2.1

Classical FTS and Imaging FTS

isFTS derives from classical (non-imaging) FTS. In a FTS system, a two-wave interferometer scans the available optical path difference (OPD) span. The scanning is usually temporal, meaning that one of the arms of the interferogram is moving. The intensity represented as a function of the OPD is called an interferogram. Unlike spectrometers based on dispersive elements such as prisms or gratings, FTS systems require a numerical treatment of the signal to recover the spectrum. It can be proven that the spectrum is the real part of the Fourier transform of the interferogram.⁷ In its most simple form, this relationship is:

with δ the OPD (m), σ the wavenumber (m^–1), S(σ) the spectrum, σ_min and σ_mαx the limits of spectral range of the instrument.

The isFTS systems we study are high-etendue systems,⁸ contrary to dispersive instruments that have an entrance slit. The field of view of a single frame is two-dimensional and limited by the size of the focal plane array (FPA). The interferometer is static and the OPD depends on the position of a ground point in the along-track direction. The interference fringes appear on the raw image itself (Fig. 1 (a)). As the carrier moves linearly over the scene, a same point on the ground is seen through all the OPDs. A sequence of raw images is taken, then registered. On the registred sequence, one ground point is on the same pixel in every image (Fig. 1 (b)). The interferogram of each pixel is extracted from the stack of registered images. Each image yields one point of the interferogram. The Fourier transform is then applied to each interferogram. The set of spectra corresponding to each point constitutes the hyperspectral cube, an object with two spatial dimensions and one spectral dimension. The hyperspectral cube can also be seen as a stack of spectral (or monochromatic) images (Fig. 1 (c)).

Figure 1. :

: Principle of isFTS (a) three raw images of the same scene taken at different moments. The yellow dot follows the same ground point; (b) the images are registered; (c) three monochromatic images at different wavelengths.

2.2

Image Registration Errors

Registration is performed by correlation methods and/or using line of sight data.⁵ The line of sight can be perturbed by an unsufficient stabilization of the platform or micro-vibrations. Registration errors cause a mixing of interferograms from different ground points which leads to a degradation of the spectra.

Figure 2 (a) illustrates this issue. It represents a spectral image taken by the instrument SIELETERS, an airborne infra-red isFTS instrument developed by ONERA.⁹ In this specific case where the micro-vibrations where not corrected by fine registration, a periodical, spatially correlated artefact can be observed on the edge of the building, only in one direction. This artefact disappears when a fine registration is applied (Fig. 2 (b)). In some cases this artefact could be mistaken for a periodical pattern present on the actual scene.

Figure 2.

Spectral images taken with the SIELETERS instrument designed by ONERA. a) Noise observed when no fine registration is applied. b) After fine registration is applied.

In the following sections, we proceed to quantify this effect and to identify on which parameters it depends. This would enable us to specify the correct registration precision so that this effect does not exceed a given value.

3.1

Image Formation Model

In this section we detail the parameters taken into account to derive an expression of the irradiance on the detector (in photoelectrons).

The optical system is made of four elements: a front afocal system, an interferometer, a lens and a detector:

We define the function h(x, y, σ) as the point spread function (PSF) of the system consisting of the lens and the detector, normalized such that ∫∫h(x, y, σ)dxdy = 1. The photonic spectral radiance (photons · s^–1 · sr^–1 · m^–2 · (m^–1)^–1) of the scene at coordinates (x, y) is written L_σ(x, y, σ).

Consequently, if G is the geometrical etendue associated with a pixel (m²sr), T₀ the constant transmission coefficient of the system, Δt the integration time (s), and (x_C, y_C) the coordinates of the optical center (or nadir point), then the number I(x, y) of electrons generated on a pixel of the FPA centered on (x, y) is given by:

3.2

Simulation programs

The main tool used for our study of the isFTS method is a simulation program consisting of a direct program and an inverse program.

The direct program simulates an isFTS acquisition by computing a sequence of raw images based on the image model (Eq. 2), geometric and spectral information on the scene and the trajectory of the carrier. The input parameters of the direct program must be chosen carefully to make sure that the sequence is invertible and to minimize non-physical numerical artefacts. The main parameters are :

The inversion program registers the images produced by the direct program, retrieves the interferograms and performs the inversion to obtain the hyperspectral cube. The interferogram is retrieved for every pixel of the scene that has been seen through the entire OPD span, in other words for all pixels of the scene that have crossed the entire field in the direction of the motion. The displacement of the carrier between two frames is assumed to be constant and known, in other words the inverse program does not take perturbations into account. This leads to degradations of the hyperspectral images. The spectrum of each pixel is obtained by computing the real part of the discrete Fourier transform of the interferogram.

