compressive sensing techniques for mm-wave nondestructive

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TAP.2017.2738034, IEEETransactions on Antennas and Propagation

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1

Compressive Sensing Techniques for mm-WaveNon-Destructive Testing of Composite Panels

Jakob Helander Student Member, IEEE, Andreas Ericsson Student Member, IEEE,Mats Gustafsson Senior Member, IEEE, Torleif Martin Senior Member, IEEE, Daniel Sjoberg Member, IEEE,

and Christer Larsson Senior Member, IEEE,

Abstract—This paper presents imaging results from measure-ments of an industrially manufactured composite test panel,utilizing two introduced algorithms for data post-processing.The system employs a planar near-field scanning set-up forcharacterizing defects in composite panels in the 50–67 GHzband, and can be considered as a complementary diagnostic toolfor non-destructive testing purposes. The introduced algorithmsare based on the reconstruction of the illuminating source atthe transmitter, enabling a separation of the sampled signal withrespect to the location of its potential sources; the scattererswithin the device under test or the transmitter. For the secondalgorithm, a L1-minimization problem formulation is introducedthat enables compressive sensing techniques to be adapted forimage retrieval. The algorithms are benchmarked against a moreconventional imaging technique, based on the Fourier Transform,and it is seen that the complete imaging system provides increaseddynamic range, improved resolution and reduced measurementtime by removal of a reference measurement. Moreover, thesystem provides stable image quality over a range of frequencies.

Index Terms—Non-destructive testing, compressive sensing,imaging, source reconstruction, millimeter wave, planar scanning.

I. INTRODUCTION

NON-destructive testing (NDT) is the science and practiceof evaluating various properties of a device under test

(DUT) without compromising its utility and usefulness [1].Applications where NDT is used are for example weldinginspection and structural evaluation of composite materials [1,2]. Composite structures, manufactured with low permittivityand low loss materials such as honeycomb or foam, areincreasingly utilized in, e.g., aircraft structural components andradomes [3], and in order to conduct NDT on such structures,the method of choice must be capable of detecting defectsrelated to low conductivity and inhomogeneity.

In recent years, millimeter wave (mm-wave) (30 – 300 GHz)imaging has attracted much attention as an electromagnetictesting technique for NDT applications [2–6]. High resolutionimaging is possible at these frequencies due to the inherentshort wavelengths. Other advantageous properties of mm-waveimaging systems are: low power consumption, compact size,

Manuscript received January, 2017; revised June 2017. The work presentedis financed by the Swedish Governmental Agency for Innovation Systems, theNational Aeronautics Research Programme, the Swedish Defence MaterielAdministration, the Swedish Foundation for Strategic Research and theSwedish Armed Forces.

The authors are with the Department of Electrical and Information Tech-nology, Lund University, Lund, Sweden, and Saab Group, Linkoping, Sweden(e-mail: [email protected]).

low weight, and that they can easily be integrated into existingindustrial scanning platforms [4]. Millimeter-wave NDT isparticularly well-suited for inspection of composite structuressince it has been proven useful for detection of materialinhomogenities, disbonds, delaminations and inclusions inlow loss dielectrics [6, 7]; all potential defects in compositestructures. For example, in [8] it was shown that subsurfacevoids in insulating foam were detectable in a wide range ofthe mm-wave spectrum.

A near-field mm-wave imaging system can either operatein reflection, such as in security applications [2, 9], or intransmission as in radome diagnostics [10] and near-fieldantenna measurements [11]. The spatial sampling of thescattered signals can be acquired either through mechanicalscanning [11] or electronic scanning by utilizing antennaarrays [2]. Much time can be saved if an electronic scan canbe performed, but at mm-wave frequencies it is non-trivial todevelop cost-effective imaging arrays of high complexity. Thiscomplexity can be reduced by performing a one-dimensional(1D) raster scan using a 1D imaging array, which providesa good compromise between complexity and scan time [12].Depending on the application, either a single frequency ora broadband electromagnetic signal is used to illuminate theobject. In this work, the focus is on transmission based near-field scanning systems. These systems traditionally utilizeeither planar, cylindrical or spherical scanning [11, 13, 14].

