signal processing of germanium detector signals david scraggs university of liverpool untf 2006
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Signal Processing of Germanium Signal Processing of Germanium Detector SignalsDetector Signals
David ScraggsDavid ScraggsUniversity of LiverpoolUniversity of Liverpool
UNTF 2006
OverviewOverview
• SmartPET
• Convolved Signals
• Wavelet Analysis
• Results
• Future Work
• Questions?
SmartPETSmartPET
• PSA assumes one charge cloud is created
• Compton scattering is most probable interaction above 200keV
• Two charge clouds in single strip possible!
PET SPECT
Convolved SignalsConvolved Signals
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Leading edge of real charge is dependent on position at which the charge carriers are Formed.
PSA gives position of interaction and LORor cone is well defined
Convolved SignalsConvolved Signals
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Leading edge is now convolution of two interactions, characterised by kink.
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1400Convolved Signal
• Discontinuity in leading edge is due to cessation of charge collection from one charge cloud
• Average interaction position
• Goal is to use PSA so convolved signals must be removed
• Signals currently analysed in time domain; not sensitive to discontinuities!
• Analyse signals in frequency domain
Convolved SignalsConvolved Signals
• Discontinuities difficult to discriminate in time domain
• Slight frequency changes are evident in frequency domain
• Fourier Transform can be used to measure frequency components
Frequency AnalysisFrequency Analysis
dtetxfX ftj2).()(
• Fourier assumes stationary signals• Detector signals are non-stationary
Frequency AnalysisFrequency Analysis
• Wavelet window function;
• Transform coefficient is integral of a convolution between the signal and wavelet;
dts
ttx
sssCWT xx
*)(1
),(,
s
t
sts
1
)(,
Wavelet AnalysisWavelet Analysis
• A mother wavelet is chosen to serve as a function for all windows in the process
• Mother wavelet is simply • Functions must satisfy certain criteria• Second derivative of a Gaussian
• Compressed or dilated version
)(t
2/2 2
)1()( tett
2]/)[(2
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1st
es
t
s
t
Wavelet AnalysisWavelet Analysis
Wavelet AnalysisWavelet Analysis
Mother Wavelet:Mexican Hat
Dilated version of mother
Wavelet TransformationWavelet Transformation
ThresholdingThresholding
• Clearly possible to alter any wavelet coefficients • Transform vector contains a range of values• Least significant components relate to the least
significant influences in the signal• Coherent structures and signal discontinuities
within the signal are identified• Can reconstruct original signal from transform• Many types of threshold• Can de-noise signals
ReconstructionReconstruction
• Inverse Wavelet Transform
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2, )(),(1
)(s
dsdtsT
Ctx s
g
Convolution IdentificationConvolution Identification
• Well distinguished convolved event
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Convolution IdentificationConvolution Identification
• Wavelet transform separates out frequencies with the signal
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Wavelet Transform
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Element No.
Wavelet Coefficient
Convolution IdentificationConvolution Identification• Signal discontinuity seen clearly at scale 2 • Two very good matches; noise also present but
very small effect at this frequency, threshold out
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Element No.
Wavelet Coefficient
Identification ResultIdentification Result
• Cs-137 Data was filtered for convolved events
• 64496 Events were convolved
• Method identified 32% or 20419 events as convolved
• A random sample of identified and non identified signals shows promising results
Identification ResultIdentification Result
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1400Pulse Train of Cs-137 Convolved Interactions
Time
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plitu
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Identified Convolved Not identified
Random sampleof pulse train
Identification ResultIdentification Result
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Pulse Train of Cs-137 Convolved Interactions
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plitu
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Pulse Train of Cs-137 Convolved Interactions
Time
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plitu
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• Identified:– Slight frequency
discontinuity near top of signal
• Not identified:– Appears smooth; could
result from two interactions close in depth
Future Work
• Coincidence data collection so that theory can be blind tested
• Remove identified convolved events from pre-reconstruction data and quantify image quality differential
SmartPET Detector NaI