PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Identifying objects in a 2D-space utilizing a novel combination of a
re-radiation based method and of a difference-image-method
Andreas Sebastian SCHMELT1; Torben MARHENKE2; Jens TWIEFEL3
1; 2; 3

Institute of Dynamic and Vibration Research; Leibniz University Hannover, Germany

ABSTRACT
To identify objects in a 2D space the feasibility of a novel attempt is investigated in this study. The acoustic
fields are simulated utilizing the 2D finite difference K-Wave toolbox in MATLAB. The generated acoustic
field is recorded at defined positions that can be understood as receiver positions. From these data, theacoustic field is recalculated using the 2D-re-radiation-method. An object-free acoustic field serves asreference. The acoustic field (at the receiver) considering the object is determined. The full acoustic field isrecalculated from the secondly recorded data. Using a difference-image-method, the calculated acoustic
fields are processed and the position, size and geometric shape is identified from the remaining diffraction
pattern. Different objects and object positions are studied. Furthermore, the influence of different excitation
signals of the transducer for the identification are investigated.
Keywords: Re-radiation, Air coupled ultrasound, Object identification

1. INTRODUCTION
For non-destructive testing (NDT), ultrasonic methods in a liquid medium are often used to getmatching impedance characteristics. Some Testing objects can't be placed in liquid media. Air-coupled
ultrasound (ACU) is applied for these type of objects as an alternative method ACU is commonly used
to identify objects or inhomogeneities of materials in several situation s, e.g. for the identification ofcans in a product line [6] or structures in a wood plate [7].
In this study, 2D simulations of airborne sound fields are performed. The MATLAB toolbox kwave is used for the simulation. The sound propagation in the 2D space is determined by ultrasonicexcitation at frequencies up to 50 kHz. Different objects are placed in this space. The simulated soundpressure signal is recorded and stored via defined points, which can be understood as microphone
positions of a B-scan.
Usually pulsed signals are used for ACU-measurements [2, 4, 6, 7], but the control electronics forthis type of signal are usually more complex and the ultrasonic transducers have a different design
than for harmonic signals. Therefore, we study here the comparability of the used methods with apulsed signal to a multiharmonic signal. The multiharmonic signal is modeled by selecting of thecorrect excitation frequencies to generate a beat signal. When setting certain se ctions to zero, there
will be got a quasi-pulsed signal in the recorded time signal. This results in almost identical results
like those with the pulsed signal. This offers the possibility to use simpler control electronics later, as
well as ultrasonic transmitters where resonant excitation and low attenuation are desired.
An innovative method to reconstruct the airborne sound fields, from recorded sound pressure
signals, is the re-radiation method [1, 2, 8, 9]. The propagation path of the sound waves can bereconstructed from the recorded data by calculating backwards in time (backward propagation) or
forwards in time (forward propagation). This method is used here to reconstruct the sound pressure
field to represent experimental conditions. Because in B-scan or A-scan experiments, one does notknow how the sound field looked like, only the recorded time data at the microphone positions are
known. Due to its mathematical origin, the re-radiation method only reconstructs sound paths; it is
not clear whether objects were on the way. Only the diffraction of the sound waves can be1schmelt@ids.uni-hannover.demarhenke@ids.uni-hannover.de3
twiefel@ids.uni-hannover.de28156