Once the signals have been acquired over a given spatialarea, the aim is to reconstruct the unknown sources on a givensurface. This is an inverse problem which has traditionallybeen solved using reconstruction in the Fourier domain, com-monly referred to in literature as time reversal, digital beamforming (DBF), aperture synthesis, back propagation, backprojection or migration technique [15, 16].

Alternatively, Compressive Sensing (CS) based processingtechniques could be used for image retrieval if the inverseproblem could be modelled as an underdetermined linearsystem [17–19]. The strategy is to formulate the relationbetween the data and unknowns as a linear mapping, modelledby a rectangular matrix, and where the unknown vector hasonly a few non-zero entries with respect to a defined basis.Since the industrial manufacturing standard is good, existingdefects in composite structures would be few and CS process-ing techniques would be adaptable to the problem at hand.Furthermore, the theory behind CS states that, if adapted toa sparse problem, the sought after solution can be recoveredusing far fewer samples than what is classically required by




the Nyquist theorem [20], allowing for a potentially reducedmeasurement time as the number of required data samplesdecreases.

The characteristics of CS have resulted in a fast develop-ment in a variety of applications related to electromagnetics[20] such as: array synthesis [21], antenna diagnostics [22],direction-of-arrival estimation [23], ground penetrating radar[24], and inverse scattering [25–27]. The state-of-the-art withinCS techniques for microwave imaging includes a variety offormulations to target specific problems and improve perfor-mance; wavelet basis functions for CS and total-variation CScan be adapted to solve problems with non-sparse scatterers(with respect to a pixel basis) [28–30]. Additional examplesare the contrast field based CS [31] applicable to weakscatterers and multi-task Bayesian CS [32].

As spatial resolution, the achievable dynamic range andoverall measurement time for acquiring data is generallyconsidered the main measures of performance of an imagingsystem [2], mm-wave imaging using CS offers a promisingfoundation for further development of electromagnetic testingtechniques for NDT applications.

In this paper, we introduce a mm-wave imaging techniquebased on CS that aims at: (1) reducing the required measure-ment time by removal of a reference measurement, and (2)improving the dynamic range compared to the conventionalimaging technique based on time reversal of the measuredfields. A 50 – 67 GHz imaging system is constructed usinga planar near-field scanning set-up, and is then applied toindustrially manufactured composite test panels. The techniquecould be seen as a complement to currently used diagnostictools used in industry such as ultrasonic testing.

The technique is developed in two steps, where first asource separation algorithm is introduced that separates thecomponents of the scattered signal related to the defects inthe DUT from the total received signal by subtracting theilluminating field. This implies that no reference measurementis needed. After the illuminating field has been subtracted, aweak scattered field from a few sparse defects is acquired.This linear inverse problem is then formulated in a CS sensein order to improve the dynamic range.

II. ALGORITHM DESCRIPTIONS

For clarity, the notation used in the rest of this paper issummarized here: boldface uppercase and boldface lowercaseare used for matrix- and vector notation, respectively.

A. Measured Signal

The measured signal is sampled at a distance z from theilluminating antenna over a finite rectangular aperture. Thesampling is performed over a uniform rectangular grid, and themeasured signal for each fixed frequency in the correspondinggrid point can be represented as the sampled signal s (acomplex number at each spatial grid point):

s(xi, yj , z) = s(x1 + (i− 1)∆, y1 + (j − 1)∆, z)

i = 1, 2, . . . , Nx

j = 1, 2, . . . , Ny.

(1)

Here, ∆ is the sample increment in the x- and y-direction,and (x1, y1) is the grid point in the lower left corner of thegrid. The aggregated samples of the signal (or field) in themeasurement plane for each fixed frequency is represented asa Nm × 1 column vector, where Nm = NxNy .

