Clustering Gammas by using Fuzzy Logic

Transcription

1 Clustering Gammas by using Fuzzy Logic A multiple Gamma event in a highly segmented Germanium detector array The Energy deposition and reconstruction The Clustering problem The Gamma kinematics Why the kinematics is not an exact tool? Why fuzzy Logic can help? What is Fuzzy Logic? Fuzzy reconstruction Preliminary Results

2 Esempio di evento con solo depositi

3 Esempio ricostruito

4 The Clustering problem Every event is a collection of N Energy depositions produced by K unknown gammas. For each dep. We know (E,x,y,z) and we can estimate (de,dx,dy,dz). Problem: Find the K groups. A Puzzle. Procedure: Group the dep. Clustering. Validate every Cluster. Trim the method parameters by maximizing the (efficiency*p/t).

5 The Gamma Kinematics 3 Points min Origin+2 Deps. P 0 = {0,0,0,0} q 12 P 1 P 2 P i = {E i,x i,y i,z i } P 3 E i [ kev ] = È E i+1 Í Î 2044 E i+1 (1- cosq) 2

6 Kinematics fails Coherent Scattering Missed deps. Unknown order Each dep. can be the sum of 2 or more single deps. 2 q 2,3 0 1 q 1,3 3 2 Deps. Reconstruction Prob.(theta) Er-Eg E1+E2=300, theta = 0.32(rad)

7 Fuzzy Logic Is an extension of the conventional Logic. Handles successfully uncertainty. Uses Linguistic Expressions (LE) as logic decision statements. Is useful if : The problem has no known mathematical description, but exists expertise on it. The Problem can be well described, but is too complex. We need to act in real time. We have to deal with noise or uncertainties disturbing the data. Introduced by Lofti Zadeh in Zadeh.L.A. Fuzzy Sets, Information and Control, Vol8(1965), 338

8 Fuzzy Kinematics q 12 ±dq q = f (d 01,d 1,2,Dx i,dy i,dz i ) E i = E q -dq q +dq i + E i 2 de i = E q -dq q +dq i - E i 2 Valid if : E - ( E + E ) de i i+1 i+2 i

9 Fuzzy Logic Procedure 1. Input data (ID) 2. Fuzzyfication Translation of ID into linguistic variables Creation of groups of values (linguistic terms) Association of terms and membership values µ (fuzzy sets, degree of truth of one statement) 3. Fuzzy Inference Implementation of Linguistic rules Aggregation (IF part of the rules) AND(A,B) -> min(µ A, µ B ) OR (A,B) -> max(µ A, µ B ) NOT(A) -> 1- µ A Weight of the rule Composition (THEN part of the rule) MAX/MIN (OR of the rules) or MAX/PROD inference(weight) Fuzzy decisions. 4. Iterate Defuzzyfication Translation of output fuzzy variables into output data (OD). (Crisp decisions and data)

10 µ 1 Fuzzy linguistic variable Example Member func. terms neg_far neg_close zero pos_close pos_far m 20 Distance [m] Distance of 19 meters is: pos_far to the degree 0.8 pos_close to the degree 0.2

11 Fuzzy Logic Application 1 Rapid Charger for NiCd Batteries The batteries can be from different manufacturers, hence showing slightly different characteristics The temperature sensor is mounted in different positions with different batteries The charge level at the beginning of charging is unknown. Many users do not completely discharge the battery prior to charging. The age and history of the battery is unknown to the charger. The number of cells in the battery is unknown The environment temperature is unknown (only the battery temp. is known

12 Fuzzy Logic Application 2 One solution Temperature Low Normal High C 60 C Charging Current Zero Small Medium Large Amps 8 deltatemp: Positiv, Negativ deltavolt: Positiv, Negativ

13 Fuzzy Logic Application 3 Rules: 1. If Voltage is High Then Current is Zero 2. If Temp is Low Then Current is Small 3. If dtemp is Positiv And Temp is Normal Then Current is Small 4. If dvolt is Positiv And Temp is Normal Then Current is Large 5. If dtemp is Negativ And Temp is High Then Current is Small

14 Fuzzy Event Reconstruction 1. Identification of the first hit of each gamma The data ambiguity has preserved the first hit? Yes, if the scattering has a medium-big angle OR a medium-big energy deposition fraction 2. Fuzzy ISODATA clustering Well established patter recognition method Based on simple physical distance Spherical events 3. Small Angle Events The first hit is missed when the scattering angle is small AND the energy deposition is small. A dedicated procedure recover those events. 4. Validation The selection criteria is the energy Balance.

