1. Method of identification of rings in spectra from RICH detectors using the deconvolution based pattern recognition algorithm V. Matoušek, M. Morháč Institute of Physics, Slovak Academy of Sciences, Slovakia RICH 2010, CassisMay 3 – 7, 2010

2. The transitions of the particles with the momentum above Cherenkov photon emission threshold are in the RICH photon detector registered as rings. These rings are far from being ideal. Recognition of rings is therefore a serious problem, especially when fast decision is required, as it is usually the case when information from RICH detector is used in triggers. There exist several algorithms for the recognition of Cherenkov rings in experimental data. They are based e.g. on triangulation method, on a Metropolis-Hastings Markov chain Monte Carlo sampling, or Hough transform method. Other pattern recognition techniques are used as well. Introduction

3. The operation of deconvolution can substantially improve the resolution in the resulting data, e.g. identification of subridges in the spectra of nuclear multifragmentation. The Gold deconvolution can be used as an efficient, high resolution, robust algorithm to identify circles in the data from RICH detectors. The relation between the input value of a linear invariant discrete system and its output can be described by convolution sum where x(i) is the input into the system, h(i) is its impulse function (response), y(i) is the output from the system, n(i) is additive noise and “ * “ denotes the operation of the convolution. An alternative approach – pattern recognition method based on Gold deconvolution method

4. In matrix form this system is where the matrix H has dimension N x M, the vectors y, n have length N and the vector x has length M, while N ≥ M (overdetermined system). For invariant convolution system the columns of the matrix H are represented by the response mutually shifted by one position where N = M + L – 1.

5. For general one-dimensional system of linear equations formula has the form One of the most efficient methods of the deconvolution is the Gold iterative algorithm. Its solution is always positive when the input data are positive, which makes the algorithm suitable for the use for naturally positive definite data, i.e., spectroscopic data. For one-dimensional data it has the form It means that the response has the same form for all columns.

6. where A = H T H, y ’ = H T y, n is the iteration number and x (0) = [1, 1, …, 1] T. For the elements of the matrix A and the vector y ’ one can write The algorithm converges to the least square estimate in the constrained subspace of positive solutions. From the computational point of view the Gold deconvolution is rather time-consuming operation. This problem is becoming relevant for large two-dimensional spectra. Therefore the implementation of the method requires further optimization.

7. Fig. 1 An example of two-dimensional spectrum of response function (circle positioned at the beginning of coordinate system) of the size 256x256 channels. The response matrix was created from these data.

8. Fig. 2 Synthetic spectrum composed of three shifted circles which are positioned quite close to each other (a) and detail of the spectrum after Gold deconvolution (2 iterations) (b). (a) (b) The aim of the use of Gold deconvolution is to concentrate as much as possible the contents of the data distributed along the perimeters of circles to their corresponding centers.

9. Fig. 3 The spectrum from Fig. 2(a) after 10 iterations of Gold deconvolution (a) and detail of the spectrum after 20 iterations of Gold deconvolution (in logarithmic scale) (b). Further increasing the number of iterations practically does not change the result. (a) (b)

10. Boosted Gold deconvolution method Iterative positive definite Gold deconvolution converges to a stable state. It is useless to increase the number of iterations; the result obtained practically does not change as it does not lead to sufficient improvement in the resolution and suppression of undesired residues in the deconvolved data. We can stop the iterations, apply a boosting operation and repeat this procedure. To change the relationship among elements of the particular solution a non-linear boosting function must be applied to it. The power function proved to work satisfactorily (boosting coefficient p ).

11. Fig. 4 Spectrum from Fig. 2(a) after boosted Gold deconvolution ( p = 2, I = 4, R = 5).

12. Fig. 5 Synthetic spectrum composed of 1 000 randomly distributed hits arranged in 50 randomly positioned circles.

13. (b) (a) Fig. 6 Spectrum from Fig. 5 after Gold deconvolution (number of iterations = 20) (a), and after convolution of the result with the response (b).

14. (a) (b) Fig. 7 Spectrum from Fig. 5 after Gold deconvolution (number of iterations = 20) (a), and after boosted Gold deconvolution (p = 5, I = 4, R = 5) (b).

15. IR p = 2 p = 3 p = 5 p = 10 p = 20 22050/0 29/08/08/0 41050/550/0 39/011/0 5850/1250/150/048/022/0 8550/7150/750/0 42/0 10450/16550/1650/0 43/0 20250/31550/18050/450/0 Table I Dependence of identification efficiency of the proposed method on number of iterations I, number of repetitions R and boosting coefficient p for the data from Fig. 5 (50 circles). It shows the number of correctly identified circles (green color) and the number of incorrectly identified false circles (red color).

16. Fig. 8 Resulting data after application of circles intersection method (CIM) to the spectrum from Fig. 5.

17. RadiusCIM algorithm Deconvolution based algorithm 3<1 s3 s 5<1 s7 s 101 s20 s 151.5 s35 s 202 s56 s 253 s1 min 23 s 305 s1 min 56 s 408 s2 min 22 s 5012 s5 min 15 s Table II. The dependence of the speed of the CIM method and the proposed deconvolution based algorithms for the data from Fig. 5 (size 256x256 channels) and for 3GHz PC on the radius.

18. (a)(b) Fig. 9 Synthetic spectrum composed of 100 000 randomly distributed hits arranged in 10 randomly positioned circles with additive noise with level of 100% of maximum value, three-dimensional view (a), and orthogonal view (b).