B. Time Reversal Technique

For a measured signal s(x, y, z2) sampled continuously ata fixed distance z = z2, the time reversal technique exploitsthe Fourier transform and its inverse in order to translate thesignal between different xy-planes [15]. The signal’s spatialspectrum at z = z2 is

S(kx, ky, z2) =

∫ ∞−∞

∫ ∞−∞

s(x, y, z2)ej(kxx+kyy)dxdy, (2)

where kx and ky are the x- and y-component of the wavenum-ber, respectively. In the spectral domain, the signal can betranslated to a xy-plane closer to the source using a simpleexponential, i.e., the signal’s spectrum at z = z1 is:

S(kx, ky, z1) = ejkzdS(kx, ky, z2). (3)

Here, kz =√k2 − k2x − k2y, d = z1 − z2 < 0 and k = 2πf/c

with f being the frequency and c being the speed of light in air.Applying the inverse Fourier transform yields the signal in thespatial domain s(x, y, z1). For discrete signals, the usage of thediscrete versions of the Fourier transform and its inverse allowsfor the time reversal technique to be efficiently implementedusing Fast Fourier Transform (FFT) algorithms. Aliasing canbe avoided by choosing the sample increment as ∆ ≤ λ/2 inaccordance to the Nyquist criterion. In the set-up consideredin this paper, the choice of ∆ can be relaxed as discussedfurther in Sec. III.

In order to retrieve the vector field components from themeasured signal, it is necessary to compensate for the probethat samples the signal. The probe correction, explained indetail in [11, 33], is based on the a priori knowledge of thereceiving characteristic of the probe. Using an open-endedwaveguide, the far-field in the spectral domain can be modelledusing formulas in [34]. The procedure can be used in order toextract either a scalar field or a vector field where the vectorfield extraction requires two independent measurements of thesame signal. These are commonly chosen as the co-polarizedand cross-polarized measurements. If the field is normalizedwith that obtained from a reference measurement, the timereversal technique can be applied for image retrieval of a DUTcontaining defects.

C. Source Separation Algorithm

A general model of the imaging set-up with appropriate fieldvectors and operators defined is seen in Fig. 1. As depictedin the figure, for a single frequency the field extracted fromthe corresponding discrete samples in the measurement planeis assembled into the Nm × 1 vector bm. The nm:th entryof bm corresponds to the field value in the spatial grid point(xi, yj) with nm = (j−1)Nx + i in relation to the introducedindices in (1). In order to expand the field in an arbitrary plane,




DUT planeb1

Measurement planebm = bm + b2

Antenna planexa

A21

A20

A10

b1

z

z2

z1

z0

d

Fig. 1. Discretized model of the imaging set-up with operators and fieldvectors as depicted.

all surfaces of interest are discretized into rectangular meshcells with rooftop functions acting as local basis functions. Theoperator Aij , constructed analogously to conventional methodof moments [35], thus maps the currents in plane j to the fieldin plane i.

The scattered field from the defects is assumed to be ordersof magnitude smaller in amplitude compared to the total fieldin the measurement plane in the sense that critical defects(e.g., delaminations and debonding) have only a minor impacton the electromagnetic properties of the DUT. The measuredfield bm can then be decomposed into the illuminating field b2

from the antenna and the slab of the DUT, and the scatteredfield bm arising only from the interior defects:

bm = bm + b2. (4)

Using (4), bm can be extracted if b2 can be estimatedappropriately. This is done by finding the currents on theantenna aperture represented by the Na×1 vector xa, using theoperator A20 (size Nm×Na). A singular value decomposition(SVD) is applied to A20:

A20 = UΣVH. (5)

Here, U and V are unitary matrices, and Σ is a diagonalmatrix containing the non-zero positive singular values of A20.The normalized singular values are truncated according toa prescribed threshold τ , selected by means of the L-curvecriterion [36]. The pseudo inverse is then constructed as [37]:

A†20 = VΣ†UH. (6)

Here, Σ† is a diagonal matrix, with all entries correspondingto singular values whose normalized value is below τ setto zero. The remaining entries contain the reciprocal of thecorresponding singular value. Applying A†20 to the measuredfield bm yields

xa = A†20bm, (7)

and the field contribution from the antenna, b2, is found as:

b2 = A20xa. (8)

After finding b2, the analysis is restricted to the scatteredfield bm, where the source separation algorithm utilizes the

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Fig. 2. Extracted scattered field bm (left) from a total field bm in themeasurement plane (right) from an example simulation using the theorypresented in Sec. II-C. The figures depict independently normalized absolutevalues of the fields in dB-scale. The relative power difference between totaland scattered field is ' 30 dB.

time reversal technique to retrieve the final image in the DUTplane. The CS algorithm introduced in the upcoming section,is instead based on the formulation of the L1-minimizationoptimization problem. The partitioning of the field as a resultof the source separation is illustrated in Fig. 2.

