**2. Fabry‐Perot hyperspectral imager**

Our hyperspectral imager (HSI) is based on a Fabry‐Perot interferometer (FPI), the optical system is represented in **Figure 1** where the scene is firstly imaged in the FPI so that the trans‐ mitted intensity is modulated by the interference, and the second image is then formed on the camera sensor by means of the relay lens.

A sequence of frames carrying the interference fringe information is acquired synchronized with the scan of the optical path delay (OPD) between the mirrors of the FPI, from contact to the maximal distance of the mirror. For each pixel of the image, the interferogram is extracted from the acquired video: as an example, we present the interferogram of a light‐emitting diode (LED) at about 635 nm in **Figure 2** where it is evident the non‐linearity of the actuators used to vary the OPD near the contact of the mirrors. The *x* axis of the interferogram is calibrated by using a laser in the optical setup. More details of the calibration technique are described in the previous work [10]. In **Figure 3**, we present the calibrated, resampled and rescaled interferogram.

adopted to separate the spectral content of the radiation impinging on the pixel: starting from the three bands of the Bayer filter camera [1], to the tens of bands with fixed bandpass filters like in the OSIRIS camera on Rosetta spacecraft [2] or hundreds or even thousands of bands of imagers based on dispersive means with gratings or prism like on the VIRTIS camera on Rosetta spacecraft [3]. In this work, we are interested in hyperspectral imagers based on inter‐ ferometers: an interferometer is placed in the optical system in front of the camera, and while the optical path delay (OPD) of the interferometer is varied, the interferogram for each pixel is acquired by the camera and the spectrum is calculated with an algorithm based on the Fourier transform. The final attainable resolution in principle is only limited by the maximal optical path delay of the interferometer. HSI based on Michelson interferometers have been implemented with success in commercial instruments by Bruker [4] and Telops [5] ensuring more than 500 bands in the infrared region and reaching a resolution of less than 1 cm−1. At INRIM, we have realized a different concept of HSI based on Fabry‐Perot interferometer (FPI) and we have validated it in different regions of the spectrum: in the UV [6], in the visible [7]

In paragraph 2 of this chapter, we will describe the principle of the reflectance spectra calcu‐ lation based on the Fourier transform. In paragraph 3, we will show the application of this technique to the field of cultural heritage in collaboration with Centre for Conservation and Restoration *La Venaria Reale* (CCR). Reflectance spectra indeed contain information useful for identifying pigments and dyes and thus for discriminating original and possibly superim‐ posed materials (e.g. pictorial retouching) that is one of the main aims of a diagnostic campaign intended at preserving artworks and guiding the conservation treatment. Moreover, reflec‐ tance spectra can be used for rendering the artworks' colour appearance under different lights in order to choose the light source most suitable for enhancing some aesthetical aspects of the objects, for increasing the visitors' satisfaction, in the meantime taking into account the preven‐ tive conservation principles and the standard recommendations for lighting in museums. On the other hand, the possibility of studying the pigments' colour appearance can be of some help when choosing materials for the conservation treatment. Finally, spectra can be converted in colorimetric values for different light sources or standard illuminants useful for calculating chromatic differences for specific purposes. Results here presented concern some real artworks of different art periods (e.g. coffins from Ancient Egypt, Italian Renaissance polychrome art‐ works) and some mockups used as references made with known pigments and binders.

Our hyperspectral imager (HSI) is based on a Fabry‐Perot interferometer (FPI), the optical system is represented in **Figure 1** where the scene is firstly imaged in the FPI so that the trans‐ mitted intensity is modulated by the interference, and the second image is then formed on the

A sequence of frames carrying the interference fringe information is acquired synchronized with the scan of the optical path delay (OPD) between the mirrors of the FPI, from contact to the maximal distance of the mirror. For each pixel of the image, the interferogram is extracted from

and in the near infrared [8] and in different applications [9].

216 Fourier Transforms - High-tech Application and Current Trends

**2. Fabry‐Perot hyperspectral imager**

camera sensor by means of the relay lens.

**Figure 1.** The scheme of the HSI: the FPI is inserted in an optical setup and the first image of the scene is formed in the FPI, where it is modulated by the interference and the second image with the interference is formed on the camera sensor. The optical path delay is changed while the image is acquired by the camera.

**Figure 2.** The interferogram extracted from the succession of frames of a LED at 635 nm. The *x* axis is the frame number. The first points of the interferogram are missing due to the penetration depth of the mirror coating.

**Figure 3.** The calibrated, resampled and rescaled interferogram from **Figure 2**.

The first points of the interferograms are missing due to the penetration depth of mirror coat‐ ings, and according to Fourier transform theory, they correspond to the cosine contributions having the longest period in the spectra calculation. By inserting a bandpass filter in the opti‐ cal setup and using the information that the spectrum has to be zero in certain regions of the electromagnetic spectrum, it is possible to find the amplitude of the missing points of the interferogram and reconstitute the original spectrum by applying the discrete Fourier transform (DFT) [10]. In **Figure 4**, we present the spectrum obtained from the DFT applied to the calibrated interferogram in **Figure 3** using the Hanning apodization function. The DFT spectrum is expressed in the frequency domain, and the spectrum of interest is in the band 400–720 nm (416–750 THz) according to the bandpass filter. The peak of the LED is at about 472 THz, corresponding to 635 nm, and since the base of the FPI is the Airy function, and not the cosine as in the Michelson interferometer, DFT creates the harmonics of the peak at 472 THz that decease as *R*n/*n*, where *R* is the reflectance of the mirrors and *n* is the order of the harmonic [11]. The interferogram in **Figure 3** is obtained with a maximal OPD of about 30 µm that corresponds to a spectral resolution of about 10 THz. The frequency interval in the spec‐ trum, as visible in the inset of **Figure 4**, is decreased below the spectral resolution by using the zero padding method [10]. A phase correction has been applied to the spectrum calculation in order to take into account the phase dispersion of the mirror coatings [11]. In the right side of spectrum are evident the aliases of the LED peak, artefacts of the DFT. The effect of aliases on the original spectrum is decreased by increasing the number of points per fringe and decreas‐ ing the reflectivity of the mirrors. Since the aliases of the LED peak have a phase dispersion that is not corrected they have deformed peaks.

Once a spectrum for each pixel is calculated, all the spectral information are stored in a hyper‐ spectral cube, a three dimensional array with the spatial information of the scene on the *x*

and *y* axis, and the spectral content on the third axis. The hyperspectral cube contains all the spectral information that can be used for the applications of the next sections.

**Figure 4.** The spectrum in frequency of the LED from the interferogram of **Figure 3** by applying the DFT. Harmonics of the fundamental peak with decreasing amplitude are present. In the inset the spectrum of the LED of interest al 472 THz (635 nm).
