Method for determining a correction value function and method for generating a frequency-corrected hyperspectral image
A method for determining a correcting quantity function k F (x, y) for calibrating an FTIR measurement arrangement with an IR detector. The IR detector includes a plurality of sensor elements, which are each located at a position (x, y), and the method includes: (a) recording interferograms IFG Rxy of a reference sample using the sensor elements of the IR detector, (b) calculating spectra R xy of the reference sample by Fourier transforming the interferograms of the reference sample for at least four sensor elements, (c) calculating correcting quantities k xy by comparing each spectrum R xy of the reference sample calculated in step b) with a reference data set of the reference sample, and (d) determining the correcting quantity function k F (x, y) using the correcting quantities k xy calculated in step c). This permits frequency shifts that occur in FTIR spectrometers with extensive detectors to be effectively corrected regardless of the position of the sensor element.
1. A method for generating a frequency-corrected hyperspectral image of a sample with a Fourier Transform infrared (FTIR) measurement arrangement including an infrared detector having a plurality of sensor elements, the method comprising, for each of the plurality of sensor elements which are respectively located at a position (x,y) of the IR detector:
recording an interferogram IFG Pxy with an equidistant sampling grid a xy with the sensor element;
Fourier transforming the interferogram IFG Pxy , to determine a spectrum S xy (ν) with a frequency axis;
wherein the spectrum S xy (ν) for each of the plurality of sensor elements is corrected with a correcting quantity function k F (x,y) for calibrating the FTIR measurement arrangement with the IR detector,
wherein the correcting quantity function k F (x,y) is determined with a method which comprises:
a) recording interferograms IFG Rxy of a reference sample with the plurality of sensor elements of the IR detector;
b) calculating spectra R of the reference sample by Fourier transforming the interferograms IFG Rxy of the reference sample for at least four sensor elements;
c) calculating correcting quantities k xy by comparing each spectrum of the spectra R xy of the reference sample calculated in said step b) with a reference data set of the reference sample, wherein the reference data set comprises a target position ν 1 of a selected absorption peak P of the reference sample;
d) determining the correcting quantity function k F (x, y) based on the correcting quantities k xy calculated in said step c),
wherein the correcting quantity function k F (x,y) is determined by stretching or compressing the spectra R xy of the reference sample calculated in said step b), or by dividing the target position v L and an actual position ν xy , wherein the interferogram IFG Pxy is recorded with the equidistant sampling grid a xy =a 0 /k F (x, y) and wherein spectra of the sample are subsequently generated by a Fourier transform of corrected interferograms, where a 0 provides a value for the equidistant sampling grid a xy when calculating values for the frequency axis.
2. The method as claimed in claim 1 , wherein the reference data set comprises a simulated spectrum S sim with a plurality of absorption peaks of the reference sample and said calculating of the correcting quantities k xy in said step c) is implemented by comparing the spectra R xy of the reference sample calculated in said step b) with the simulated spectrum S sim .
3. The method as claimed in claim 2 , wherein the correcting quantities k xy are determined by maximizing a correlation, by varying the correcting quantities k xy between the simulated spectrum S sim (ν) and the spectra R xy (ν/k xy ) stretched or compressed by 1/k xy .
4. The method as claimed in claim 1 , wherein the correcting quantity function k F (x,y) is given by
k
F
(
x
,
y
)
=
k
c
cos
(
arctan
(
(
c
y
-
y
)
2
+
(
c
x
-
x
)
2
f
eff
)
)
where c y , c x , f eff and k c are parameters for matching the correcting quantity function k F (x,y) to the calculated correcting quantities k xy .
5. The method as claimed in claim 4 , wherein the correcting quantity function k F (x,y) is matched to the calculated correcting quantities k xy by minimizing an error function:
Σ xy ( k F ( x,y )− k xy ) 2 .
6. The method as claimed in claim 4 , wherein the parameters required for the matching are determined by setting up equations with the correcting quantities k xy calculated in said step c) for the at least four sensor elements to produce a system of equations and by solving the system of equations through curve fitting.
7. The method as claimed in claim 1 , wherein the correcting quantity function k F (x,y) is given by
k F ( x,y )= a *( x 2 +y 2 )+ b*x+c*y+d
where a, b, c and d are parameters for matching the correcting quantity function k F (x,y) to the calculated correcting quantities k xy .