D. Compressive Sensing Algorithm

The CS algorithm utilizes the SPGL1 Matlab solver basedon the general basis pursuit denoise (BPDN) L1-minimizationproblem [38, 39]. For a single frequency, the target imagecan be acquired from the scattered field bm, by finding thecorresponding scattered field b1 in the DUT plane. The initialoptimization problem is then formulated as:

minimize ||b1||1subject to ||A21b1 − bm||2 ≤ σ.

(9)

The choice of the user-defined threshold σ is discussed in thefinal part of this section. In order to reconstruct properties ofthe defects rather than the scattered field b1, we introduce thescattering amplitudes s as

b1 = B1s, (10)

with

B1 = diag(b1) =

b1,1 0 . . . 00 b1,2 . . . 0...

.... . .

...0 0 . . . b1,N1

representing the incident field on the panel at z = z1 from theantenna, see Fig. 1. N1 spatial grid points are assumed in theDUT plane. Using xa and the operator A10 (size N1 ×Na),and, b1 can be found through

b1 = A10xa. (11)

The scattering amplitudes s comprise N1 unknowns, inducedby the scatterers in the DUT plane. In a pixel basis, only fewnon-zero elements would be needed to represent the defectsif their physical size is on a wavelength scale. This wouldcorrespond to defects no larger than a few centimeters whenconsidering the 60 GHz band; a reasonable scenario since




industrial manufacturing processes rarely results in large-sizefabrication errors. The optimization problem then reads:

minimize ||s||1subject to ||A21B1s− bm||2 ≤ σ.

(12)

The incident field, expressed in B1, is not uniformly dis-tributed over the DUT plane which results in a non-uniformweighting of the scattering amplitudes in this formulation. Tore-compensate for this effect the final formulation is obtainedas:

minimize ||s||1subject to ||B2,invA21B1s−B2,invbm||2 ≤ σ,

(13)

where B2,inv represents the magnitude of the inverse of theincident field in the measurement plane:

B2,inv =

|b2,1|−1 0 . . . 0

0 |b2,2|−1 . . . 0...

.... . .

...0 0 . . . |b2,Nm

|−1

.1) Choice of the threshold σ: For a known field bm, an

estimate of the scattering amplitudes s is retrieved by timereversing bm using the adjoint operator A∗21:

B1s = A∗21bm. (14)

By applying A21 to B1s, the estimated scattered field in themeasurement plane bm is found as:

bm = A21B1s = A21A∗21bm. (15)

Finally, the threshold is chosen as

σ = ||bm − bm||2 = ||(1−A21A∗21)bm||2. (16)

This choice of σ is stable, and the performance of the algo-rithm would not be compromised by small fractional changesto this value.

2) Comment on implementation: Given that the measure-ment and DUT plane are discretized identically, it follows thatA21 is a block Toeplitz matrix. Due to the large number of gridpoints necessary for a reasonable resolution, it is not desirableto compute the complete matrix operator. Rather, A21 can beextended to a circulant matrix due to its Toeplitz characteristic,and the general linear operations A21x and A∗21b can insteadbe evaluated efficiently in a matrix free way using an FFTalgorithm [37, 40].

III. PERFORMANCE

The three implemented algorithms are:1) The time reversal technique, as described in Sec. II-B.

This technique employs time reversal of the field bm

in the measurement plane with DUT present and afree space reference. The final image is obtained afternormalization with the free space field back-propagatedto the DUT plane.

