Theoretical and practical analysis of spatial and spectral calibration of static Fourier transform infrared spectrometers

: Static Fourier transform spectrometers require, especially in the infrared, a spatial calibration step. Unfortunately, the superposition of fringes on the measured images has a major impact on spatial calibration and therefore on the returned spectra. We first study how to pre-process images so that spectral errors are minimized. Then, we develop a spectrum formation model that is used to correct those spectral errors. The performance, evaluated on synthetic data, is remarkable and theoretically justifies the use of this calibration concept.


Introduction
Fourier Transform Infrared Spectrometers (FTIR) are unparalleled tools for extracting as much spectral information as possible above the visible spectrum [1]. But the FTIR temporal exploration of interferometry range limits the accessible temporal resolution to a few seconds without giving up too much spectral and radiometric resolution. Furthermore, FTIR moving parts are difficult to use in vibrationally challenging environments such as vehicles or aircrafts.
Thus instead of temporally exploring the optical path difference (OPD) it makes sense to spatially spread the OPD over a detector array.
This instrument concept has been the subject of numerous studies since the first article of 1965 by Stroke&Funkhouser [2] but has seen few real working instruments [3-9] and even less evidence of high signal to noise spectrum measures [10,11]. Our goal here is to shed light on the potential reasons for these difficulties and to propose remedies.
The origin of all the difficulties lies in the derived nature of spectrum measurements, which means that stringent conditions are necessary to comply with the requirements of the Fourier transform used to generate the spectra.
Since spatial or angular displacement is necessary to explore all available OPDs, the nonuniformity of the array must be corrected as well as the resulting spatial sampling on the scene. It is therefore necessary to design an interpolation scheme. And the extreme sensibility of the Fourier Transform also means that the non-linearity of the real OPD must be evaluated, possibly corrected or taken into account.
Finally, usually an asymmetric interferogram is acquired to increase the spectral resolution, but this step can also generate spectral errors.
Thus, the measurement sought is the result of many processes, all of which are subject to errors with unacceptable consequences on the resulting spectrum.
We will start here by showing how to calibrate the detector array without using the cumbersome and perilous solution of modifying the instrument to block light in one of the arms of the interferometer.
We will then develop a theoretical model of the spectral errors induced by this spatial calibration and the best way to correct it.
From a practical point of view, we will show how to process real interferogram images before using them to retrieve the observed spectra.
The analysis will also be based on synthetic data in order to compare our results with the underlying truth, which is difficult to find on real acquisitions.

Detector spatial calibration behind fringes
Spatial calibration of visible detector arrays is not, by far, as important as for Infrared detector arrays. Mainly because the most efficient infrared detectors are still today a ternary mixture, HgCdTe, which requires extreme mixing accuracy. We will therefore explicitly study the case of infrared, but a few decibels SNR improvement in visible matrices is certainly possible and worthy of the effort required.
We started working on this subject twenty years ago [12,13], then applied the oversampled design to build a Cryogenic Infrared Stationary Fourier Transform radiospectrometer called Mistere [14]. Later, we developed an airborne Imaging Infrared Static Fourier Transform Spectrometer, called Sieleters [9] which comes in two flavors; MWIR and LWIR.
We provide a quick overview of the performance of these instruments for reference purposes in Table 1. In both cases, the interferometric stage superimposes fringes on an image (in Sieleters case) or a uniform field (in Mistere case). But both will have the same type of image when illuminated by blackbodies used to calibrate them. Thus the same calibration procedure can be used for both instruments.
The standard spatial calibration procedure [15] is based on the assumption that all focal plane detectors must behave in the same way when illuminated by a collimated light field. Then, to enforce this behavior, an affine transformation is calculated for each elementary detector from two or more different intensities to resemble as much as possible a reference detector output. Unfortunately this strategy cannot be used directly because there will always be fringes in the focal plane now. Thus there are two main possible solutions: -We can block one of the interferometric optical paths to remove the fringes. This is the best solution as long as you can make sure that it does not introduce any perceptible stray light and that you compensate for the loss of half the light intensity. Since good calibration strategy involves bracketing the expected intensity by your calibration sources. It must also be compatible with the integration time of your experiments and the physical integrity of the instrument.
-If the first solution is not practical you will have to use signal processing solutions to reconstitute the signal as if there were no fringes. Then the calibration can be performed as usual.
Since Sieleters and Mistere are two cryogenic instruments, we did not take the risk to include the means to block one optical path by inserting a screen, either motorized or controlled from the outside. These options were not chosen because of the difficulties and hazards associated with the operation of cryogenic temperature moving parts (~80K), the increased complexity and thermal "pollution" created by an engine in operation or by the thermal bridge induced by a would-be handle.
A digital solution therefore seems to be the best solution, the impact of which will be explored further on.

Frequen
The goal here because in th spectrometers possible that s  As these possible, the i segments in 2 small part of response to a Thus if ca we will obtain But we must errors. (1)). The g obtain a continu nges were pres ges taken by w see Fig. 1 S BB (T,ν y ) being the blackbody spectrum at temperature T convoluted by the spectrometer response function in the hypothesis that fringes are perfectly horizontal, L BB (T) being the integrated luminance of the blackbody over the spectral window, δ y being the optical path difference (OPD) on the y axis of the detector. This OPD is in practice measured on the detector by means of a laser. By design the OPD is almost perfectly linear in the detector plane.
By stating that the OPD is linear we recognize the real part of a Fourier Transform in the above equation and knowing that the spectrum is real we can write the fringe-polluted image: fringes BB BB BB

