IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007 1857 Image Reconstruction of Defects in Metallic Plates Using a Multifrequency Detector System and a Discrete Geometric Approach E. Cardelli , A. Faba , R. Specogna , and F. Trevisan Department of Industrial Engineering, Perugia University, 06124 Perugia, Italy Polo Scientifico Didattico di Terni, 05100 Terni, Italy Center for Electric and Magnetic Applied Research, 06124 Perugia, Italy Department of Ingegneria Elettrica, University of Udine, I-33100 Udine, Italy We present an inversion procedure for the image reconstruction of defects in metallic plates, using a multifrequency eddy-current system. The solution of the eddy-current forward problem is achieved by means of a discrete geometric approach, while the inverse problem is resolved with an iterative linearization algorithm based on sensitivity data. In particular, we propose a suitable measurement point on the region under test using a probe coil exited by means a multifrequency signal, in order to improve the amount of usable data and the accuracy of the inverse procedure. Index Terms—Discrete geometric approach, inverse problems, multifrequency eddy currents, sensitivity analysis. I. INTRODUCTION E DDY-current inspections is one of the most interesting approaches in electromagnetic nondestructive evaluation of metallic materials. The presence of a defect produces an impedance variation of the probe coil and, therefore, in the voltage or current at the coil leads, that we can use to locate the defect and also to estimate its shape and depth, so that we can verify the integrity of the plate [1]–[4]. To this aim, we use 3-D image reconstruction algorithms based on an inversion procedure elaborating experimental data together with solu- tions of proper forward problems. The results presented in this paper have been obtained using simulated data instead of the experimental ones. The direct model is based on a discrete geo- metric approach for electromagnetic field, by means of integral quantities associated with the oriented geometric elements of a pair of interlocked cell complexes [5]–[13]. II. MULTIFREQUENCIES DETECTOR SYSTEM In general, the 3-D image reconstruction of defects in a ma- terial has been derived by means of suitable detection systems consisting of several probe coils [14], [15]. Indeed the multi- probe coil systems provide many data to be considered in the sensitivity analysis. Nevertheless, they cannot be enough to as- sure an accurate resolution of the inverse problem which re- mains highly “ill-posed.” Moreover, in most of practical ap- plications, only a finite portion of the sample under inspection is available to be tested. In this case, increasing the number of probe coil positions could lead to redundant data that are useless in the inversion procedure. Therefore, in this work, we present Digital Object Identifier 10.1109/TMAG.2007.892525 a multifrequency excitation for the probe coil, in order to re- duce the “ill-posed” characteristic of the problem and to im- prove the inversion procedure, in terms of capability to distin- guish the depth of the defect. In particular, in this paper, we show some results obtained using two values of frequencies 500 Hz and 1 kHz, associated with two different depth of inspection, and with 13 different positions of the probe coil on the metallic plate. For each frequency, we impose a unitary excitation cur- rent and so we calculate the variation of the voltage at the probe coil leads. III. FORWARD PROBLEM We introduce in the domain of interest (containing the conductive region ) a pair of interlocked cell complexes: the primal , based on simplexes, and its barycentric dual , [5]–[8]. The mutual interconnections between the cell complex are described by the incidence matrices: between edges and nodes, between faces and edges, and between volumes and faces. The matrices and describe the mutual interconnections of . The arrays of degree of freedoms can be associated univocally to the elements of or . We have that is the array of voltages on primal edges, is the array of fluxes on primal faces, is the array of magnetic voltages on dual edges and is the array of currents on dual faces. The physical laws of the eddy-current problem, can be written exactly, as follows: (Gauss’ law); (Ampère’s law); (Faraday’s law); and (continuity law). In addition, the discrete magnetic and Ohm’s constitutive equations are, respectively, , where are two square matrices ( and , respectively). These matrices can be derived in a geometric way as described in [8]–[13]. To solve the discrete eddy-current problem [10], [11], we search for an array of scalar potential values on primal nodes of the conducting region 0018-9464/$25.00 © 2007 IEEE