4.1

Theoretical Analysis

In this section we determine the expression of the intensity perturbation on the hyperspectral cube as a function of the position of the pixel and the wavelength in the case of a single registration error.

We consider a scene consisting of a rectangle of monochromatic uniform radiance I₀ on a dark background as illustrated on Fig. 3. A sequence of N instantaneous images is generated by displacing the scene in the direction by a constant step equal to Δ_y.

Figure 3.

a) Four instantaneous images from a sequence with a single registration error at image k. The scene is a rectangle of intensity I₀ on a dark background. The displacement step is Δ_y. Note that the fringes are omitted in this model. b) The same four images after registration.

At the image 0, the position of the optical center in the registered frame is C₀(x_C,0, y_C,0). At the image k, the expected position without perturbations of the optical center is C_k(x_C,0,y_C,0 + kΔ_y). If a registration error of small amplitude (ϵ_x, ϵ_y) occurs at image k, the position of the optical center becomes C_k(x_C,0+ ϵ_x, y_C,0+kΔ_y+ϵ_y).

The OPD map is linear: δ(y) = py with p a constant slope. We note δ_M,k the OPD associated with the point M at the image k. We define O(x₀, y₀) as the reference point so that δ_O,0 = δ₀.

At image number k, a mixing of interferograms of different points from the scene occurs. The interferogram of point M, which should be periodical since the emission is monochromatic, presents an error at the point number k as illustrated on Fig. 4.

Figure 4.

Perturbed interferogram (a) as a function of the image number (b) as a function of the OPD

The expression of the interferogram at the point O is :

with D the Dirac distribution, I_P is the intensity of the perturbation (which depends on, the expression is calculated below) and Ῑ(δ) the undisturbed interferogram.

The optical path difference associated with the point M(x, y) at image k, situated at the distance y – y_O from the point O, is δ_M,k = δ_O,k + p(y – y_O). If we consider that the contrast introduced by the fringes is negligible compared to the constrast of the scene, we can neglect the fringes on the instantaneous images (see Fig. 3). Then, the undisturbed interferogram is the same at the points M and O. This approximation is valid when the spectrum is broad, in which case the contrast of the fringes is low except when the OPD is close to zero. We will see below that this approximation is also reasonably valid in the case of a monochromatic spectrum.

The expression of the interferogram at M is :

The spectrum at M is the real part of the Fourier transform of the interferogram :

with the undisturbed spectrum.

We can note :

where P(x, y, σ) is the perturbation :

Let us now determine the quantity I_P,k.

On the image k, the intensity at M is :

with I(x, y) the intensity of the initial scene.

Let us define the vectors :

Then the perturbation is :

4.2

Illustration of the Impact of the Single Perturbation on the Hyperspectral Image

On a vertical edge of the rectangle on Fig. 3, and .

Equation 3 becomes :

The perturbation presents a periodical sinusoidal behaviour in the direction of the movement with a frequency σ, the same as the frequency of the fringes on the raw images. This result corroborates the images taken by SIELETERS that present a ”crenel” effect on vertical edges (Fig. 2).

The possibility to control the geometry of the initial scene in the simulation program enabled us to identify the areas of the images that are most affected by the perturbation. The simulation was performed with an initial scene composed of a uniform rectangle on a dark background, emitting a monochromatic spectrum at σ_e. A sequence of 160 images was generated. A single registration error was introduced at image 80 (when the entire rectangle is in the field of view of the camera) such that ϵ_x = ϵ_y = –0.4 FPA pixels. We extracted the monochromatic image at σ_e from the hyperspectral cube (Fig 5).

Figure 5.

Monochromatic images from a hyperspectral cube simulated with a monochromatic rectangle as the initial scene. Left : monochromatic image at σ_e without perturbations. Centre : monochromatic image at σ_e with a single perturbation. A periodical noise appears on the vertical edges. Right : Perturbation P(x, y, σ_e)

As in the SIELETERS image, the simulated monochromatic image shows a spatially correlated noise in the direction of the movement. Fig. 5(c) shows that the perturbation is more important on the edges of the rectangle, i.e. in the areas of high radiance gradient.