reconstructed. One way to identify objects is to look for the places where the sound pressure drops,
because the sound waves first have to diffract around the object, creating a sound shadow behind theobject. Due to the impedance jump from air to the most of other stuffs, e.g. like plastic, wood or steel,the most acoustic energy will be reflected from the solid surface and not transmitted. In this way,delaminations in wooden panels are detected in [9]. The authors from [2] were also able to estimate
the position of plastic beads in this way.
Difference-image-methods are often used to identify moving objects in videos out of many frames.
The challenge is to find a good threshold for the identification. In [3] the authors present techniques
to identify motions in a set of frames and shadow detection algorithms using difference -image
methods. They show the possibility of the identification depends on the correct threshold estimation
for the evaluation of the difference images.
With the approach proposed here, the object-free reconstructed airborne sound field is used as areference image and analyzed with the object-afflicted reconstructed airborne sound fields using a
difference-image method. This makes it possible to use the entire reconstructed sound field and not
only the sound pressure drops at the object position from the re -radiation method for object
identification. From the difference images using suitable threshold values, border images are
generated from which not only the size and position of the ob ject can be determined, but also its
geometric shape. For this, three fundamental geometric shapes made of PMMA are used. Furthermore,peak border images of the mean absolute deviation are determined, on the basis of which it is also
possible to determine the position, shape and size of the objects. The new approach is to use
difference-image methods of a reconstructed airborne sound field to identify object size, position andgeometric shape. This could improve e.g. the product inspection during the process on a productionline.

2. Methods
2.1 Simulation setup
For the simulation of the acoustic fields the 2D finite difference toolbox for MATLAB named KWave, is used. K-Wave use for the calculation a pseudospectral method in the k-space. The twoadvantages rather than other finite difference or finite element methods are first the efficiently
amplitude calculation of the Fourier components using the fast Fourier transform and that the basic
functions are sinusoidal so that theoretically only two grid points per wavelength are needed opposing
to other methods which need six or ten grid points [5]. In Figure 1 on the left side the two excitation
signals for the whole simulation time of 1.5 ms are depicted. At first is a sine pulse with a frequencyof 50 kHz and ten oscillations at a maximum amplitude of 0.2 m/s. The transient response and decay
behaviour is modelled with an exponential function and a damping coefficient of δ= -50000 1/s. The
second is a multi-harmonic sine signal. This signal consists of two sine signals with frequencies of 48
kHz and 50 kHz each of them has an amplitude of 0.1 m/s. Thus, the resulting beat signal has anamplitude of 0.2 m/s and a beat frequency of 2 kHz. The transient response is the same as for the firstexcitation signal. The simulated airborne sound field (see Figure 1, right) is about 0.24 m x 0.33 m
with a grid resolution of dx= dy= 1.65e-4 m which is less than λ/40. At all boundaries 20 elements are
configured for as perfectly matched layers (PML), which are not depicted in the diagrams. The PML’s
has an anisotropic absorption of 2 Np/s. The surrounding medium is air with a sound velocity of c=
343 m/s and a density of ρ= 1.24 kg/m³. The media are modelled as sound absorbing media following
the frequency power law:(1)ߙ ൌ ߙ଴  ߱ ௛  Ǥ
The air has a power law prefactor of α 0= 0.01 (rad/s) -h Np/m. With the exponent for the frequency
power law h= 2 for the whole simulated airborne sound field. In this study, objects in the far field areinvestigated. The transducer is modelled as a velocity source in y direction. At the receiver positions
the sound pressure is recorded. The objects have material properties of PMMA with a density of ρ obj=
1190 kg/m³, a sound velocity of cobj= 2540 m/s and a frequency power law coefficients of α0,obj= 1(rad/s) -h Np/m. In the simulation only longitudinal waves are considered, no shear waves.

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Figure 1 – The excitations signals (left) and the simulation space (right)
2.2 Re-radiation
With the re-radiation method, it is possible to calculate the propagation path of sound waves
backwards or forwards in time from a single plane measurement. For each calculated plane on thepath of the sound waves, the amplitudes can be depicted. This gives interpretable images of the
maximum sound pressure fields. Using the Rayleigh integral for an arbitrary-pulsed scalar field in a
3D space is the fundament for the re-radiation method, which is used here. The derivation of the reradiation equation follows [1] and [2]. The distance from the receiver positions to the propagated pointis:(2)ܴ ൌ ඥሺ‫ ݔ‬െ ‫ ݔ‬ᇱ ሻଶ ൅ ‫ ݕ‬ଶ ൅ ሺ‫ ݖ‬െ ‫ ݖ‬ᇱ ሻଶ.
With the wave number k, the imaginary unit i and the propagation exponent V, with V= 1 forforward propagation or V= -1 for backward propagation, the free space Green’s function G, which isan important part of the re-radiation method, is:
௏୧௞ோ
 Ǥ
Ͷߨܴ
The arbitrary-pulsed scalar field becomes the following form:߲‫ ܩ‬ᇱ ᇱ‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ǡ ߱ሻ ൌ ʹ න ‫݌‬ሺ‫ ݔ‬ᇱ ǡ Ͳǡ ‫ ݖ‬ᇱ ǡ ߱ሻ‫  ݖ ݔ‬Ǥ
߲‫ݕ‬
‫ܩ‬ൌെ

