IEEE - Aerospace and Electronic Systems - March 2021 - 29

Br€uggenwirth et al.

Figure 12.
SSVA super-resolution flowchart.

Figure 11.
Range profiles properly aligned (left) and the resulting wellfocused ISAR image (right).
Raw Data: Image #1.
Signal Processing: " Re-alignment method " .

c E;i
Dt

(
)
arg min X
jjxD ðt þ Dt; iÞj À xM ðt; i; NW Þj :
¼
Dt
t
(9)

Finally, the Phase Gradient Algorithm (PGA),
described in [5], is applied. The results are presented in
Figure 11, where the same subset of range profiles of
Figure 9 is shown to be well aligned along the slow-time
dimension, and a well-focused ISAR image is obtained,
with comparable quality with respect to the results
obtained with the " entropy-based alignment. "

SUPER-RESOLUTION METHODS
SUPER SPATIALLY VARIANT APODIZATION
In this section, the SSVA SR technique is described with
some example results.
SSVA is a SR technique introduced by Stankwitz and
Kosek in [6]. It exploits the nonlinear properties of the
spatially variant apodization (SVA) [7] technique in order
to improve the image resolution. SVA is a nonlinear adaptive FIR filtering approach that is directly applied to ISAR
images with a sinc-like point spread function (PSF) and
allows suppressing sidelobes while preserving the PSF's
mainlobe. The nonlinearity of the SVA causes a bandwidth widening that is exploited in order to perform a controlled extrapolation of the complex signal in the
frequency/slow-time domain and, therefore, improves the
image resolution.
The SSVA technique is implemented in the framework of the SPERI project as described in [6], and a block
diagram of the process is shown in Figure 12.
The first step of the algorithm applies the SVA to the
input ISAR image (II in Figure 12).
In a second step, an Inverse Discrete Fourier Transform (IDFT) is applied to the result; at this point, the output of the IDFT, namely S SVA , is no longer band-limited.
In fact, it has a larger extent in the frequency/slow-time
domain with respect to the original complex signal S , and
its amplitude is tapered.
MARCH 2021

In the third step, the signal amplitude tapering is compensated by means of the Bandwidth Extrapolation (BWE) process, which is performed in the following two substeps:
1) inverse filtering operation in order to equalize the
magnitude taper;
2) the filtered signal portion within the original bandwidth is replaced with the original complex signal.
The inverse filter is created by defining a 2-D sinc
function with its sidelobes forced to zero:

W ðp; qÞ ¼

sincðxÞT sincðyÞ if jx j 1 and jyj
0
otherwise

1
(10)

where p and q represent the vertical and horizontal
dimensions in the image domain, x goes from ÀN=2 to
N=2 À 1 with a Dx step and y goes from ÀM=2 to
M=2 À 1 with a Dy step.
N and M are the original sizes of the complex signal
(before zero-padding) along the two coordinates (time and
frequency), and Dx and Dy are calculated as the ratio
between the original and the size of the zero-padded complex signal along the same coordinates.
A 2D-DFT is then performed in order to generate the
inverse filter wðm; nÞ. The magnitude tapering effect of
the SVA operator is inverted as follows:
S SVAÀBWE ðm; nÞ ¼

S SVA ðm; nÞ
:
jwðm; nÞj

(11)

In order to avoid singularities in the new data, this operation is performed within the first two nulls of wðm; nÞ and
S SVAÀBWE ðm; nÞ is set to zero outside this interval.
In order to complete the Bandwidth Extrapolation
step, the original complex signal is replaced within its
original bandwidth:
S SVAÀBWE ðm; nÞ

S SVA ðm; nÞ
if ðm; nÞ 2 B
¼
S SVAÀBWE ðm; nÞ otherwise

(12)

where B is the supporting time-frequency domain of the
original complex signal.
The final step of the algorithm is to apply the DFT on
S SVAÀBWE ðm; nÞ in order to obtain the final I SSVA image.

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