15 Fuzzy Event reconstruction Step 1 For each dep. pair i,j having medium-big scattering angle q or medium-big energy deposition fraction (E i /E 0 ) (Big Angle): The i_first_- j_second F i, j linguistic function has threes term and the membership function is calculated using: PC 0,i Compton probability, Klein-Nishina (E i, cosq) If(E 0 E i +E j ) within de 0 term Equal : Fe PF i,j Photo peak Prob If(E 0 > E i +E j ) within de 0 term Bigger: Fb PC i,j Compton probability, If(E 0 < E i +E j ) within de 0 term Smaller: Fs PF i,j Photo peak Prob.

16 Fuzzy Event reconstruction Fuzzy Rules Rule 1: One dep_j is second to a dep_i, if exist one i such that: Fe i,j (Equal) OR Fb i,j (Bigger) is true Rule 2 IF dep_ j is NOT second to any dep_i THEN j is a first Dep. Fuzzy inference: IF [1-max(max (Fe i,j, Fb i,j ) ] is true for any i THEN dep_j is first For 30 gamma events of 1.3 MeV is TRUE the 72% of the interactions and the 95% of the Peak-accepted ones

17 Fuzzy Event reconstruction Step 2 Create K Clusters. One for each first hit. Apply the fuzzy ISODATA iterative method minimizing the following merit function: J 2 =  j ŒN m 2 2 k, j d k, j d 2 k,j = distance(center of cluster k, point j) After each iteration m the center of cluster is recalculated by { x m +1, y m +1,z m +1 } = m 2 k, j {x m, y m,z m } j  j ŒN Stop when dj 2 < limit. ( < 10 iterations) For each deposit with µ k,j > 0.5, assign it to cluster k.

18 Fuzzy Event reconstruction Step 4 Validate Cluster if: Ê f 1 n, E ˆ Á 0 Ë de E - E 0 s Ê f 0 2 n, E ˆ Á 0 de de 0 Ë 0 Where n is the number of deposits belonging to the Cluster, E s is the Cluster Energy, f 1 and f 2 are empirical values.

19 Terminology Photo peak efficiency: e T = e G e Ph e R e D e G = geometrical Fraction of the emitted gammas that have interacted with the detector e Ph = photo peak Fraction of the interacting gammas that have deposited the full energy in the active part of the detector e R = reconstruction Fraction of full energy gammas that are successfully reconstructed e D = Doppler correction Fraction of fully reconstructed gammas that are correctly Doppler corrected (first hit localization) P/T = PeakToTotal Fraction of fully successfully reconstructed gammas over the validated ones FH = First hit percentage In the results the efficiency will be e Ph e R

20 Results The used data is a set obtained by means of Geant4 With: Spherical Shell of 15 cm inner and 24 cm outer radii Events with 30 gammas of 1332 kev Packing individual depositions if the distance is < 0.5 cm. Smearing the center of gravity of the packed deposition by randomly displacing it by 0.5 cm max around each coordinate, as a function of Dep. Energy. No single hit gammas considered. Optimization of parameters by maximizing the product of P/T * e Ph e R e Ph e R = 26%, P/T = 73%, FH = 94% Errors ±1% absolute

21 Fuzzy Spectra

22 Original Spectra

23 Work in Progress 1. Ignored depositions + Backscattering (0.6, 80) Hits with a LOW membership function in ANY Cluster. Belong to gammas that escape identification of the first hit or FAR members of a backscattering. 2. Loosely bound clusters (0.9, 73) Rejected Clusters with a first hit having a LOW membership function. Most likely are single hit gammas or random nearby depositions. 3. Merging of fuzzy connected Clusters. (1, 75) 4. Inclusion of Single hit gammas. (1.5, 50) So far: e Ph e R = 30%, P/T = 70%, FH = 92% Further work: 5. Gammas with a BIG number of depositions. Collection of 2 or more gammas: 2 gammas: Total-Total, Total-Partial, Partial-Partial More: discard 6. Tracking of more realistic emission Spectra and detector shape