Image x y T gain x y S T y gain x y L T offset x y
We will remove fringes from blackbody images by frequency filtering, since they are inherently sinusoidal. Thus the Fourier Transform is the natural tool to do it. We note by Π (gate function) the frequency filter designed to remove fringes frequencies. Then we have in Fourier space: We have seen in Fig. 1 and Fig. 2 that the observed spectrum is strongly concentrated in the 2D Fourier space, since it is a 1D segment (hence the ν y -only dependence of the blackbody spectrum). On contrary, the Fourier transform of both gain and offset are much more spread out, obeying a 1/f law [16]. Thus by virtue of the convolution of S BB (T,ν y ) by the Fourier Transform of the gain, the blackbody spectrum energy will be spread over the gain frequencies, widening the frequency area in need of removal.
Let's assume that the chosen filter is wide enough to remove completely any fringes influence in the resulting blackbody image. Thus we can give an expression of the affine spatial transformation computed by using these filtered blackbody images on two temperatures: Since fringes exist on a narrow frequency domain and that the Fourier Transform of the gain is strongly concentrated at the origin (1/f), we will neglect the blurring effect of this transformation. And because by definition the difference of spectrum exists only on the frequency we filter, we will have: , .
In practice, since the calibration temperatures are very close, a few tens of degrees, the shape of spectral luminances are very similar. Thus, when weighted by the integrated luminance of the blackbody at the other temperature, the multiplicative of the gain in Eq. (7) will be very small. So: Thus any raw image of homogenous spectrum S(ν y ) will be spatially corrected by the resulting affine transformation:

S T y S T y S T y S T y gain x y x y
We can also overlook the difference of fringes in the previous equation which are filtered by Π. Thus: . , x y (11) To form an image we have to use and process single columns, thus we will Fourier Transform the following quantity at a given column x 0 : We don't take into account here the errors made by symmetrizing the partial interferogram often recorded in such instruments, like in our Sieleters design among others [17]. Thus: We expect the multiplicative factors A and B to have similar behaviour and magnitude, but since L is the integrand of S we can also expect to be able to neglect the A part of this last equation. In any case it would have been difficult to build a model of the convolution.
Thus our tentative corrective model will be: Now, the most reliable calibrated spectral source available in infrared being the blackbody, we need to know if a blackbody spectral calibration of this approximate model would give good enough results on non-blackbody sources of radiation.
To this end we have simulated the whole process with synthetic but realistic spatial gain and offset.

Testing quality of spectral calibration on fringe filtered blackbody images
We chose to closely model the Infrared band III of Sieleters with synthetic gain and offset given Fig. 3. Although this simple model cannot explain the classical [18] error divergence for luminance values outside the calibrating luminances, it is widely used to perform correction of spatial noise. It is thus used here.   [-0.22e18,0.17e18] from 15°C to e model outsid (see Fig. 4 The first s to-border con overlapped su mixing functi frequency con Fig. 8 This first step has already direct effects on the image spectrum since it almost completely wipes out the vertical part of the classical "cross" shape plaguing careless image Fourier Transform (see Fig. 8). Fig. 8. Left, norm of the Fourier transform of the raw image of Fig. 7. Right, norm of the Fourier transform of the "flattened" image of Fig. 7. Notice that low frequencies are missing as well as the vertical part of the "cross". The horizontal part is weakened although not erased. But this step can't completely reduce border discrepancies; the usual solution is to build a larger image using four axial symmetries on each border. Gibbs-like oscillations will then be spatially located on new borders and can be expected to spare the "useful" content. This solution enforces a C 0 property on the original image but is not C 1 .
This mirroring does work on the current simulation but is not suitable for real instruments in which the fringes can be tilted, accidentally or voluntarily as for Mistere (see Fig. 1 and Fig. 2). We need a solution that can extend the inclined fringes to a larger image. Several solutions were tried, including an autoregressive model but we finally opted for a Fourierbased solution.
In this new proposal, a conventional mirroring solution is used as a seeding solution, then the fringes will be extended by a Fourier space manipulation involving zeroing fringes frequencies (see Fig. 10).

Selecting fringe frequencies
To do this we must of course remove as few frequencies as possible but also avoid destroying too much information by doing so. It is therefore necessary to find a rule for detecting anomalies. To do this we must find a model of the Fourier Transform of blackbody images when they are free of fringes. Under the affine hypothesis we can write the blackbody image as the sum of two different images (see Eq. (15)).

Image x y T gain x y L T offset x y
It is our experience that gain and offset behave like classical images [16], apart from being noisier. Thus we can expect that their Fourier Transform will be mostly made of a quickly decreasing 1/fp signal and on most frequencies by the Fourier transform of a white noise.
And since any orthogonal transform will output a gaussian white noise from an inbound gaussian white noise, the real and imaginary part of the Fourier Transform of a white noise should have an homogeneous statistics.
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Evalua
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Let's first ex calibration (se   Fig. 16. Left, the spectral parameter which multiplies the luminance of Eq. (14). Right, the spectral offset of Eq. (14). No clear correlation seems to exist between these parameters.
Since the spectral calibration of black bodies is almost perfect, it is preferable to check the spectral restitution of different spectra shapes (see Fig. 17). Fig. 17. Left, the calibrated spectra obtained with the parameters shown Fig. 16. Right, the difference with input spectra.
No errors can be seen without building the difference with the original input spectrum, on the contrary calibration errors are directly visible on the corrected spectrum when using a generic spectral affine model of correction (see Fig. 18). Fig. 18. Left, the calibrated spectra obtained with an affine spectral calibration. On the right, the difference with input spectra.
We can also compute the root mean square of this error by using all columns (see Fig. 19). These resu often in exces [9,14].

Conclusio
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