(3)

(4)

ௌ

The derivation of the Green’s function with V= -1 (because only the backward propagation is ofinterest in this work) is:‫ݕ‬߲‫ି ܩ‬୧௞ோ ͳൌ൬ ൅ ݇൰  Ǥ
ܴ
߲‫ݕ‬
Ͷߨܴ ܴ

(5)

Inhomogeneous waves vanish for R>> λ/(2π), so the term 1/R can be neglected. For large distancesfrom the transducer surface the term y/R؆ 1. This simplification increases the calculation speed
significantly. With a finite measurement plane there is a pixel size hp and with the simplified derivationof the Green’s function Gs the pressure field can be implemented as:‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ‫ݖ‬ǡ ߱ሻ ؆ ʹ ݄୮ଶ  ‫݌‬ሺ‫ ݔ‬ᇱ ǡ Ͳǡ ‫ ݖ‬ᇱ ǡ ߱ሻ ‫ܩ כ‬ୱ ሺ‫ ݔ‬െ ‫ ݔ‬ᇱ ǡ ‫ݕ‬ǡ ‫ ݖ‬െ ‫ݖ‬Ԣሻ.
(6)
Here means * the convolution of the pressure at the receiver positions with the derivative Green’s
function. The convolution can be interpreted as the summation of all proportion measuring points toeach single calculated point of propagation plane. For the 2D pressure field the case of a line excitation
a homogenous pressure field in z-direction is assumed. To convert equation 5, accordingly thesimplified Green’s function must be integrated over the z-direction:ஶ‫ܩ‬ୱ ሺ‫ ݔ‬െ ‫ݔ‬Ԣǡ ‫ݕ‬ሻ ൌ න ‫ܩ‬ୱ ሺ‫ ݔ‬െ ‫ ݔ‬ᇱ ǡ ‫ݕ‬ǡ ‫ ݖ‬െ ‫ݖ‬Ԣሻ‫ ݖ‬ൌ െିஶ ݇ ‫ݕ‬Ͷ ඥሺ‫ ݔ‬െ ‫ ݔ‬ᇱ ሻଶ ൅ ‫ ݕ‬ଶ‫ܪ‬ଵଶ ቀ݇ඥሺ‫ ݔ‬െ ‫ ݔ‬ᇱ ሻଶ ൅ ‫ ݕ‬ଶ ቁ Ǥ

(7)

Here is H 12 the tabulated Hankel function of the first order and second kind. So, the 2D pressure
field can be calculated by:8158‫݌‬ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ߱ሻ ؆ ʹ ݄୮ଶ  ‫݌‬ሺ‫ ݔ‬ᇱ ǡ Ͳǡ ߱ሻ ‫ܩ כ‬ୱ ሺ‫ ݔ‬െ ‫ݔ‬Ԣǡ ‫ݕ‬ሻ(8)The free space the back propagated pressure field of the two signals is depicted in figure 2. A
typical behaviour of a piston transducer is depicted for the sine pulse and for the beat signal. The
pressure field is normalized to the maximum pressure which was recorded at all receivers (see figure
1, right) over the time in the object free space ps (t) for the individual excitation signals. For the beatsignal, a part of 0.25 ms to 0.75 ms of figure 1 left is used for the calculation. It arrives at the receiver
positions at 0.89 ms to 1.39 ms. The rest of the beat signal is filled with zeros. So, the beat is used asa quasi-pulse signal. In figure 2 are the two signals depicted. They result in nearly the same pressure
field.

