The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
Figure 3: Single-frame acquisition of landscape around 
Florence city 
2.1 Interferogram dispersion 
The raw interferogram of the energy coming from a pixel of the 
observed scene is pre-filtered due to the finite pixel dimension 
p, and uniformly sampled in a region limited by the detector 
size D . The measured interferogram i(x) may be expressed 
as: 
i(x) 
i(OPD(x)) * rect(-) 
P 
combi—)rect{—) (1) 
P D 
x being the pixel position. It can be easily shown that the 
relationship between OPD and the entering ray direction & is 
linear, as long as the device FOV is not superior to a few 
degrees: 
In view of Eq.3 the minimum wavelength A rnm we can 
reconstruct avoiding signal aliasing is A min = 2SOPD , 
SOPD being the optical path difference subtended by two 
adjacent pixels. Otherwise speaking, the greatest wavenumber 
(i.e. the shortest wavelength) one can observe without aliasing 
is that wavelength for which not less than two detector elements 
cover one fringe cycle. All wavelengths longer than this limit 
will have their fringe cycles sampled by more than two detector 
elements. 
2.2 Spectrum recovery 
As known the spectrum of any pixels have to be reconstructed 
by means of an inverse cosine transform; i.e. the real part of the 
inverse Fourier transform. A possible critical point for 
performing this transformation is constituted by a not perfect 
knowledge of the factor y _ , which is essential for 
/ 
computing the OPD corresponding to a generic position x ■ As 
long as some error affects the knowledge of the factor y the 
calculated inverse transform will lack of accuracy, possibily 
originating a wrong estimate of the pixel spectrum. A 
preliminary issue is allowing some theoretical modelling that 
can account for this type of uncertainty. Let us suppose that the 
value y available for the factor y be erroneous, then the 
retrieved spectrum /(£) differs from the true one /(A:) as 
stated by the following equations: 
7(k) = Rej ii ld (x)e 2 ^dx 
7(k) = Rej fi ld (x)e 2 ^’ !C dx 
k being: 
(4) 
(5) 
OPD{s) = —{j~ 70 )p = —x (2) 
/ / 
a being the proportionality constant between OPD and & , and 
/ the focal plane distance. The constant a is related to the 
maximum digitised optical path difference OPD max and to the 
maximum angle i9 max . 
The raw interferogram is constituted by points (x, ZW(jc)) , 
which indicate the pixel position on the detector and the 
corresponding electronic signal expressed in digital number. 
According to Shannon’s theorem the interferogram sampling 
frequency k s should be grater than the full bandwidth A max of 
the concerned signal: 
k 0 =■ 
2SOPD 
>k r 
(3) 
k=k^-=> X = A^- 
r r 
It can be easily shown that the : 
I(k) = I(k) (7) 
Therefore, an error in the interferogram dispersion gives rise to 
a corresponding wavelength scale error in the spectrum domain. 
Let us note that the difference /(&) — /(A:) may become large 
around narrow absorption lines due to the atmosphere or the 
observed target. A second trouble that can affect the inverse 
transform procedure is connected with the exact knowledge of 
the interferogram’s centre. It is evident that an error affecting 
the central position of the interferogram produces a well-known 
cosine-like modulation of the inverse cosine transform.