Method for correcting the stacking phenomenon applied to X-ray spectrums acquired using a spectrometric sensor
A method correcting a measured spectrum of X radiation, according to a number of channels Nc, each channel i corresponding to an energy range between E i and Ei+ΔE i , including: determining function δt i,j (k) determining size of temporal deviation Δt interval separating two interactions with energy E i and E j , stacking of which leads to a detected energy value E k ; determining, from the function δt i,j (k), probability function P i,j (k) that an event counted in a channel k corresponds to a stack of two interactions, respectively of energies E i and E j ; determining, from the probability function P i,j (k), a stack spectrum as a part of the measured spectrum that corresponds only to the stacks alone; and calculating or estimating at least a first corrected spectrum, by the difference between the measured spectrum and the stack spectrum.
1. A method for correcting a measured spectrum of X radiation according to a number of channels Nc, each channel i corresponding to an energy range between E i and E i +ΔE i , comprising:
determining, using processing circuitry, a function δt i,j (k), which defines a size of a temporal deviation Δt interval, such that two interactions with energy E i and E j temporally separated by Δt are stacked and considered to have a detected energy value E k ;
determining, using the processing circuitry, from the function δt i,j (k), a probability function P i,j (k) that an event counted in a channel k corresponds to a stack of two interactions, respectively, of energies E i and E j ;
calculating, using the processing circuitry, from the probability function P i,j (k), a stack spectrum, which is a part of the measured spectrum that corresponds only to the stack; and
adjusting, using the processing circuitry, the measured spectrum based on a difference between the measured spectrum and the calculated stack spectrum thus forming at least a first corrected spectrum accounting for perturbations present in the measured spectrum due to a stacking phenomenon.
2. The method according to claim 1 , the stack spectrum being calculated from the measured spectrum, exposure time, dead time data, and a minimum duration, separating two photons, below which there is stacking of the two photons and only one of the two photons is detected.
3. The method according to claim 2 , the dead time being determined by simulation or experimentally.
4. The method according to claim 1 , further comprising iteratively calculating N it (N it ≧1) corrected spectrums, each iteration calculating a corrected spectrum of order (n) from a corrected spectrum of order (n−1) or from the measured spectrum if there is no preceding corrected spectrum of order (n−1), by calculating a difference between the corrected spectrum of order (n−1), or the measured spectrum, and the stack spectrum, wherein 1≦n≦N it , n is an integer, and N it corresponds to a number of iterations of said calculating.
5. The method according to claim 4 , further comprising, iterated N it times, with N it ≧1:
calculating, using processing circuitry, a mean stacking probability, as a function of a preceding corrected spectrum Sp cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum, and exposure time and dead time data;
estimating, using processing circuitry, the stack spectrum, as a function of the preceding corrected spectrum, or of the measured spectrum if there is no preceding corrected spectrum, and the exposure time; and
calculating, using processing circuitry, the corrected spectrum, using the difference between a preceding corrected spectrum or the measured spectrum, and the estimated stack spectrum.
6. The method according to claim 5 , the mean stacking probability being calculated using formula:
P
mean
=
1
-
(
1
-
2
×
T
dead
T
expo
)
∑
j
=
1
N
max
Sp
cor
(
n
-
1
)
(
j
)
where Sp cor(n-1) (j) is the value, for the channel j, of the preceding corrected spectrum SP cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum.
7. The method according to claim 1 , the stack spectrum being calculated using formula:
Emp
(
k
)
=
∑
i
=
1
Nc
∑
j
=
i
Nc
P
i
,
j
(
k
)
.
8. The method according to claim 7 , wherein:
P
i
,
j
(
k
)
=
1
-
[
1
-
2
×
δ
t
i
,
j
(
k
)
/
T
expo
]
Sp
cor
(
n
-
1
)
(
E
i
)
×
Sp
cor
(
n
-
1
)
(
E
j
)
Sp cor(n-1) (j) being the value, for the channel j, of the preceding corrected spectrum Sp cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum.
9. The method according to claim 1 , δt i,j (k) being determined from an inverse function of a stacking function F Ei,Ej (Δt), the stacking function establishing a relationship between a temporal shift of two stacked interactions of energies E i and E j and the energy attributed to that stack.
10. The method according to claim 9 , further comprising estimating the function F Ei,Ej (Δt) by simulation or experimentally, or by a decreasing affine function of the energies E i and E j .
11. The method according to claim 1 , the corrected spectrum being divided by a factor depending on a mean probability of the corrected spectrum, the mean probability being a stacking probability of an incident photon with at least one other photon.