Figure 2 – The backpropagated pressure field of the sine pulse (left) and of the beat signal (right)
2.3 Difference-image
In [3] the authors have shown several threshold methods for the evaluation of difference -images.
Here we use a very similar calculation of the threshold hysteresis. For the difference-image method
is the object free 2D pressure filed of the re-radiation p no,obj(x,y,ω) and the 2D pressure field with anobject p obj (x,y,ω) been used. Both in dB normalized to the maximum pressure which was recorded at
all receivers over the time in the object free space ps (t). At first a background image must be calculated
to make the results better interpretable. The background image B xy is:ଶ‫݌‬୭ୠ୨ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ߱ሻ‫݌‬୬୭ǡ୭ୠ୨ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ߱ሻ(9)‫ ୷୶ܤ‬ൌ ୶୷ǡଷ୶ଷ ൭ቈʹͲ ଵ଴ ቆቇ െ ʹͲ ଵ଴ ቆቇ቉ ൱൫‫݌‬ୱ ሺ‫ݐ‬ሻ൯൫‫݌‬ୱ ሺ‫ݐ‬ሻ൯
A 2D-median-filter med xy,3x3 with a 3x3 neighbourhood is used to smooth the background image.
With this image the difference-image D xy is calculated of:‫݌‬୭ୠ୨ ሺ‫ݔ‬ǡ ‫ݕ‬ǡ ߱ሻ‫ ୷୶ܦ‬ൌ ቤʹͲ ଵ଴ ቆቇ െ ‫ ୷୶ܤ‬ቤǤ
(10)൫‫݌‬ୱ ሺ‫ݐ‬ሻ൯In a further step, the difference-image can be processed to a peak border image of the medianabsolute deviation:‫ ܦܣܯ‬ൌ ୶୷ǡଷ୶ଷ ൫ห‫ ୷୶ܦ‬െ ୶୷ǡଷ୶ଷ ൫‫ ୷୶ܦ‬൯ห൯Ǥ(11)
To get well interpretable pictures in MATLAB the colormap ‘grey’ is used. For the median absolutedeviation the colour axis has its limits from 0 to the minimum of the median absolute deviation whichis greater than zero min xy(MAD>0). It has been shown empirically that these are good values. For thedifference-image the colour axis has his limits from the minimum of the difference-image min xy(D xy)
to the maximum max xy(D xy). Only these pixels are displayed which follows the pixel wise calculated
threshold R with the upper and lower threshold to get a borderline image of the difference-image with
1.0035<R≤1.005:ܴൌඩሺʹሻǤͳ ൅ ʹ‫୷୶ܦ‬൰ ൬ͳ ൅ ‫୷୶ܦ‬(12)

3. Results of the using sine pulse signal
In this section, the results from the sine pulse signal as excitation are discussed. For that, theairborne sound field without and with an object is simulated the sound pressure signals at the receivers