2) The source separation algorithm. This algorithm utilizessource separation to estimate the scattered field bm

in the measurement plane (Sec. II-C), and thereafter

MeasurementsSDUT21 , SFS

21

Time gating Probecorrection

AlgorithmSelection

SourceSeparation

SourceSeparationTime reversal

NormalizationEy/E

FSy

Time reversalCompressive

sensing

Imaging results

Ey

S21

3)2)1)

Fig. 3. Block diagram depicting the post processing steps used in thiswork. The top row corresponds to processing that were carried out for allmeasurement data. The three parallel schemes at the bottom correspond to 1)the time reversal technique, 2) the source separation algorithm and 3) the CSalgorithm. The free space measurement data, denoted by FS, is associatedwith the dashed line.

employs time reversal of bm (Sec. II-B) in order toretrieve the final image.

3) The CS algorithm. This algorithm utilizes source sep-aration to estimate the scattered field bm in the mea-surement plane (Sec. II-C), and thereafter employs thegeneral BPDN L1-minimization optimization problemformulation in (13), in order to retrieve the final image(Sec. II-D).

A block diagram describing the algorithms can be seen in Fig.3. As depicted, the top row of blocks represents measurementpost-processing procedures and thus only apply to measureddata (Sec. III-B). When executing the time reversal routine inalgorithm 1) or 2), interpolation of the data can be employedusing zero padding of the signal’s spatial spectrum. For allresults presented in this section, sampling is carried out inthe measurement plane at z2 = 530 mm, with ∆ = 5 mmand Nm = 512 = 2 601, resulting in a total finite squareaperture of 250×250 mm2. The DUT plane is at z1 = 470 mmwith N1 = 2032 = 41 209 number of spatial grid points. Thedistance between the measurement plane and the DUT planeis d = 60 mm. The antenna aperture plane is at z0 = 0 mm.

Note that in the frequency band of interest 50 – 67 GHz, thesample increment is ∆ ≈ λ and thereby the Nyquist criterionis violated. However, no aliasing is introduced that affects




yx

z 0

z 1

z 2

1 23

(a) Side view.

y

x z

1

2

3

(b) Top view.

Fig. 4. Simulation set-up in FEKO for synthetic data generation. The side-view is depicted in (a), and the top-view in (b). The dimensions of the resistivesheets are not to scale. Highlighted regions in (a) denote the measurement-,DUT- and antenna plane in descending order.

the final image due to the fact that the geometrical set-uplimits the possible incident plane waves under consideration,or equivalently restricts the (kx, ky)-spectrum. Consequently,any aliasing that occurs falls outside of the wavenumber regionof interest.

A. Synthetic Data

The above described algorithms are employed on syntheticdata, extracted from simulations using FEKO. This discardsany deterministic measurement errors that might appear whenmeasured data is considered, and consequently provides aproof of concept of the algorithms themselves.

The Ey-component (co-polarization) in the aperture of astandard gain horn antenna at 60 GHz is used as the illu-minating source, and three rectangular resistive sheets withR = 100 Ω represent the DUT. The set-up can be seen in Fig.4. With labelling according to Fig. 4b, the dimensions of theresistive sheets are 1.5×10 mm2, 5×2.5 mm2 and 1×1 mm2.

Retrieved images using the time reversal technique, sourceseparation algorithm and the CS algorithm can be seen inFig. 5. As seen, for the depicted color range ([−60, 0] dB),the CS algorithm provides the best dynamic range as onlyscattering amplitudes related to the physical dimension of theresistive sheets have non-zero entries, i.e., the sparse solutionto the problem is found. The time reversal technique gives littleinformation whereas the source separation algorithm managesto detect the defects, with noticeable low power noise present.

Due to the large difference in dynamic range for the differentalgorithms, the images from the time reversal technique andsource separation algorithm are shown in Fig. 6 for customizedcolor ranges. It is seen that the source separation algorithmprovides a dynamic range of around ' 20 dB. As Figs. 5 and6 (a) depict the time reversed field normalized with a freespace measurement (|Ey|/|EFS

