Millimetre-wave (mmW) imaging is an emerging technique for non-destructive testing. Since many polymers are transparent in this frequency range, mmW imaging is an attractive means in the testing of polymer devices, and images of relatively high resolution are possible. This contribution presents an algorithm for the precise imaging of arbitrarily shaped dielectric objects. The reconstruction algorithm is capable of automatically detecting the object's contour, followed by a material-sensitive reconstruction of the object's interior. As an example we examined a polyethylene device with simulated material defects, which could be depicted precisely.
Throughout the whole process chain, quality management is a fundamental task in industrial production. The monitoring of devices and materials is a core issue in order to guarantee a consistent quality of the products. For the detection of material defects the interior of devices is of essential interest. When the device under test (DUT) is not transparent to the human eye, wave-based imaging, employing electromagnetic (EM) or acoustic waves, can be applied. There is a variety of wave-based techniques utilized for non-destructive testing (NDT). Among them are microwave and terahertz radar, ultrasound, X-ray tomography and many more.
In this contribution we present an imaging system employing millimetre waves. Millimetre waves have become an emerging technique in recent years. Due to miniaturization advances in semiconductor technology, leading to considerable cost reduction, they have become attractive for a huge field of applications ranging from NDT to security screening and others (Ahmed et al., 2012; Agarwal et al., 2015). They offer a number of specific advantages compared to the other techniques named above: though they cannot provide the resolution of X-ray, they have the advantage of a non-ionizing radiation. Furthermore, mmW imaging is less costly than employing EM waves of higher frequencies (e.g. X-ray or Terahertz). Sound- or ultrasound-based imaging on the other hand is a cost-efficient solution for many NDT applications. However, it usually requires the DUT to be immersed in water or another coupling medium – for air-coupled ultrasound typically is not able to properly penetrate into the inside of solids due to the very high difference in the acoustic impedances (Hillger et al., 2015).
From their frequency range and the corresponding wavelengths, millimetre waves offer a good compromise between penetration depth and resolution. The resolution, both laterally and in the range direction, lies in the range of a few millimetres, depending on system parameters like bandwidth or aperture size, but also on the DUT's material (Ahmed et al., 2012).
For the data acquisition a synthetic aperture radar (SAR) is utilized. The SAR technique originates from remote sensing – therefore SAR processing algorithms originally were based on the assumption of a free-space propagation of the electromagnetic wave. This assumption still holds when screening dielectric devices which exhibit a relative electrical permittivity equal (or very close) to one. However, when applying such algorithms to a scenario in which the wave propagates through a material with a refractive index significantly greater than one, the reconstruction is based on false assumptions and the reconstructed image will be of low quality or even faulty. The reasons for this are the change in phase velocity and the resulting refraction of the wave occurring at the material boundary in the case of non-normal incidence.
The reconstruction algorithm presented in this paper takes into account the effects named above. It can therefore be applied not only to surroundings that exhibit a free-space-like behaviour, but also to subsurface imaging of refractive materials, which includes many polymers, too.
The article is outlined as follows: first a brief review of the theory of electromagnetic wave propagation in heterogeneous media will be given. Then, the image formation concept will be presented. Starting from the automated detection of the DUT's surface, two approaches for the handling of the material inhomogeneity are shown. The concept was evaluated by measurements – the results are presented in Sect. 4. Eventually a conclusion will sum up the main issues of the article.
The propagation of an electromagnetic wave can be described by the Helmholtz
equation for the electric field strength in the time domain
Snell's law can be derived from Fermat's principle (Hecht, 2002). Both Snell's law and Fermat's principle and their notations originate from optics, but can be applied to electromagnetic waves of other frequency ranges, too.
Fermat's principle states that a wave travelling from a point
Since synthetic aperture (SA) data are time- and space-dependent, the data
processing can operate in the time and space domain or in the temporal and
spatial frequency domain, which is obtained from a multidimensional Fourier
transform of the measured data. A straightforward way to compress the SA data
is by employing matched filtering in the time–space domain (Cumming and
Wong, 2005). Here, for each pixel, that is, each possible target
There is a number of reconstruction algorithms employed for SAR processing,
like range-Doppler, chirp scaling or
Therefore, matched filtering, which can be applied to any geometry, is still the most commonly used technique for subsurface imaging of irregularly shaped objects and will be used in this contribution, too.