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positions are recorded (see section 2.1). Out of these data’s the airborne sound field is calculated withthe re-radiation method (see section 2.2). Using the proposed difference-image method (see section2.3) D xy- and MAD -images are used for the object identification. At first the object position, then thesize and after that the geometry identification is discussed. In figure 3 are in the first column the
difference-images (D xy-images) including the placed object. So, the reader knows where the object isplaced. In these images only the pixels are shown which are in the upper threshold (see section 2.3).
In the middle column the D xy-images without the object and only the pixels are shown which are in
between the upper and lower threshold (see section 2.3). The right column are the peak border imagesof the median absolute deviation (MAD-image). The images in each row belong together.
The first three rows are for the object position identification in the sound field. In these rows acircular object with a diameter of 20 mm are placed. In the first row are the object centre is at y= 30
mm and x= 0 mm. That the object is at x= 0 mm can be clearly seen out of the symmetry of these three
images. The y-position can be determined out of the turbulence line in image c). The re-radiation
method can only calculate correct up to the object, because the equat ion doesn't consider objects inthe sound field, so, these turbulences line will be produced. So, in the MAD-image is a turbulence line
(this line is once shown in image c) on the right side in red as example). Out of this line the vertical
position of the centre of a circular object can be determined to y= 30 mm ± 2 mm. The diameter canbe determined from the D xy-image b) of the border’s in the middle of the image (red arrow) to d= 19.5mm ± 1 mm. In the same way the size and position of the object in the second row can be determined
to d= 19.5 mm ± 1 mm and y= 90 mm ± 2 mm and x= 0 mm. In third row, the object identification ispossible of utilizing the MAD-image. Due to the free sound path and that the object is not in the wayof the main lobe there are only some peak borders. Some turbulences can be seen on the right but its
not possible to identify the position clearly, it’s only possible to identify the existence of an object. In
the D xy-images g) and h) it is clearer where the object is, because only the sound path from thetransducer to the object make big differences and only these can be seen in these images. It can be
determined that the object is at x= 100 mm ± 2 mm and that the diameter is d= 19 mm ± 1 mm. The yposition is determined to y= 30 mm ± 2 mm.
The fourth and fifth row are for the size determination in comparison to row one. The object in the
fourth row has a diameter of d= 10 mm and a centre position of x= 0 mm and y= 30 mm. In the fifth
row the object has a diameter of d= 80 mm and a centre position of x= 0 mm and y= 30 mm.
The x-position is clear to determine, because the images are symmetrically so x= 0 mm. The yposition of the centre can be determined by the turbulence line in the MAD-images l) and o) to y= 30
mm ± 2 mm. Like it is shown in row one in image b) the diameter can be determined from the widthof the borders of the D xy-images k) and n) to d= 9.5 mm ± 1mm mm in k) and d= 79.5 mm ± 1 mm inn).
The sixth and seventh row are for the geometry identification of the objec t. In row sixth is aquadratic object with a side length of a= 20 mm placed with the centre position x= 0 mm and y= 30
mm. In the seventh row is an equiangular triangle placed with a side length of 20 mm and a centre
position of x= 0 mm and y= 30 mm.
From the borders of the D xy-image q) can be determined that the object is not circular, it must havevertical borders, because the borders in the image are vertical and not circular like in the images above.
From the distance of the vertical borders can one side length determined to a= 19 mm ± 1 mm. The
upper end of the quadratic object is at the borders which are positioned on the vertical lines inclining
away from the middle to the left and right in image q). The lower end is at the turbulence line in the
MAD-image. In contrast to the circular objects is the turbulence line not a marker for the centre of thequadratic object. This is due to the geometric shape and to the re-radiation method. The sound waves
path is with a circular object more like a sound lens and with the quadratic object more like a pistontransducer. So, the other side length of the quadratic object can be determined to a= 20 mm ± 2 mm.
Due to the symmetric behaviour of the images the x position is determined to x= 0 mm and the yposition can be determined from the side length and the upper or lower edges to y= 30 mm ± 2 mm.
The x position of the triangle in the seventh row can be determined due to the symmetric shape of theimages to x= 0 mm. The lower edge of the triangle is at the turbulence line in the MAD-image and thetriangle stands on four peak borders in the D xy-images s) and t). So, the lower edge can be gooddetermined to y= 40 mm ± 0.5 mm. The peak of the triangle is placed between two difference borders
and on the left and right of the triangle there are two difference dots which delimit the triangle andbarely touch it. Out of this information the triangle can be build.

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Figure 3 –The left column are D xy-images with the upper threshold with placed PMMA objects a), d)
and g) with a circular object and a diameter of 20 mm. j) with a diameter of 10mm, m) with adiameter of 80 mm, p) is quadratic with a side length of 20 mm and s) is an equiangular triangle
with the side length of 20 mm. The middle column are D xy-images with the upper and lower
threshold. The right column are the MAD-images. The images in a row belong together.