y |), the shadowing effect fromthe resistive sheets is clearly seen as the amplitude aroundthe defects is lower than the surrounding medium. However,both the source separation algorithm and CS algorithm acton the scattered field using (4), and consequently the imagesin Fig. 5 (b), (c) and Fig. 6 (b) show the resistive sheetsas the maximum amplitude in the DUT plane. Truncationeffects, i.e. sidelobes, arising from the finite aperture of themeasurement plane can be seen for both the time reversaltechnique and source separation algorithm, most clearly seenin Fig. 5 (b) as the vertical- and horizontal lines springingfrom the resistive sheets. The sidelobes can be suppressedby applying an appropriate windowing function, reducing theeffect of the finite aperture of the measurement plane, with thecost of degraded resolution. However, these sidelobes do notappear for the CS algorithm, due to the usage of operators fortransforming between xy-planes. Moreover, the CS algorithmmanages to resolve the 1 × 1 mm2 (0.2λ × 0.2λ) defect,whereas this defect is not detected by neither the time reversaltechnique nor the source separation algorithm.

B. Measurements

Measurements were carried out in the microwave laboratoryat Lund University, Sweden. The illuminating source was cho-sen as a 50 – 67 GHz standard gain horn antenna, and an open-ended waveguide was used as the scanning probe. A singleplanar scan was conducted, with both horn and probe beingy-polarized. The transmitted signal was measured using anAgilent E8361A network analyzer, and a low noise amplifier(LNA) was added on the receiver side. The transmit power was−7 dBm, the intermediate frequency (IF) bandwidth was setto 300 Hz and a linearly spaced frequency sampling with 201points across the whole operational band of the horn antennawas used.

In order to acquire the necessary data as input for thealgorithms, post-processing routines were employed on themeasured S21, as seen in the top row of blocks in Fig.3. First, multipath scattering components of the signal aresuppressed by applying an appropriate window function inthe time domain, here chosen as a Hanning window. Aninteresting remark is that even with a free space path loss of' 68 dB/m at 60 GHz, and additional losses due to reflectionand surrounding absorbers, the data filtering resulted in aconsiderable improvement of the purity of the S21 over themeasurement aperture. Secondly, probe correction is appliedin order to extract the field components from the S21 asdescribed in Sec. II-B. Using one planar scan, only a singlescalar component may be extracted, here chosen as the co-polarized component of the electric field, i.e., Ey accordingto the alignment of the horn and the probe. It should be notedthat the probe correction is not limited to a specific field or




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Fig. 5. Retrieved images at 60GHz for the synthetic data using the three different algorithms. (a) depicts |Ey|/|EFSy |, were EFS

y denotes the freespace reference measurement, (b) depicts |Ey|, and (c) depicts the absolute value of the scattering amplitudes s. All figures are in normalized dB-scale(20 log10(|u|/max(|u|)), where u is the parameter being plotted) with color range [−60, 0] dB.

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Fig. 6. Retrieved images at 60GHz for the synthetic data using the timereversal technique and source separation algorithm with customized colorrange. (a) depicts |Ey|/|EFS

y | were |EFSy | denotes the free space reference

measurement, and (b) depicts |Ey|. Both figures are in normalized dB-scale(20 log10(|u|/max(|u|)), where u is the parameter being plotted) with colorrange: (a) [−5, 0] dB, and (b) [−20, 0] dB.

component [33]. Rather, there exists a flexibility in which fieldand which component to extract. As mentioned in Sec. II-B,an additional planar scan with the probe rotated 90 wouldallow for the extraction of the complete vector fields, with thecost of doubling the total measurement time.

The measurement set-up is shown in Fig. 7. The DUT isan industrially manufactured composite test panel provided bySaab Aerostructures with dimension 300× 300× 3 mm3. Thepanel consists of a 2 mm thick low permittivity over-expandedNomex honeycomb core sandwiched between two 0.5 mmsheets of TenCate EX-1515, a cyanate ester quartz fabric

(a)

PNA

LNA

Stationary horn Scanning probe

DUT

(b)

Fig. 7. Photo (a), and block schematic (b) of the measurement set-up. Receiv-ing antenna is a 50 – 67GHz standard gain horn antenna, and the transmittingscanning probe is a rectangular open-ended waveguide. Surrounding absorbersare seen in black (a).

