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52 3. Standard destriping techniques and application to MODIS
Figure 3.9 – (Left) Noisy image from Terra MODIS band 27 (Right) Module of its
fourier transform, showing lobes in the vertical axis of the fourier domain
of stripes but inevitably removes fine structures of the signal and, therefore compromisies
further quantitative analysis. To minimize the loss of high-frequency information, the
filtering procedure shall take into account the periodic nature of stripes prior to the design
of a band-pass filter. Stripe-related frequencies are determined from the spectrum of the
noisy image I s , which we consider as a two dimensional vector of size M × N. Its discrete
fourier transform is given by :
I F s (u, v) =
M∑
N∑
i=1 k=1
( ( jui
I s (i, k)exp −
2πM + jvk ))
2πN
(3.16)
where j is the imaginary unity (j = √ −1). Periodic stripes on images translate in the
fourier spectrum as a signature taking the form of multiple lobes, concentrated along the
vertical axis (figure 3.9b). An accurate estimation of periodic stripe frequencies requires a
distinction from the frequencies related to the true signal structures. For instance, the frequency
power spectrum estimated on the striped signal values, barely reveals the presence
of periodic noise (figure 3.10a). The spectral components of stripes can be highlighted
using the averaging method of
3.5.2 Band-pass filtering
The spectral analysis of MODIS data described above, identifies the frequencies to be
reduced. [Simpson et al., 1995] proposed several possible implementations of Finite Impulsional
Response (FIR) filters. The approach we retain here is based on a two-dimensional
filter in fourier space composed of wells centered at stripe frequencies. The band-pass filter