8. The method as claimed in claim 7 , wherein the correcting quantity function k F (x,y) is matched to the calculated correcting quantities k xy by minimizing an error function:
Σ xy ( k F ( x,y )− k xy ) 2 .
9. The method as claimed in claim 7 , wherein the parameters required for the matching are determined by setting up equations with the correcting quantities k xy calculated in said step c) for the at least four sensor elements to produce a system of equations and by solving the system of equations through curve fitting.
10. A method for generating a frequency-corrected hyperspectral image of a sample with a Fourier Transform infrared (FTIR) measurement arrangement including an infrared detector having a plurality of sensor elements, the method comprising, for each of the plurality of sensor elements which are respectively located at a position (x,y) of the IR detector:
recording an interferogram IFG Pxy with an equidistant sampling grid a xy with the sensor element;
Fourier transforming the interferogram IFG Pxy , to determine a spectrum S xy (ν) with a frequency axis;
wherein the spectrum S xy (ν) for each of the plurality of sensor elements is corrected with a correcting quantity function k F (x,y) for calibrating the FTIR measurement arrangement with the IR detector,
wherein the correcting quantity function k F (x,y) is determined with a method which comprises:
a) recording interferograms IFG Rxy of a reference sample with the plurality of sensor elements of the IR detector;
b) calculating spectra R xy of the reference sample by Fourier transforming the interferograms IFG Rxy of the reference sample for at least four sensor elements;
c) calculating correcting quantities k xy by comparing each spectrum of the spectra R xy of the reference sample calculated in said step b) with a reference data set of the reference sample, wherein the reference data set comprises a target position ν 1 of a selected absorption peak P of the reference sample;
d) determining the correcting quantity function k F (x, y) based on the correcting quantities k xy calculated in said step c),
wherein the correcting quantity function k F (x,y) is determined by stretching or compressing the spectra R xy of the reference sample calculated in said step b), or by dividing the target position v L and an actual position ν xy , wherein the interferogram IFG Pxy is recorded with the equidistant sampling grid a xy =a 0 and wherein spectra of the sample are subsequently generated by a Fourier transform of interferograms, where a 0 *k F (x,y) provides a value for the equidistant sampling grid a xy when calculating values for the frequency axis.
11. The method as claimed in claim 10 , wherein the reference data set comprises a simulated spectrum S sim with a plurality of absorption peaks of the reference sample and said calculating of the correcting quantities k xy in said step c) is implemented by comparing the spectra R xy of the reference sample calculated in said step b) with the simulated spectrum S sim .
12. The method as claimed in claim 10 , wherein the correcting quantities k xy are determined by maximizing a correlation, by varying the correcting quantities k xy , between the simulated spectrum S sim (ν) and the spectra R xy (ν/k xy ) stretched or compressed by 1/k xy .
13. The method as claimed in claim 10 , wherein the correcting quantity function k F (x,y) is given by
k
F
(
x
,
y
)
=
k
c
cos
(
arctan
(
(
c
y
-
y
)
2
+
(
c
x
-
x
)
2
f
eff
)
)
where c y , c x , f eff and k c are parameters for matching the correcting quantity function k F (x,y) to the calculated correcting quantities k xy .
14. The method as claimed in claim 13 , wherein the correcting quantity function k F (x,y) is matched to the calculated correcting quantities k xy by minimizing an error function:
Σ xy ( k F ( x,y )− k xy ) 2 .
15. The method as claimed in claim 13 , wherein the parameters required for the matching are determined by setting up equations with the correcting quantities k xy calculated in said step c) for the at least four sensor elements to produce a system of equations and by solving the system of equations through curve fitting.
16. The method as claimed in claim 10 , wherein the correcting quantity function k F (x,y) is given by
k F ( x,y )= a *( x 2 +y 2 )+ b*x+c*y+d
where a, b, c and d are parameters for matching the correcting quantity function k F (x,y) to the calculated correcting quantities k xy .
17. The method as claimed in claim 16 , wherein the correcting quantity function k F (x,y) is matched to the calculated correcting quantities k xy by minimizing an error function:
Σ xy ( k F ( x,y )− k xy ) 2 .
18. The method as claimed in claim 16 , wherein the parameters required for the matching are determined by setting up equations with the correcting quantities k xy calculated in said step c) for the at least four sensor elements to produce a system of equations and by solving the system of equations through curve fitting.