We compared the theoretical and simulated results for a uniform rectangle with a single perturbation, using Eq. 3 for the strongly approximated theoretical calculation. The theoretical and simulated perturbations are represented on Fig. 6 and show good agreement. On Fig. 6(c) we represented the difference between the theoretical and simulated results. The image presents a pattern with the exact periodicity of the fringes on the instantaneous image, which indicates that the discrepancies are caused by the inadequacy of the approximated model that neglects the fringes.

Figure 6.

P(x, y, σ_e) with a single perturbation. a) Theoretical perturbation. b) Simulated perturbation. c) Difference between theoretical and simulated perturbation.

5.1

General case theory

If more than one image is perturbed, the expression of the perturbation becomes:

and δ_O,k = δ₀ + kpΔ_y, thus:

5.2

White noise

In the simulation, the white noise is introduced using a random integer generator with uniform distribution. The initial scene is oversampled with respect to the detector by a factor r = 10, thus allowing displacements by a distance as small as a tenth of a FPA pixel. To simulate the perturbations ϵ_x and ϵ_y, we generate random numbers in the discrete set {–5/r; –4/r;…; 0; 1/r; 4/r}, and add them to the expected position of the nadir point C_k (x_C,k, y_C,k).

We performed Monte-Carlo simulations to determine the regions of the scene that undergo the largest degradations. The target is a tilted wedge of orientation 12°, emitting a monochromatic spectrum at the wavelength σ_e. We simulated a movement of the carrier in the direction and generated sequences of 100 perturbed images. The sequence was then processed by the inversion model without taking the perturbations into account. We generated several realizations of these sequences and computed the standard deviation for each pixel of the monochromatic image at σ_e.

We performed the simulation for perturbations in both the along-track (Figure 7) and across-track directions (Figure 8).

Figure 7.

Monochromatic image with a white noise in the (along-track) direction. The movement of the carrier was in the direction. Left : one realization of the white noise. Right : Standard deviation for 10 realizations.

Figure 8.

Monochromatic image with a white noise in the (across-track) direction. The movement of the carrier was in the direction. Left : one realization of the white noise. Right : Standard deviation for 10 realizations.

The pseudo-periodical noise on the quasi-vertical edge can be observed with both along-track and across-track perturbations. The noise is spatially correlated even though the perturbations are temporally uncorrelated.

The standard deviation figures show that intensity estimation errors are more important in the areas of high intensity gradient (i.e. on the edges), and even more so on the edges that are approximately perpendicular to the direction of the perturbations. For instance, on Fig. 7, the random noise is in the vertical direction and the most perturbed edge is the quasi-horizontal one, whereas in 8 the noise is in the horizontal direction and the quasi-vertical edge is more perturbed. This is consistent with Eq. 3.

5.3

Sinusoidal perturbation

The case of the sinusoidal perturbation is interesting since it reproduces micro-vibrations of the carrier. For variations in the direction , the amplitude of the perturbation at image k has the form :

where K is the period of the perturbation in number of images, not necessarily an integer.

We note :

From Eq. 4 we obtain :

Using the exponential notation so that :

We note is the sum of a geometric progression of common ratio e^iΦ :

The function takes non negligible values when |1 – е^іΦ|tends to zero, i.e. when Φ = 2πn with n an integer.

Consequently, is non negligible for values of σ such that : . At these values of σ the spectrum presents parasitic peaks.

Let Δ_δ = pΔ_y denote the sampling step of the interferogram. According to the Shannon-Nyquist theorem, the estimated spectrum is periodical with a periodicity of and the maximum measurable wavenumber is . Hence if we only represent the period such that n = 0, two parasitic peaks will appear :

A simulation was performed with a scene emitting a monochromatic spectrum at σ_e = 2.5 · 10⁶ m^–1 with a sinusoidal perturbation in the direction of the movement (the direction). We chose the values : K = 10, p = 1 · 10^–7 m/pixel, Δ_y = 1 FPA pixel. We expected to find the peaks σ₀₊ = 1 · 10⁶ m^–1 and σ_0– = –1 · 10⁶ m^–1. The spectrum of a point of that scene is represented on Fig. 9.

Figure 9. :

: Spectrum of a monochromatic emission at σ_e = 2.5 · 10⁶ m^–1 with a sinusoidal perturbation of period K =10 images.

The peaks at ±σ_e are visible as well as the peaks at σ_0±. Moreover, another set of parasitic peaks appear at ±σ_e + σ_0±, namely at ±3.5 · 10⁶ m^–1 and ±1.5 · 10⁶ m^–1. These peaks do not appear in our approximated theoretical model since we neglected the fringes of the instantaneous images.