If the contour of the object, which is the material boundary between the two media, is not known – or if it is known but its orientation towards the aperture plane is not – then the material boundary must be determined prior to the actual reconstruction. In the following, a way to extract the contour directly from the measurement data is shown.
Therefore, we first reconstruct the space between the aperture and the boundary. Here we can assume a free-space propagation of the EM wave, which means that no refraction has to be taken into account. The resulting image will be defocused, but the boundary will be reconstructed at its true position. Since the boundary will be the strongest reflection in most cases, one way to extract it directly is to search for the brightest pixel in each column. This approach is often used in ground-penetrating radar (Walker and Bell, 2001; Feng et al., 2010). It is convenient because it does not require additional measurements and there are effective algorithms existing for a free-space reconstruction.
Evidently however the error made in estimating the boundary will affect the
quality (i.e. the signal-to-noise ratio, SNR) of the further reconstruction:
due to the imaging principle of interfering single measurements it is
essential that the resulting phase error caused by the estimation uncertainty
is less than
This section addresses the matched filtering for heterogeneous surroundings.
A model of the scenario with the applied nomenclature is shown in Fig. 1.
Here, the indices A, T, and B denote the coordinates of the respective
antenna, boundary point and target. For clarity the indices
We use a monostatic synthetic aperture radar, transmitting a signal
Model and nomenclature.
According to the antenna's isotropic directivity pattern, some part of the
radiated electromagnetic field will be radiated in such a way that after
traversing medium one and being refracted at the boundary it will actually
meet the target (
In Eq. (11), any phase offset due to the scatterer's reflection properties
will be neglected, for it is a constant and thus will not influence the
reconstruction process. Furthermore, since the reconstruction method will
only evaluate the phase, the signal's amplitude is not considered further.
Accordingly, the signal hypothesis becomes
Throughout this article we assume the object under test to consist of one non-magnetic, dielectric, frequency-independent and lossless material whose electric permittivity is known. For many NDT applications these are valid assumptions.
In the following, two means of finding the optical path in a two-media system are described. One is based on Snell's law, and the other one is based on Fermat's principle.
In order to determine the optical path, we search for that point within the
boundary which is the true point of transit between the two media.
Therefore, we discretize the boundary into a distribution of points. Then,
the resulting optical path lengths from the antenna to the assumed target
are a function of the boundary distribution
Note that in order to improve the efficiency of this ray tracing, it is sufficient to consider only those boundary points which lie between the respective antenna and target positions. It is obvious that boundary points lying beyond cannot minimize the optical path.
Like before, we discretize the boundary into a distribution of points. From
the geometry in Fig. 1 it can be seen that the
incident angle is
Likewise, for
Note that, of course, this approach is also applicable to a scenario with a
planar material boundary. Then, in Eqs. (17) and (18) the difference angle
Again, it is sufficient to consider only those boundary points between the current antenna position and target.
To sum up the concept, a flowchart of the complete image reconstruction procedure is shown in Fig. 2.
Measurements were conducted to demonstrate the algorithm's feasibility. Here,
we examined a polyethylene (PE) object into which two holes were drilled.
Polyethylene displays a relative permittivity of
The object under test is depicted in Fig. 3. It was constructed to be invariant along the vertical direction, thus allowing for a 2-D reconstruction. Its surface was chosen to be non-planar and non-symmetric in order to demonstrate the algorithm's capacity to reconstruct rather complex objects.
Flowchart of the image reconstruction process.
Sketch of the object under test and its cross section with all relevant measures.
Employed measurement set-up.
Measurement parameters.
