De Zaeytijd, Jürgen

Franchois, Ann

MAES, T

English

Ghent University Academic Bibliography

2-D real-time quantitative microwave imaging ofreinforcement bars: simulations and experimentsJ. De Zaeytijd ∗ T.Maes ∗ A. Franchois †Abstract — This paper treats the non-destructivelocalisation of reinforcement bars. It introduces aquantitative algorithm using a Gauss-Newton-typeoptimization to determine the radius and positionof one reinforcement bar in a homogeneous mediumfrom data collected with a bi-static linear measure-ment setup. The use of the Linear Sampling methodas a preprocessing step and a fast analytical 2Dmodel for the direct problem results in a robustand quasi-realtime microwave imaging system. Re-constructions from both simulated and experimentaldata are presented.1 IntroductionFor repair and renovation of concrete structures, in-formation on the location and geometry of the re-inforcement bars is a necessity. The position of therebars is important since most processes of corro-sion are directly related to an insufficient concretecover or incorrect placement of the reinforcement.The diameter of the bars determines the bearing ca-pacity of concrete elements. There exists a stronginterest in non-destructive techniques to determinethese important parameters. Recently a near-fieldSynthetic Aperture Radar (SAR) technique was de-veloped in our department [1]. This technique,however, produces only qualitative images, that al-low for the detection and localisation of the rebars,but with a limited accuracy in terms of position,and with almost no information about the diame-ter of the bars. Moreover, image artefacts are of-ten present. Its main advantage is its speed, whichmakes it suited for real-time applications. In thispaper we present a quantitative microwave inver-sion method for the detection of metal bars in freespace. This is the first step towards a completelyquantitative imaging algorithm for rebars in con-crete. The data are collected with a bi-static linearmeasurement setup which measures the scatteredfield in reflection only. For the inversion, a Gauss-Newton minimization algorithm [2] is applied to aleast squares datafit cost function. The evaluationof this cost function and its derivatives is very effi-cient due to the use of an analytical forward modelthat explicitely uses the facts that the bars have a∗INTEC, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium, e-mail:jurgen.dezaeytijd@intec.ugent.be†INTEC-IMEC, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgiumcircular cross-section and can be modeled as per-fectly electrically conducting (PEC) at microwavefrequencies. To avoid the pitfall of multiple localminima of the cost function, the Linear SamplingMethod (LSM) [3, 4] is used as a preprocessing stepthat finds a good initial estimate for the optimiza-tion algorithm. The resulting method is robust andprovides quasi-realtime reconstructions of the rein-forcement bars. It is validated on both simulatedand measured scattering data.2 Theory and algorithm2.1 Direct problemWe consider a Perfectly Electrically Conducting(PEC) cylinder with circulair cross section placedin a homogeneous background medium. The ge-ometry of sources and scatterer is assumed to bez-invariant, so a 2D approximation can be used.This assumption is inspired by the vertical positionof the reinforcement bar and the use of only TM-excitations in the measurements. Scattering by thissimple configuration can be treated analytically [5]allowing for a fast simulation of the scattered fieldand its derivatives.The major difficulty is presented by the linearmeasurement setup, which requires the incorpora-tion of the antenna pattern in the calculations, be-cause the scatterer is not always in the main lobeof the antenna. Without incorporation of the an-tenna pattern in the direct problem, the simulatedscattered field will never match the measured data.After experimental determination of amplitude andphase of the pattern it can be incorporated in theanalytical calculations.2.2 Inverse problemIt is known that the inverse scattering problem isill-posed, i.e. the existence, uniqueness and stabil-ity of the solution are not simultaneously guaran-teed. To address the nonexistence, caused by noiseand modeling errors, the solution is redefined as theminimizer of a cost function. A least squares costfunction is used:Q(x0, y0, a) = ‖E(x0, y0, a)−Em)‖2 (1)1-4244-0767-2/07/$20.00 ©2007 IEEEwhere E(x0, y0, a) is a vector that contains the sim-ulated field scattered from a bar (position (x0, y0)of the center, radius a) and Em contains the mea-sured field data. This cost function proved to havemultiple local minima which makes it difficult for alocal optimization method to converge succesfully.2.2.1 Linear SamplingA good initial guess can avoid the problem of mul-tiple minima. To this end the Linear SamplingMethod (LSM) is applied to the measured data[3, 4]. The LSM solves a linear ill-posed problem todetermine the support of the scatterer. To handlethe ill-posedness of the linear problem a Tichonovregularization is employed. The image acquired bythe method is then used to estimate the positionand radius of the rebar.2.2.2 Gauss-Newton methodSubsequently the Gauss-Newton method is used tofind the minimizer of the costfunction (1). This lo-cal method will converge to the correct global min-imum if it is started from the initial LSM estimate.The Gauss-Newton algorithm adjusts the vector ofreconstruction variables u = [x0, y0, a] in each iter-ation with the update ∆u = [∆x0,∆y0,∆a] calcu-lated as:∆u = −<[JHJ]−1<[JH∆E] (2)with J the Jacobian matrix of the scattered fields Eand ∆E the vector with ∆E = E−Em. In practicewe have found it advisable to perform the optimiza-tion in two steps: first the radius is kept fixed at theLSM value, while the x0- and y0-values are updatedand in a second step all parameters are updated si-multaneously. This seems to prevent problems withnegative values for the radius of the bar and avoidsthe use of a constrained optimization.3 Experimental setupTo perform the measurements a linear, fully auto-mated positioning system is used. Attached