12. A device for correcting a measured X radiation spectrum, comprising a number of channels Nc, each channel i corresponding to an energy range between E i and E i +ΔE i , comprising:
at least one processing circuit configured to:
determine a function δt i,j (k), which defines a size of a temporal deviation Δt interval, such that two interactions with energy E i and E j temporally separated by Δt are stacked and considered to have a detected energy value E k ;
determine, from the function δt i,j (k), a probability function P i,j (k) that an event counted in a channel k corresponds to a stack of two interactions, respectively, of energies E i and E j ;
calculate, from the probability function P i,j (k), a stack spectrum, which is a part of the measured spectrum that corresponds only to the stack; and
adjust, using the processing circuitry, the measured spectrum based on a difference between the measured spectrum and the calculated stack spectrum thus forming at least a first corrected spectrum accounting for perturbations present in the measured spectrum due to a stacking phenomenon.
13. The device according to claim 12 , the at least one processing circuit comprising at least one processor.
14. The device according to claim 12 , the stack spectrum being calculated from the measured spectrum, exposure time, dead time data, and a minimum duration, separating two photons, below which there is stacking of the two photons and only one of the two photons is detected.
15. The device according to claim 14 , wherein the at least one processing circuit is further configured to determine the dead time by simulation.
16. The device according to claim 12 , wherein the at least one processing circuit is further configured to iteratively calculate N it (N it ≧1) corrected spectrums, each iterative calculation being of a corrected spectrum of order (n) from a corrected spectrum of order (n−1) or from the measured spectrum if there is no preceding corrected spectrum of order (n−1), to use a calculated difference between the corrected spectrum of order (n−1), or the measured spectrum, and the stack spectrum, wherein 1≦n≦N it , n is an integer, and N it corresponds to a number of iterations of the calculation.
17. The device according to claim 16 , wherein the at least one processing circuit is further configured to carry out the following steps, iterated N it times, with N it ≧1:
calculate a mean stacking probability, as a function of a preceding corrected spectrum Sp cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum, and exposure time and dead time data;
estimate the stack spectrum, as a function of the preceding corrected spectrum, or of the measured spectrum if there is no preceding corrected spectrum, and the exposure time; and
calculate the corrected spectrum, using the difference between a preceding corrected spectrum or the measured spectrum, and the estimated stack spectrum.
18. The device according to claim 17 , wherein the at least one processing circuit is further configured to calculate the mean stacking probability using formula:
P
mean
=
1
-
(
1
-
2
×
T
dead
T
expo
)
∑
j
=
1
N
c
Sp
cor
(
n
-
1
)
(
j
)
where SP cor(n-1) (j) is the value, for the channel j, of the preceding corrected spectrum SP cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum.
19. The device according to claim 12 , wherein the at least one processing circuit is further configured to calculate the stack spectrum using formula:
Emp
(
k
)
=
∑
i
=
1
Nc
∑
j
=
i
Nc
P
i
,
j
(
k
)
.
20. The device according to claim 19 , wherein the at least one processing circuit is further configured to calculate the stack spectrum using the formula:
Emp
(
k
)
=
∑
i
=
1
Nc
∑
j
=
i
Nc
P
i
,
j
(
k
)
with:
P
i
,
j
(
k
)
=
1
-
[
1
-
2
×
δ
t
i
,
j
(
k
)
/
T
expo
]
Sp
cor
(
n
-
1
)
(
E
i
)
×
Sp
cor
(
n
-
1
)
(
E
j
)
where Sp cor(n-1) (j) is the value, for the channel j, of the preceding corrected spectrum SP cor(n-1) , or of the measured spectrum if there is no preceding corrected spectrum.
21. The device according to claim 12 , the function δt i,j (k) being determined from an inverse function of a stacking function F Ei,Ej (Δt), the stacking function associating a pair of photons with respective energies (E i , E j ) with the energy measured as a function of a shift Δt between respective arrival moments of the pair of photons, such that:
δ t i,j ( k )= F EiEj −1 ( E k+1 )− F EiEj −1 ( E k ).
22. The device according to claim 21 , wherein the at least one processing circuit is further configured to estimate the function F Ei,Ej (Δt) by simulation, the function F Ei,Ej (Δt) being estimated by a decreasing affine function of the energies E i and E j .
23. A spectrometry device, comprising:
a radiation sensor;
an electronic processor configured to process signals coming from the sensor and to form a spectrum of a radiation having interacted with the sensor; and
a device for correcting a measured X radiation spectrum, comprising a number of channels Nc, each channel i corresponding to an energy range between E i and E i +ΔE i , comprising at least one processing circuit configured to:
determine a function δt i,j (k), which defines a size of a temporal deviation Δt interval, such that two interactions with energy E i and E j temporally separated by Δt are stacked and considered to have a detected energy value E k ;
determine, from the function δt i,j (k), a probability function P i,j (k) that an event counted in a channel k corresponds to a stack of two interactions, respectively, of energies E i and E j ;
calculate, from the probability function P i,j (k), a stack spectrum, which is a part of the measured spectrum that corresponds only to the stack; and
adjust, using the processing circuitry, the measured spectrum based on a difference between the measured spectrum and the calculated stack spectrum thus forming at least a first corrected spectrum accounting for perturbations present in the measured spectrum due to a stacking phenomenon.