Compressive Acquisition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 28

a b c

Cross−range frequency Cross−range Cross−range frequency

d e

Cross−range Cross−range

Fig. 3.8 SAAB 9000 car ISAR example. (a) Full measured data. (b) Traditional reconstruction
from the full data. (c) Jittered slow-time undersampled phase histories (40% of data used). (d)
Traditional reconstruction from the compressive measurements in (c). (e) Reconstructed image by
solving ( 3.13 )

where η is the measurement noise with η 2 ≤ ε . Then, the reflectivity map x can

1 minimization problem

be recovered via α by solving the following

α α 1 s. t.

α 2 ≤ ε .

α rec = arg min y −F Ω B (3.13)

3.3.3 Numerical Examples

In this section, we present some simulation results that show the effectiveness
of random and jittered undersampling in slow-time axis for SAR imaging. The
comparison is made with the traditional PFA algorithm [30]. The SPGL1 algorithm
[8]isusedtosolve( 3.13 ). Figure 3.8 shows the results obtained when the jittered
undersampling is applied to the ISAR data collected on a SAAB 9000 car [103].
In this experiment, only 40% of the data is used. As can be seen from Fig. 3.8 ,the
reconstructed image from the compressed measurements, shown in Fig. 3.8 (e), is
identical to the one reconstructed from the full measurements, shown in Fig. 3.8 (a).
Figure 3.8 (d) shows how the traditional reconstruction fails to recover the ISAR
image from the compressive measurements shown in Fig. 3.8 (c).