Pileup correction method for a photon-counting detector

1. A method for determining a parameter vector that includes a plurality of parameters of a detector pileup model of a photon-counting detector, the detector pileup model being used for pileup correction for a spectral computed-tomography scanner, the method comprising:

setting values of the plurality of parameters, the plurality of parameters including a dead time parameter and individual probabilities of different pileup events, the individual probabilities including a probability of single photon events, a probability of double quasi-coincident photon events, and a probability of at least three quasi-coincident photon events;

determining, using (1) a detector response model, (2) an incident spectrum, and (3) the set values of the parameter vector, a plurality of component spectra, each component spectrum corresponding to one of the individual probabilities of the different pileup events;

summing the plurality of component spectra to generate an output spectrum;

calculating, based on the output spectrum and a measured spectrum, a value of a cost function;

updating at least one of the values of the parameter vector; and

repeating the determining, summing, calculating, and updating steps until a stopping criteria is met, so as to determine a parameter vector that optimizes the cost function.

2. The method of claim 1 , wherein the setting step comprises setting the dead time parameter based on the scanner geometry.

3. The method of claim 1 , wherein the calculating step comprises:

calculating the value of the cost function using:

ψ 2 ( a )=∫ dE[S out ( E, a )− S m ( E )] 2

wherein ψ( a ) is the cost function, a is the parameter vector, is the output spectrum, S m (E) is the measured spectrum, and E is energy.

4. The method of claim 1 , wherein

the plurality of parameters further include a time threshold to distinguish between peak pileup events and tail pileup events; and

the determining step comprises determining, for the double quasi-coincident photon events, a peak pileup component spectrum and a tail pileup component spectrum.

5. The method of claim 1 , wherein the repeating step comprises repeating the determining, summing, calculating, and updating steps until the cost function falls below a predetermined threshold or for a predetermined number of iterations.

6. The method of claim 1 , wherein the updating step comprises updating the parameter vector a according to an exhaustive search method.

7. The method of claim 1 , wherein the updating step comprises updating the parameter vector a according to a nonlinear least-squares method.

8. The method of claim 1 , wherein the determining step comprises:

determining, as one of the plurality of component spectra, a first component spectrum for single photon events, the first component spectrum being determined as:

S 0 ( E )= e −nτ d ∫∫dz 0 dE 0 χ 0 S in ( E 0 ) e −μ CZT (E 0 )z 0 μ CZT ( E 0 )

where S 0 (E) is determined from the detected energy, E, which is defined as

E=ν p ( t TOF 0 :z 0 ,E 0 ),

S in is the incident spectrum, E 0 is the incident energy, z 0 is an interaction point, n is an incident count rate, μ CZT (E) is a linear attenuation value, τ d is the dead time parameter, ν p is an output voltage of a pre-amplifier; and χ 0 is a probability of single photon events.

9. A method of performing pileup correction for a spectral computed-tomography scanner, the scanner including a photon-counting detector, the method comprising:

determining, using the method of claim 1 , a parameter vector that includes a plurality of parameters of a detector pileup model of the photon-counting detector;

performing a scan using the scanner to generate a measured spectrum for the photon-counting detector; and

determining, using the detector pileup model and the measured spectrum, an incident spectrum for the photon-counting detector.

10. A device for determining a parameter vector that includes a plurality of parameters of a detector pileup model of a photon-counting detector, the detector pileup model being used for pileup correction for a spectral computed-tomography scanner, the device comprising:

a processor configured to

set values of the plurality of parameters, the plurality of parameters including a dead time parameter and individual probabilities of different pileup events, the individual probabilities including a probability of single photon events, a probability of double quasi-coincident photon events, and a probability of at least three quasi-coincident photon events;

determine, using (1) a detector response model, (2) an incident spectrum, and (3) the set values of the parameter vector, a plurality of component spectra, each component spectrum corresponding to one of the individual probabilities of the different pileup events;

sum the plurality of component spectra to generate an output spectrum;

calculate, based on the output spectrum and a measured spectrum, a value of a cost function;

update at least one of the values of the parameter vector; and

repeat the determining, summing, calculating, and updating steps until a stopping criteria is met, so as to determine a parameter vector that optimizes the cost function.