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4. Results of the using beat signal
In this section, the results from the beat signal as excitation are discussed. For that, the airborne soundfield without and with an object is simulated the sound pressure signals at the receivers positions arerecorded (see section 2.1). Out of these data’s the airborne sound field is calculated with the reradiation method (see section 2.2). Using the proposed difference-image method (see section 2.3) D xyand MAD -images are used for the object identification. In figure 4 the results of the beat signal are
depicted. In the first column are the D xy-images with the placed object, so, the reader knows where
the object is placed. In these images only these pixels are shown which are in the upper threshold (seesection 2.3). In the middle column the D xy-images without the object and only these pixels are shown
which are in between upper and lower threshold (see section 2.3). The right column are the peakborder images of the median absolute deviation (MAD-image). The images in a row belong together.
In the first row the same circular object like in figure 3 in the first row is placed. It has a diameterof d= 20 mm at x= 0 mm and y= 30 mm. On the first look the images looks nearly equal to the images
with utilizing the sine pulse signals, however there are some differences. The position and size of thecircular object can be determined like with the sine pulse signal in section 3. In comparison with thefirst row of figure 3 the sound path of the main lobe doesn’t have big differences. The differences areoutside of the main lobe, outside of the area where the object is placed. For y>50mm the borders of
the D xy-images have changed and there are some new, but there are not important for the determinationof the object. On the left and right edge of the D xy-images the borders look more rippled, but this is
not an important difference for the object identification. The MAD-image has more turbulences in thepeak borders, but the horizontal turbulence line could even good determined.
The second row of figure 4 depict the results for the quadratic object. It’s the same quadratic object
like in figure 3 row six. It has a side length of a= 20 mm and it’s placed at x= 0 mm and y= 30 mm.
The object position and geometry identification can be done like in section 3 with the sine pulse signal.
The largest difference to the result of the sine pulse signal is, that in the bottom of the D xy-images d)and e) are some border areas. These borders are far away of the place where the object is, so there are
not important for the object identification. Like in the images with the circular object, the borders
where the object is not, looks more rippled than with the sine pulse signal as excitation signal. These
ripples make the peak borders in the MAD-image more turbulence.
In the third row of figure 4 the same equiangular triangle is placed like in figure 3 in the seventhrow. It has a side length of 20 mm and a centre position of x= 0 mm and y= 30 mm. The position andgeometric shape can be determined like in section 3, but in these images are more differences than in
the images before with the other geometric shaped objects. It determined out of the borders in the D xyimages g) and h) that there is a triangle object. The four border peaks in these images, at the triangle
position, can also be determined. The upper peak of the triangle is also placed between two difference
borders. On the left and right of the triangle there are no longer two difference dots which delimit thetriangle and barely touch it. This makes the reconstruction of the triangle less accurate. Above thetriangle in image g) in comparison to figure 3 image s) there are no longer rhombic shaped difference
peaks. It’s the same in figure 4 i) in comparison to figure 3 v). There are vertical lines now. Just as
with the other geometric shaped objects, the borders in the Dxy -images g) and h) looks more rippled
than with the sine pulse signal.

5. Conclusion
We have shown, with these model-based results, the good applicability of the re-radiation method
in conjunction with a difference-image method to identify objects in a 2D sound field. With a sine
pulse signal, the position identification and the determination of the geometric shape works well. With
the Dxy-images, it is possible to determine the position up to an a ccuracy of 1 mm. The precision of
the determination of the geometric shape of the object is clearly possible. The results of the beat signal
show nearly equal results to the sine pulse signal. This open the opportunity to use resonant working
transducers and to use electronic equipment for harmonic signals, which are not so expensive like forpulse signals, and to use the good characteristics of the pulsed signal. The results must be verified infuture experiments but our results provide a new way to object description or identification from therecording of airborne sound fields.

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Figure 4 – The left column are D xy-images with the upper threshold with placed PMMA (blue)
objects a) with a circular object and a diameter of 20 mm. d) is quadratic with a side length of 20mm and g) is an equiangular triangle with the side length of 20 mm. The middle column are D xyimages with the upper and lower threshold. The right column are the MAD-images. The images in a
row belong together.

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