on twoseparate rails the transmitting and receiving anten-nas move along the x-axis. The antennas are posi-tioned at different heights to prevent collision dur-ing movement. In this setup exponentially taperedhorn antennas are used in the frequency range of9-11Ghz. After calibration of the antenna-cablesand alignment of the antennas, a rebar is placedin front of the antennas. Two data sets are col-lected, one without rebar (free space measurement)and one with the bar in place (total field measure-ment). For every measurement, the transmitter isFigure 1: Overview of simulation parameterssuccessively placed in N equidistant positions andfor each transmitter position, the scattered field ismeasured in the same N points by the receiver (seeFigure 1).4 CalibrationIn theory the scattered field is obtained by sub-tracting the free space data from the total fielddata. Due to a variety of factors the measurementsare not completely reproducible and the crosstalkbetween transmitter and receivers is not entirelyeliminated. A rigorous calibration method utiliz-ing known theoretical properties of the scatteredfield is applied on the data. This method removesthe crosstalk peaks without any a priori informa-tion on the bar. The key problem still remainingis normalizing the measurements to the level of thesimulated values. Due to a lack of reproducibilitya method using a reference measurement cannot beused at this time.5 Results5.1 Simulated dataTo test the performance of the proposed method,we first conducted a parameter study for follow-ing parameters: the position of the bar (x0, y0) ona test grid, the distance D between the antennaarray and the center of this grid, the radius a ofthe bar and the number of antenna-positions N(see Figure 1). The antennas were always spaced2 cm apart. The simulations were performed foromnidirectional antennas (line sources) and wherecorrupted with Gaussian noise, corresponding to aSignal to Noise Ration (SNR) of 50 dB. The re-construction error is defined as ‖u − u0‖, where uand u0 are the reconstructed and actual parametervectors respectively. It was found that the methoddoes not converge optimally in certain situations.One factor is the influence of the minimal angleθmin under which the bar is seen from the anten-nas (Figure 1). This angle should be smaller thenapproximately 60 degrees to keep enough depth res-olution in the measurements. This can be achievedby increasing N or decreasing D. It was furthershown that the Linear Sampling grid has to includethe rebar completely. Placing the bar at the edgesof the grid results in faulty intial estimates and canlead to convergence in a wrong local minimum. Themean error over all test cases that comply with therestrictions above was only 0.07 mm. The mean ex-ecution time was 13 s (on a 2.4Ghz processor) andthe mean number of iterations was 7.To investigate the influence of measurementnoise, the following two test cases were selected :N = 51, D = 0.8 m, a = 6 mm, position 1 =(0, 0) and position 2 = (0.15, 0.05). The SNR val-ues were varied from 15 to 50 dB. Figure 2 showsthe reconstruction error as a function of the SNR.In the worst case scenario (test case 1, SNR = 15dB) we found an error = 0.5 mm, 7 iterations andan execution time of 16 s. It can be concluded thathigh noise levels do not significantly affect the min-imizer of the cost function (although the minimalvalue is altered) and that the LSM can handle highnoise levels.To investigate the influence of the antenna pat-tern on the performance of the algorithm, the sametwo test cases were selected and again the recon-struction error was calculated for different SNR val-ues (Figure 3). By adding the pattern, executiontime rises drastically due to a more complex ana-lytical calculation. The pattern narrows the mea-surement angle and by doing so lowers the depthresolution. The maximum error (test case 2, SNR= 15 dB) is 5.8 mm with 41 iterations and executiontime of 206 s, but generally, for SNR values largerthan 20 dB, it can be concluded that the algorithmyields mm-precision and has execution times in theorder of minutes.5.2 Experimental dataThe algorithm was also applied to real measure-ments taken with our setup. After calibration wewere able to succesfully find a solution. The rebarwith radius a = 6 mm was placed in the vincin-ity of the origin (0, 0) of the grid at D = 1 m andwith N = 51. The algorithm found a position of(−0.0007, 0.0097) and a radius of 6.9 mm. The es-Figure 2: Reconstruction error vs. SNR withoutradiation pattern.Figure 3: Reconstruction error vs. SNR with radi-ation pattern.timated SNR of the measured data was 15 dB. Sincethe bar was positioned manually without great pre-cision, especially in the y-direction, this result canbe called satisfactory.6 ConclusionsTo solve the 2D inverse scattering problem of find-ing a reinforcement bar in a homogeneous space,an algorithm, consisting of a Linear Sampling pre-processing step and an iterative Gauss-Newton op-timization, was developed. A parameter studyshowed that mm-precision can be attained for awide range of configurations with execution timesin the order of minutes. It has also been shownthat the method can be applied to experimentaldata with good outcome.7 AcknowledgmentsJu¨rgen De Zaeytijd is a Research Assistant of theFund for Scientific Research - Flanders.References[1] A. Franchois, P. Lewyllie, L. Taerwe, “2-D near-field SAR for non-destructive testing of rebarsin a concrete wall“, Int. J. Appl. Electrom.Mech., Vol. 19, No. 1-4, pp. 333-338, 2004[2] J. E. Dennis Jr., Robert B. Schnabel, Numeri-cal methods for unconstrained optimization andnonlinear equations, Prentice Hall, New Jersey,1983[3] D. Colton, H. Haddar, M. Piana, “The lin-ear sampling method in inverse electromagneticscattering theory”, Inverse Problems, Vol. 19,No. 6, pp. S105-S137, 2003[4] R. Aramini, M. Brinone, M. Piana, “Applica-tions of a No-Sampling approach to the LinearSampling Method”,Proceedings of ACES 2007,Verona, Italy, pp. 1947 - 1954, 2007[5] J. Van Bladel, Electromagnetic fields, Springer-Verlag, New York, 1985 

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