11. The device of claim 10 , wherein the processor is further configured to set the dead time parameter based on the scanner geometry.

12. The device of claim 10 , wherein the processor is further configured to

calculate the value of the cost function using:

ψ 2 ( a )=∫ dE[S out ( E, a )− S m ( E )] 2

wherein ψ( a ) is the cost function, a is the parameter vector, S out (E, a ) is the output spectrum, S m (E) is the measured spectrum, and E is energy.

13. The device of claim 10 , wherein

the plurality of parameters further include a time threshold to distinguish between peak pileup events and tail pileup events; and

the processor is further configured to determine, for the double quasi-coincident photon events, a peak pileup component spectrum and a tail pileup component spectrum.

14. The device of claim 10 , wherein the processor is further configured to repeat the determining, summing, calculating, and updating steps until the cost function falls below a predetermined threshold or for a predetermined number of iterations.

15. The device of claim 10 , wherein the processor is further configured to update the parameter vector a according to an exhaustive search method.

16. The device of claim 10 , wherein the processor is further configured to update the parameter vector a according to a nonlinear least-squares method.

17. The device of claim 10 , wherein the processor is further configured to determine, as one of the plurality of component spectra, a first component spectrum for single photon events, the first component spectrum being determined as:

S 0 ( E )= e −nτ d ∫∫dz 0 dE 0 χ 0 S in ( E 0 ) e −μ CZT (E 0 )z 0 μ CZT ( E 0 )

where S 0 (E) is determined from the detected energy, E, which is defined as

E=ν p ( t TOF 0 :z 0 ,E 0 ),

S in is the incident spectrum, E 0 is the incident energy, z 0 is an interaction point, n is an incident count rate, μ CZT (E) is a linear attenuation value, τ d is the dead time parameter, ν p is an output voltage of a pre-amplifier; and χ 0 is a probability of single photon events.

18. An apparatus for performing pileup correction for a spectral computed-tomography scanner, the scanner including a photon-counting detector, the apparatus comprising:

the device of claim 10 ,

wherein the processor is further configured to

cause the scanner to perform a scan to generate a measured spectrum for the photon-counting detector, and

determine, using the detector pileup model and the measured spectrum, an incident spectrum for the photon-counting detector.

19. A method for determining an output spectrum from a parameter vector that includes a plurality of parameters of a detector pileup model of a photon-counting detector, the detector pileup model being used for pileup correction for a spectral computed-tomography scanner, the method comprising:

setting values of the plurality of parameters, the plurality of parameters including a dead time parameter and individual probabilities of different pileup events, the individual probabilities including a probability of single photon events and a probability of double quasi-coincident photon events;

determining, using (1) a detector response model, (2) an incident spectrum, and (3) the set values of the parameter vector, a plurality of component spectra, each component spectrum corresponding to one of the individual probabilities of the different pileup events, wherein the determining step includes determining, as one of the plurality of component spectra, a first component spectrum for single photon events, the first component spectrum being determined as:

S 0 ( E )= e −nτ d ∫∫dz 0 dE 0 χ 0 S in ( E 0 ) e −μ CZT (E 0 )z 0 μ CZT ( E 0 )

where S 0 (E) is determined from the detected energy, E, which is defined as

E=ν p ( t TOF 0 :z 0 ,E 0 ),

S in is the incident spectrum, E 0 is the incident energy, z 0 is an interaction point, n is an incident count rate, μ CZT (E) is a linear attenuation value, τ d is the dead time parameter, ν p is an output voltage of a pre-amplifier; and χ 0 is a probability of single photon events; and

summing the plurality of component spectra to generate the output spectrum.