Gapped-Data APES - Spectral Analysis of Signals: The Missing Data Case - page 30

512 × 512 WFFT spectrum of the full data. In Fig. 3.2(d) we show the 512 × 512

×

APES spectrum of the full data obtained by using a 2-D filter matrix of size 16 25.

Fig. 3.2(e) shows the WFFT spectrum obtained by setting the missing samples to
zero.Fig. 3.2(f ) shows the GAPES spectrum with an initial filter of size 10

8.
Comparing Fig. 3.2(f ) with 3.2(d), we can see that GAPES still gives very good
spectral estimates as if there were no missing samples.
GAPES applied to SAR data: In this example we apply the GAPES al-
gorithm to the SAR data. The “Backhoe Data Dome, Version 1.0” consists of
simulated wideband (7-13 GHz), full polarization, complex backscatter data from
a backhoe vehicle in free space. The 3-D computer-aided design (CAD) model of
the backhoe vehicle is shown in Fig. 3.3(a), with a viewing direction correspond-
ing to (approximately) 45 ◦ elevation and 45 ◦ azimuth. The backscattered data has
been generated over a full upper 2

×

steradian viewing hemisphere, which is also
illustrated in Fig. 3.3(a). We consider a 48

π

48 HH polarization data matrix col-
lected at 0 ◦ elevation, from approximately a 3 ◦ azimuth cut centered around 0 ◦ az-
imuth, covering approximately a 0.45 GHz bandwidth centered around 10 GHz. In
Fig. 3.3(b) we show the SAR image obtained by applying WFFT to the full data.
Fig. 3.3(c) shows the image obtained by the application of APES to the full data
with a 2-D filter of size 24

×

36. Note that the two closely located vertical lines
(corresponds to the loader bucket) are well resolved by APES because of its su-
per resolution. To simulate the gapped data, we create artificial gaps in the phase
history data matrix by removing the columns 10-17 and 30-37, as illustrated in
Fig. 3.3(d). In Fig. 3.3(e) we show the result of applying WFFT to the data where
the missing samples are set to zero. Significant artifacts due to the data gapping can
be observed. Fig. 3.3(f ) shows the resulting image of GAPES after one iteration.
(Further iteration did not change the result visibly.) To perform the interpolation,
we apply 2-D GAPES with an initial filter matrix of size 20

×

96 grid.
After the interpolation step, the spectrum of the so-obtained interpolated data ma-
trix is computed via 2-D APES with the same filter size as that used in Fig. 3.3(c).
We can see that GAPES can still resolve the two vertical spectral lines clearly.

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