In this contribution we compare the back-projection algorithm with our
recently developed modified range migration algorithm for 3-D terahertz
imaging using sparse multistatic line arrays. A 2-D planar sampling scheme is
generated using the array's aperture in combination with an orthogonal
synthetic aperture obtained through linear movement of the object under test.
A stepped frequency continuous wave signal modulation is used for range
focusing. Comparisons of the focusing quality show that results using the
modified range migration algorithm reflect these of the
back-projection algorithm except for some degradation along the array's axis
due to the operation in the array's near-field. Nevertheless the highest
computational efficiency is obtained from the modified range migration
algorithm, which is better than the numerically optimized version of the
back-projection algorithm. Measurements have been performed by using an
imaging system operating in the

The use of the effective aperture of a sparse
multistatic array reduces the number of required transmitters (

In order to achieve real-time operation, computational efficient DBF
algorithms are mandatory. The range migration algorithm (RMA)
is a very computational efficient algorithm
since it is mainly implemented
using the fast Fourier transform (FFT) algorithm.
It is widely used for synthetic aperture radar (SAR) imaging
and is established for monostatic and bistatic
configurations

In this contribution we compare a modified RMA with the back-projection (BP) algorithm and the fast-factorized back-projection (FFBP) algorithm by taking the focusing quality and the asymptotic computing complexity as criteria.

This paper is organized as follows.
In Sect.

Schematic of a generic imaging setup using the effective aperture concept in combination with a synthetic aperture.

A schematic of the exemplary imaging scenario is depicted in
Fig.

Assuming that the measured signal
stems from a set of point sources distributed within a volume
to be inspected, this algorithm projects the measurement data back
to their sources. Details
on the algorithm can be found in

This is a numerically optimized implementation of the BP algorithm.
The optimization is based on reducing the computational
costs of the summation in Eq. (

From Eq. (

For a correct reconstruction of the object sampling constraints in the space
domain have to be fulfilled. These constraints and the expected spatial
resolution are discussed in the following paragraphs. For an extended
discussion we refer to

The spacing

Along the

The range resolution is determined by the signal modulation bandwidth

Firstly, we compare the focusing quality of the BP algorithm, FFBP
algorithm and the RMA using numerical simulations. Assuming that the required spatial resolution is 4

3-D image reconstruction using the BP algorithm.

3-D image reconstruction using the FFBP algorithm.

3-D image reconstruction using the modified RMA algorithm.

Focusing quality comparison of the modified RMA with the BP and FFBP algorithms.

The implementation of the two latter algorithms requires the definition of a
discrete 3-D rectangular volume, which covers the expected extent of the
target. The distances between each two points of the volume is set to half of
the expected resolution along each dimension. Two perpendicular sections from
the image reconstruction obtained using the BP algorithm are depicted in Fig.

For a quantitative assessment of the imaging quality, we take a closer look
at the focusing along the

Reconstruction time comparison of the modified RMA with the BP and FFBP algorithms.

Secondly, we compare the computing costs of the three discussed algorithms.
Using the asymptotic computational complexity of
the implementation steps of each algorithm we can estimate
its computational burden.
For the RMA the highest computational burden is caused by
the 3-D Stolt interpolation. Since this is implemented using
a complex 3-D linear interpolation, its asymptotic operation
count

Multistatic sparse line array composed of 12

A-sandwich GFRP with a Rohacell core with a size of
(20

We experimentally investigated the imaging performance of the algorithms
using measurement data from the imaging system presented in

3-D terahertz image reconstruction of the GFRP sample:

As measurement sample we used an A-sandwich (20

We have compared the back-projection algorithm (BP) and
its numerical optimized implementation, the
fast-factorization back-projection (FFBP) algorithm, with a
modified range migration algorithm (RMA) for 3-D terahertz imaging
using a sparse multistatic line array in combination with
a synthetic aperture. As criteria for comparison we took
the focusing quality and the computational complexity.
We have shown that the computational costs can be significantly reduced using the FFBP algorithm with competitive imaging quality to the BP algorithm.
On the other side the modified RMA yields the highest
computational efficiency at the costs of a minor focusing
degradation along the array's axis due to the operation
in the array's near-field.
We used an undersampled sparse multistatic
line array to inspect the inner structure
of a glass fibre reinforced plastics
(GFRP) sample in the frequency range 75 to 110

No data sets were used in this article.

The relation given in Eq. (

The authors declare that they have no conflict of interest.

This work was supported by FhG Internal Programs under Grant No. Attract
018-692 158, the FhG pilot project: Fraunhofer innovations for cultural
heritage and the Innovation Center of Applied System Modeling for
Computational Engineering (ASM4CE) in Kaiserslautern, Germany. The authors
would also like to thank Andreas Keil and Georg von Freymann
for the fruitful discussions. Finally our thank goes to John D. Hunter
for providing the 2-D graphics environment matplotlib