Pulse characterization with Wavelet transforms combined with classification using binary arrays

Overview Wavelet Transformation Creating binary arrays out of and how to deal with them An estimator for adequate events "Hamming distance" Correlation of adequate events How to handle multiple hits

Concept overview wavelet transformation, "binarization" binary data selected class agents selected representation selection of class estimation of adequate events out of selected classes correlation between adequate events and classifier agent binary representation from selected classes from s data pool of known position + binary representation one wavelet describes new and binary representation subtraction of the found wavelet coefficients from pulse

What is wavelet transformation and how can we use it? The wavelet transformation is a relatively new concept (about 10 years old) It provides a time-frequency representation: Ψ = 1 τ s s x ( t) = ψ ( x) = x( t) ψ ( t ) dt Haar wavelet Wavelet transformation wavelet transformation is basely a convolution ψ ( dataset t τ s t 1 0 ) Wavelet Function t ψ ( 1 2 0 ) t ψ ( 1 ) 1 2 t 0 t 1 ψ ( ) ψ ( ) 1 4 1 4

how does the transformation look like? signal Haar wavelet folded HP = high pass LP = low pass

For better accuracy a wavelet of higher order is used: "Daubechies" Wavelet" order 1 order 2 order 3 order 4

Daubechies wavelet transformation Daubechies Wavelet order 5 signal folded

Transformation of wavelet coefficient to binary array wavelet coefficient := a i threshold := µ Binary representation := pos i Binary representation := neg i Example: µ => If µ > a i >= 0 => pos i = 0 If -µ < a i <= 0 => neg i = 0 If µ < a i => pos i = 1 If µ > a i => neg i = 1 a i => pos i =1 ; neg i = 0 a i => pos i =neg i = 0 0 -µ a i => pos i =0 ; neg i = 1 µ:= 2.5 a 4-4 8-7 0 2-5 7 2 union of binary arrays posi negi 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 B = pos neg

Concept overview wavelet transformation, "binarization" binary data "B" selected class agents selected representation selection of class estimation of adequate events out of selected classes correlation between adequate events and classifier agent binary representation from selected classes from s data pool of known position + binary representation one wavelet describes new and binary representation subtraction of the found wavelet coefficients from pulse

Classification of binary arrays to position depending classes and creating agents to represent the position depending classes 2 D representation of detector shape with area definition agent to represent area definition 0 1 0 1...

Concept overview wavelet transformation, "binarization" binary data "B" selected classes selected representation selection of class estimation of adequate events out of selected classes correlation between adequate events and classifier agent binary representation from selected classes from s data pool of known position + binary representation one wavelet describes new and binary representation subtraction of the found wavelet coefficients from pulse

How to find the right class agent? transformed binary pulse transformed binary pulse B 1 0 1 0 1 1 0 1 1 B 1 0 1 0 1 1 0 1 1 & agent 1 agent 2 1 0 0 0 1 1 0 0 1 1 0 0 1 0 1 0 0 1 = 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 0 0 1 & = agent 1 equal solution => significant agent 2 not equal solution => insignificant position of interaction agent 1 agent 2

selected class Hamming Distance to estimate adequate binary arrays from a selected class The Hamming distances of the selected class have to be calculated The Hamming distance is used to count the number of flipped bits in a fixed-length binary word B 1 1 0 0 1 1 1 & The Hamming distance 3 => the smaller the similar 1 1 1 0 0 1 0

Test of Hamming distance length of Hamming array : 400 bit upper limit of Hamming distance: 60 bit pulse to find Pulses selected out of a data pool with about 3500 events, 18 have been found.

Test of Hamming distance length of Hamming array : 400 bit upper limit of Hamming distance: 150 bit pulse to find Pulses selected out of a data pool with about 3500 events, 280 have been found.

Correlations between selected events and s Events with a Hamming distance lower than a fixed limit are taken The correlation will be calculated from these events The event with the best correlation will be selected

Multiple hits and how to handle them "transitive of wavelets" As the wavelet transformation is a transitive operation it is allowable to swap them! * := transformation operator ( ) wavelet pulse a + pulse b = * wavelet * pulse a + wavelet * pulse b that means it is possible to subtract from each other to get a new pulse

Multiple hits and how to handle them "transitive of wavelets" the wavelet by a wavelet will subtracted from the found wavelet and fed back in to the algorithm selected classes selection of class selected representation estimation of adequate events out of selected classes correlation between adequate events and one wavelet describes new and binary representation subtraction of the found wavelet coefficients from pulse

Status and further development Scanned data to be transformed in to the wavelet coefficient to create a database with wavelets and their binary representation Analysis of s in combination with in order to find best position range for the classes and their agents to build the database Analysis of the different correlation ideas and selection of the most suitable one Selection of the thresholds: Hamming distance stopping of the algorithm