented in CATS and applicable for relatively fast tracks.

The ENN algorithm employs a segment model for track description: a track is considered as a sequence of neighboring straight-line segments connecting nodes of the ENN -- points in 3D space which can change their positions during iterations of the method. The ENN which is implemented in CATS consists of two interacting arms (lower and upper ENNs in Fig. 4.7), each arm includes $M$ nodes. The initial positions of the nodes on both ENN arms are chosen to fully encompass a given sequence of drift circles around the wires fired $\{w_i\}$, $i=1,\dots,N$.

Figure 4.7: Left: segment track model in a two-arm elastic neural net, right: ENN node dynamics -- for details see text.

During an iteration an ENN node $i$ moves under the influence of three forces:

• the retracting force $F_1$ that tries to improve the smoothness of the track by minimizing the breaking angle $\phi_i$ between the two neighboring segments sharing the node;
• the attracting force $F_2$ that pulls the node to the nearest drift circle $w_i$;
• the attracting force $F_3$ that brings together opposite nodes $i$, $i+M$ on the lower and upper arm.

All three forces, $F_1$, $F_2$, and $F_3$, depend on the mutual position of a node, its counterpart in the other arm and the nearest drift circle. It is assumed that the changes in the node's positions, $u$, due to the different forces are described by the following equations:

$$\Delta u_1=\beta\phi_i,\qquad \Delta u_2=\alpha\rho_i,\qquad \Delta u_3=\gamma \left(u_{i+N}-u_i\right),$$

where $\rho_i$ is the residual between the node and the nearest drift circle, $\alpha$, $\beta$ and $\gamma$ are coefficients linearly changing during iterations:

$$\alpha=\alpha_1j, \qquad \beta=\beta_1\left(1-\beta_2j\right), \qquad \gamma=\gamma_1j.$$

Here $j$ is the number of the iteration, $\alpha_1$, $\beta_1$, $\beta_2$, and $\gamma_1$ are adjustable parameters of the algorithm.

In each iteration the positions of all nodes are updated according to the equation:

$$u_i^{(j+1)}=u_i^{(j)}+(\Delta u_1)_i^{(j)}+(\Delta u_2)_i^{(j)}+(\Delta u_3)_i^{(j)}, \quad i=1,\dots 2M.$$

This approach is very general, it can, in principle, be applied to a wide spectrum of track reconstruction problems from reconstruction of straight, high-momentum tracks to fits of low-momentum hard scattered, even broken, tracks (the latter is the case in [#!CPC!#,#!NIM!#]). However, the algorithm can be simplified and made significantly faster, if the specific circumstances of the experiment are taken into account. CATS is dealing with relatively fast straight-line tracks. Most of the tracks which are of physics interest have momenta well above 1 GeV. Thus it is feasible to substitute a deformable segment-wise double ENN by a single straight-line rigid template.

The template's dynamics is described as a motion under the influence of an attraction force pulling towards the drift circles. In order to suppress the influence of outlying hits the force depends on the distance between the template and a drift circle decreasing for large distances. Let $R_k=(x_k,y_k,t_{xk},t_{yk})$ be a vector of the template's parameters at $k$-th iteration. The algorithm assigns two weight coefficients $C_i^-$ and $C_i^+$ to each drift circle:

$$C_i^-(R_k)=f(\rho_i^-(R_k)), \qquad C_i^+(R_k)=f(\rho_i^+(R_k)),\qquad i=1,\dots,N,$$

where $f(\cdot)$ is the truncated normal density function, $\rho^+(\cdot)$, $\rho^-(\cdot)$ are the residuals between the template defined by the vector $R_k$ and the drift circles corresponding to L/R equal $+1$ and $-1$. Using weights the vector $R$ is updated as follows:

$$R_{k+1}=R_k+\beta_k\left(R_m-R_k\right),$$

where $\beta$ is a coefficient approaching unity during the iteration process. The vector $R_m$ is a solution of the following auxiliary optimization problem:

$$R_m=\arg\min_{R}\sum_{i=1}^N\left(C_i^-(R_k)\left(\rho_i^-(R)\right)^2+ C_i^+(R_k)\left(\rho_i^+(R)\right)^2 \right).$$

Studies on simulated data have shown that this algorithm provides remarkably high efficiency of the L/R ambiguity solution and very low level of noise hits. This is very important since an insufficient quality of L/R resolution and noise contamination in tracks can deteriorate the accuracy of the estimates produced by the Kalman refit of the reconstructed tracks.