prepreg [41]. The panel was assembled using TenCate EX-1516, an adhesive developed for bonding solid, honeycomb orfoam core structures used in aircraft and space applications[41], and covered with a fluoropolymer film. The defects, in-serted into the Nomex honeycomb core, were selected in orderto investigate the detection limits of the imaging methods formetallic and dielectric objects with different shapes and sizes.A picture of the inner layers of the panel before assembly isseen in Fig. 10. Defects in the right column are metallic whilethe remaining defects consist of the same adhesive used for




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Fig. 8. Retrieved images at 60GHz for the measured data using the three different algorithms. (a) depicts |Ey|/|EFSy |, were EFS

y denotes the freespace reference measurement, (b) depicts |Ey|, and (c) depicts the absolute value of the scattering amplitudes s. All figures are in normalized dB-scale(20 log10(|u|/max(|u|)), where u is the parameter being plotted) with color range: (a) [−40, -35] dB, (b) [−20, 0] dB and (c) [−60, 0] dB.

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Fig. 9. Retrieved images at 61GHz for the measured data using the three different algorithms. (a) depicts |Ey|/|EFSy |, were EFS

y denotes the freespace reference measurement, (b) depicts |Ey|, and (c) depicts the absolute value of the scattering amplitudes s. All figures are in normalized dB-scale(20 log10(|u|/max(|u|)), where u is the parameter being plotted) with color range: (a) [−30, -25] dB, (b) [−20, 0] dB and (c) [−60, 0] dB.

Dielectric defectsConducting defects

Fig. 10. Photo (left) and schematic (right) of the panel under test as seenfrom the probe. The size of the panel is 300× 300× 3mm3.

assembling the panel. The largest dimension of all defects arebetween 2− 20 mm, i.e. ranging in size from sub-wavelengthto a few wavelengths of the illuminating signal. With thegeometrical settings presented earlier, one planar scan took 4 hto conduct, i.e., the total measurement time for retrieving thenecessary data for the different algorithms is 8 h, 4 h and 4 hfor the time reversal technique, the source separation algorithmand the CS algorithm, respectively.

Retrieved images at f = 60 GHz using the time reversaltechnique, the source separation algorithm and the CS algo-rithm can be seen in Fig. 8. Note the customized color scalefor every image, motivated by the large difference in dynamicrange achieved by the different algorithms. The image qualityobtained in the synthetic case is maintained for the measureddata. As for the synthetic case, the CS algorithm provides thebest dynamic range. The source separation algorithm and CSalgorithm provide stable images regardless of the choice offrequency, demonstrated by the similarity of the equivalentimages at f = 61 GHz shown in Figs. 9 (b) and (c). Errorsarising from the measurement set-up can be seen to affect theimage quality of Figs. 8 and 9 (a) to a larger extent, perhapsexplained by the exploitation of data from two independentmeasurements (one with DUT present and one free spacereference). Whereas the dynamic range is maintained, thecolor scale is different for the different frequencies in (a) inorder to provide the best image. The effect of the weightingcompensation introduced by the operator B2,inv can also beseen in Figs. 8 and 9 (c), as the scattering amplitudes for alldefects are of the same magnitude unlike as in (a) and (b).

IV. CONCLUSION

A complete imaging system using a planar near-field scan-ning set-up operating in the 50 – 67 GHz band has been




presented. Its potential as a complementary tool for industrialNDT diagnostic purposes has been validated through measure-ments on an industrially manufactured composite test panelusing two introduced post-processing algorithms. The sourceseparation algorithm can be regarded as a stepping-stone inthe development of the more efficient CS algorithm, yet bothalgorithms provide noticeable enhancements compared to themore conventional time reversal technique. These enhance-ments include reduction of measurement time by removal of areference measurement, increased dynamic range, and stableimage quality over a range of frequencies. All three are criticalmeasures of system performance.

It should be noted that sparsity is a relative concept in CS,and a scatterer can be sparse in some basis but not sparse withrespect to another one. The pixel basis used in this work isconvenient due to the small physical size of the defects underconsideration.

Further development of the CS algorithm is ongoing, mainlyfor reduction of measurement time while maintaining imagequality. The single frequency problem formulation provided in(13) allows for a straightforward extension to the multiple fre-quency scenario, in which data from several fixed frequenciescan be utilized in order to solve for the scattering amplitudes.Potentially, this would allow for a reduction in spatial sampleswithout compromising image quality. This implementation iscurrently in progress.