The employed measurement set-up is shown in Fig. 4. It consists of one
horizontal and two vertical traversing units, on which a pair of antennas is
mounted. The traversing units allow for a movement along a distance of
approximately 1.1 m in the vertical direction and 0.65 m in the horizontal
direction. It is therefore possible to span a synthetic aperture of those
dimensions. The two vertical units can be moved separately, which also makes
multistatic measurements possible. For the measurements presented in this
paper we employed a quasi-monostatic set-up, i.e. transmitter and receiver
antenna were in close proximity; they were mounted with a spacing of 2 mm
between them. The antennas were two H-polarized horn antennas with a physical
aperture of 2.45 mm
In order to generate a 2-D image, a line aperture is sufficient. From the
spatial sampling theorem the spacing between the antenna array positions
must not exceed
The system is calibrated by two calibration measurements (Gumbmann, 2011):
Load standard: an empty space measurement with no reflecting object to
eliminate crosstalk between the antennas; short standard: a measurement with a metal plate placed in front of the
aperture at a defined reference distance
The raw data
As a first step, a free-space reconstruction of the calibrated data was
performed. Figure 5 shows the reconstruction of the object assuming
free-space propagation throughout the whole domain. As a reference for the
reconstructed image, a sketch of the DUT's geometry is depicted also. For
the sake of simplicity, the object is depicted in a local coordinate system,
starting from
The reconstruction image is normalized to its maximum intensity value. For clutter reduction, all pixels exhibiting a value below 10 % of the maximum value are set to zero. The object's contour is illustrated by dashed lines.
The boundary contour was estimated by a column-wise maximum search as
described in Sect. 3.2. The respective points are depicted in white in
Fig. 5. For a comparison the estimated boundary is shown in Fig. 6 together
with the true contour. It can be seen that the estimation is a good
approximation to the real contour: on the upper plateau the estimated values
and the analytical ones match perfectly. The error made by the estimation is
depicted in Fig. 7. Its mean value is 0.655 mm; the maximum error is
2.14 mm. The estimation displays an uncertainty that is determined by the
system's range resolution. For the W-band, whose bandwidth
Figure 8 shows the image obtained with the described material-sensitive reconstruction algorithm based on the estimated material boundary. Here, Fermat's principle was used for the ray tracing. The concept based on Snell's law was evaluated in the conference paper (Ullmann et al., 2017). Again, the image is normalized to the maximum intensity and all values below 0.1 are neglected. As before, the reference sketch can be seen on the right side of the figure.
Comparison of the detected material boundary (solid line) and its analytical contour (dashed line).
Estimation error
Note that in the reconstructed images only the targets' upper and lower
boundaries are visible. They correspond to the material discontinuities at
which reflections occur. From Fig. 8 it can be seen that in contrast to the
free-space reconstruction, both targets are reconstructed at their true
positions. The improvement in the localization is because in the free-space
case the propagation velocity is assumed too high. Consequently, since the
velocity is proportional to the traversed way, an overly long distance along
the range direction is reconstructed. With the adapted algorithm this error
is not made. Furthermore, with the developed method the targets and the
material boundary are focused more precisely. Since the lateral resolution
depends on the wavelength
Comparing the resolution of free-space reconstruction
(
The actual resolution in the reconstructed images can be estimated from
Fig. 9. Here, the point that displayed the maximum intensity at the upper
left air inclusion (
From Fig. 9 the respective resolutions can be estimated to
Note that, when instead examining the lower right target (
This article presents methods for the detection of
subsurface material defects by means of
millimetre-wave synthetic aperture radar imaging. The proposed reconstruction
algorithm first detects the shape of the object's surface automatically. In
the actual reconstruction it takes into account the effects occurring at the
surface material discontinuity, namely refraction and the change in phase
velocity, thus allowing for a precise reconstruction of the object under
test. The required ray tracing through the heterogeneous surrounding can be
accomplished using different approaches. Here we presented a way based on
Snell's law and one based on Fermat's principle. Both methods are feasible
(see Fig. 8 in this paper and Fig. 5 in Ullmann et al., 2017); however,
compared to the ray tracing based on Fermat's principle, the method based on
Snell's law requires the extra step of determining the difference angle
An experimental verification was conducted by reconstructing a polyethylene object with air inclusions. Here, the proposed algorithm displayed a higher resolution compared to a conventional free-space reconstruction since the lateral resolution depends on the wavelength and the range resolution depends on the phase velocity.
The underlying measurement data are not publicly available but can be requested from the authors if required.
The authors declare that they have no conflict of interest.
This article is part of the special issue “Sensor/IRS2 2017”. It is a result of the AMA Conferences, Nuremberg, Germany, 30 May–1 June 2017.
The authors would like to thank the European Regional Development Fund (ERDF)
for funding parts of the research activities. This work is part of the
project “Advanced Analytics for Production Optimization” (E