19. (a) (b) Fig. 10 Spectrum from Fig. 9 after boosted Gold deconvolution (p = 20, I = 5, R = 4) (a) and convolved data (b).

20. (a)(b) Fig. 11 Synthetic spectrum composed of 20 000 randomly distributed hits arranged in 9 regularly closely positioned circles (coordinates in both dimensions are 100, 110, 120) (a), and orthogonal view (b).

21. Fig. 12 Spectrum from Fig. 11 after boosted Gold deconvolution (p = 5, I = 2, R = 10).

22. Fig. 13 Synthetic spectrum composed of 200 randomly distributed hits arranged in 6 randomly positioned circles with added 500 randomly distributed noise hits and 3 peaks (σ x = 2, σ y = 2) (a). Deconvolved spectrum after complete boosted deconvolution (p = 20, I = 5, R = 4) (b).. (a)(b)

23. (a)(b) Fig. 14 An example of two-dimensional ring response function with constant profile (width = 10 channels) (a), and Gaussian profile (b).

24. Fig. 15 Synthetic spectrum composed of randomly distributed hits arranged in 6 randomly positioned circles with radius changing from 21 to 26 and with identified centers of the circles (a). Spectrum constructed by the same way with 6 randomly positioned ellipses and with their identified centers (b). (a)(b)

25. Positions of circles in original data – x, y Positions of identified circles using ring response with constant profile – x, y Positions of identified circles using ring response with Gaussian profile – x, y 50, 5548, 5148, 53 217, 110218, 106218, 108 123, 192 126, 192 66, 16663, 16564, 165 156, 123154, 119157, 121 217, 171217, 172219, 170 Table III Positions of circles in original data (Fig. 15) and identified centers of circles in deconvolved data using ring response function with constant profile (Fig. 14(a)) and Gaussian profile (Fig. 14(b)).

26. (a) (b) Fig. 16 An example of simulated spectrum from GEANT4 (a), and rings after convolution with the response (b).

27. (a)(b) Fig. 17 An example of experimental RICH spectrum (HADES RICH detector) (a), three identified rings after convolution with the response function with marked centers (b).

28. So far we assumed only circle shape of the objects to be identified. The newly proposed method is much more universal. It allows identification of the objects of any shape. Let us take, e.g. Lissajous curves: x = a sin (ωt) y = a sin (ωt – π/3) x = a sin (ωt) y = a sin (2ωt – π/4)

29. Fig. 18 A synthetic spectrum of 500 randomly distributed hits arranged in 10 randomly generated Lissajous curves according to the expression in the previous slide and 500 randomly distributed noise hits (a). Convolved data of identified centers after complete boosted deconvolution (p = 20, I = 5, R = 4) (b). (a) (b)

30. Fig. 19 A synthetic spectrum of 500 randomly distributed hits arranged in 10 randomly generated Lissajous curves according to expression in previous slides and 500 randomly distributed noise hits (a). Convolved data of identified centers after complete boosted deconvolution (p = 20, I = 5, R = 4) (b). (a) (b)

31. (a) (b) Fig. 20 A synthetic spectrum of 500 randomly distributed hits arranged in 10 randomly generated curves of irregular shape and 500 randomly distributed noise hits (a). Convolved data of identified centers after complete boosted deconvolution (p = 20, I = 5, R = 4) (b).

32. A deconvolution based pattern recognition algorithm, which allows to automatically recognize rings in two-dimensional spectra from RICH detectors was presented. We have used boosted Gold deconvolution algorithm, which makes it possible to concentrate the contents of one ring into one point located in its centre. We have presented various properties of the proposed algorithm, e.g. the influence of the number of iterations (I ), number of repetitions (R) and boosting coefficient (p) on the efficiency of the circle identification, etc. The above described algorithm is rather simple and allows its realization on PC computers. Though the procedure is fully automatic, due to large variability of the data, some intervention of the user and tuning of some parameters is required. The method was implemented in Data Acquisition, Processing and Visualization system (DAqProVis) developed at the Institute of Physics, Slovak Academy of Sciences. Conclusions

33. Some relevant publications 1.M. Morháč, J. Kliman, V. Matoušek, et al., Efficient one- and two- dimensional Gold deconvolution and its application to γ-ray spectra decomposition, Nucl. Instr. and Meth. A 401 (1997) 385. 2.M. Morháč, Deconvolution methods and their applications in the analysis of γ-ray spectra, Nucl. Instr. and Meth. A 559 (2006) 119. 3.P. Bandžuch, M. Morháč, J. Krištiak, Study of the Van Cittert and Gold iterative methods of deconvolution and their application in the deconvolution of experimental spectra of positron annihilation, Nucl. Instr. and Meth. A 384 (1997) pp. 506-515. 4.M. Morháč, V. Matoušek, J. Kliman, Efficient algorithm of multidimensional deconvolution and its application to nuclear data processing, Digital Signal Processing 13 (2003) 144. 5.M. Morháč, J. Kliman, V. Matoušek, et al., Integrated multiparameter nuclear data analysis package, Nucl. Instr. & Meth. A 389 (1997) 89. 6.M. Morháč, V. Matoušek, I. Turzo, et al., DaqProVis, a toolkit for acquisition, interactive analysis, processing and visualization of multidimensional data, Nucl. Instr. & Meth. A 559 (2006) 76.