ACKNOWLEDGEMENT

The authors would like to thank Mikael Petersson at SaabAerostructures for manufacturing the test panel. The authorswould also like to thank Carl Gustafson and Iman Vakili forassisting with the measurements.

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Jakob Helander received his M.Sc. in engineeringphysics from Lund university in 2014. He is cur-rently a Ph.D. student at the Electromagnetic TheoryGroup, Department of Electrical and InformationTechnology at Lund university. He was the supervi-sor for the winning team of the 2015 IEEE Antennasand Propagation Society Student Design Contest.His research interests include antennas, inverse scat-tering problems, imaging and computational electro-magnetics, mainly hybrid MoM methods.

Andreas Ericsson received his M.Sc. degree fromLund University, Sweden, in 2013. He is currentlyworking toward his Ph.D. degree with the Electro-magnetic Theory Group at the Department of Elec-trical and Information Technology, Lund University.His research interests involves antennas, periodicstructures, radar cross section, and electromagneticscattering.

Mats Gustafsson received the M.Sc. degree in En-gineering Physics 1994, the Ph.D. degree in Electro-magnetic Theory 2000, was appointed Docent 2005,and Professor of Electromagnetic Theory 2011, allfrom Lund University, Sweden. He co-founded thecompany Phase holographic imaging AB in 2004.His research interests are in scattering and antennatheory and inverse scattering and imaging. He haswritten over 80 peer reviewed journal papers andover 100 conference papers. Prof. Gustafsson re-ceived the IEEE Schelkunoff Transactions Prize Pa-

per Award 2010 and Best Paper Awards at EuCAP 2007 and 2013. He servedas an IEEE AP-S Distinguished Lecturer for 2013-2015.

Torleif Martin (S’96–M’00–SM’08) received theM.Sc. degree in engineering physics from UppsalaUniversity, Uppsala, Sweden, and the Ph.D. de-gree in theoretical physics from Linkoping Instituteof Technology, Linkoping, Sweden, in 1989 and2001, respectively. Between 2005 and 2008 he wasa Deputy Research Director. He was involved inresearch projects at FOI including array antennas,EMC and SAR, with special interests in compu-tational electromagnetics, in particular the FDTDmethod. Since 2008, he has a position as Principal

Engineer in Survivability at Saab Aeronautics, Saab AB, Linkoping, Sweden,where he is mainly working with research and development projects. He wasappointed Adjunct Professor in 2014 at Lund University, Lund, Sweden, andreceived the 2002 ACES outstanding journal paper award.

Daniel Sjoberg received the M.Sc. degree in en-gineering physics and Ph.D. degree in engineering,electromagnetic theory from Lund University, Lund,Sweden, in 1996 and 2001, respectively. In 2001,he joined the Electromagnetic Theory Group, LundUniversity, where, in 2005, he became a Docentin electromagnetic theory. He is currently a Profes-sor and the Head of the Department of Electricaland Information Technology, Lund University. Hisresearch interests are in electromagnetic propertiesof materials, composite materials, homogenization,

periodic structures, numerical methods, radar cross section, wave propagationin complex and nonlinear media, and inverse scattering problems.

Christer Larsson received the B.Sc. degree inPhysics from Uppsala University, Uppsala, Swedenin 1981 and the Ph.D. degree in Physics from theRoyal Technical Institute, Stockholm, Sweden in1990. He was appointed Adjunct Professor in 2007and Docent in 2010, both at Lund University, Lund,Sweden. Dr. Larsson has held the position as SeniorSpecialist in Radar Signatures and is now Managerfor Radar sensors at Saab Dynamics, Linkoping,Sweden. His research interests are in radar cross sec-tion, radar imaging, compressive sensing and in the

electromagnetic properties of materials with a special focus on measurementtechniques. Dr. Larsson has published 21 peer reviewed journal papers, 1book chapter and more than 30 conference papers. He received the best paperaward at the AMTA 2015 symposium. He serves on the board of the AMTAfor 2017-2019.

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