Un metodo non iterativo per un’ampia classe di problemi inversi in elettromagnetismo Guglielmo Rubinacci Dipartimento di Ingegneria Elettrica e delle Tecnologie dell’Informazione Università degli Studi di Napoli Federico II Antonello Tamburrino Dipartimento di Ingegneria Elettrica e dell' Informazione "Maurizio Scarano" Università degli Studi di Cassino e del Lazio Meridionale ECE, Michigan State University, East Lansing, MI, USA 1

Outline  Statement of the problem  Non-Iterative Imaging Methods: a short overview  The Monotonicity based Non-Iterative Method:  Electrical Resistance Tomography  Eddy Current Testing (f.d., large skin-depth)  Eddy Current Testing (f.d., small skin-depth)  Eddy Current Testing (pulsed ECT)  Microwave imaging  Conclusions 2

Statement of the problem -Low frequency (or DC) electromagnetic imaging of the interior of conductive body; -Starting points: -commercial systems: elementary data processing capabilities; -non-linear and ill-posed inverse problem. Electrical Resistance Tomography (DC regime) Eddy Current Tomography (Quasi-static regime) V c V V c V v1v1 i1i1 v2v2 i2i2 … iMiM V c V V c V v1v1 i1i1 v2v2 vMvM iMiM i2i2... … vMvM 3

Statement of the problem V c V c V Detection (output: binary) V c V c V Sizing (output: few parameters) Pre-defined (parametric) shapes (x c, y c,  x,  y)  x  y ..  (x c, y c ) V c V V c V Imaging (output: many parameters) Arbitrary shapes and topologies V 4

Start Set k=1 and p k = initial guess Read m Compute F(p k ) and … (Forward problem) Stop rule is satisfied? Stop Update solution and increase k Visualize/process p k Y N Start Sample domain V c at points r k, k=1,…,N Read m Compute  (r k ) for k=1,…,N (indicator function) Stop Visualize/process  rkrk VcVc Overview: Iterative vs. Non-Iterative I.M. 5

IterativeNon-Iterative Computational costHighLow! A priori informationStrongLittle Input data“Few” input dataMany input data Recovered informationFull reconstructionPartial information (shape reconstruction) Underlying approximation None: full non-linear method F. Cakoni, D. Colton and P. Monk, The linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, CBMS-NSF Regional Conference Series in Applied Mathematics, Overview: Iterative vs. Non-Iterative I.M. 6

Linear Sampling Method (LSM) Factorization Method (FM) MUltiple Signal Classification (MUSIC) Monotonicity Method (MM)  Stronger mathematical foundation More difficult to be extended Small and non- interacting anomalies Provide an indicator function related to subset of the space rather than points LSM: D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), pp FM: A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), pp MUSIC: A. J. Devaney, Super-resolution processing of multi-static data using time-reversal and MUSIC, preprint in MM: A. Tamburrino, G. Rubinacci, “A new non-iterative inversion method for Electrical Resistance Tomography”, Inverse Problems, vol. 18, pp , December Overview: Iterative vs. Non-Iterative I.M. 7

Impedance Analizer PC (Inversion Algorithm) V c V Resistance Analyzer PC ( Inversion Algorithm ) V c V  i >  b v1v1 i1i1 v2v2 i2i2 vMvM iMiM Phase 2 (  =  i ) Phase 1 (  =  b ) … Electrical Resistance Tomography 8

V c V V c V v1v1 i1i1 v2v2 i2i2 vMvM iMiM ERT model and data (Currents are applied and voltages measured) The problem is to determine  from  (  ) Neumann-to-Dirichlet map 9

V c V V c V v1v1 i1i1 v2v2 i2i2 vMvM iMiM … ERT model and data 10

Monotonicity proof 11

Phase 1 Phase 2 VcVc DβDβ Phase 1 Phase 2 VcVc DβDβ DαDα D. G. Gisser, D. Isaacson and J. C. Newell, “Electric current computed tomography and eigenvalues”, SIAM J. Appl. Math., vol. 50, pp , A. Tamburrino, G. Rubinacci, “A new non-iterative inversion method for Electrical Resistance Tomography”, Inverse Problems, vol. 18, pp , December  i >  b Larger anomaly  larger ohmic losses prescribed electrodes’ currents) Negative s.d. matrix Monotonicity 12

Phase 1 Phase 2 VcVc DβDβ Phase 1 Phase 2 VcVc DβDβ DαDα B. Harrach, M. Ullrich, Monotonicity-based shape reconstruction in electrical impedance tomography SIAM J. Appl. Math., Vol. 45, No. 6, pp. 3382–3403,  i >  b Monotonicity 13

Phase 1 Phase 2 VcVc V Phase 1 Phase 2 VcVc kk A. Tamburrino, G. Rubinacci, “A new non-iterative inversion method for Electrical Resistance Tomography”, Inverse Problems, vol. 18, pp , December Inversion: underlying idea 14

kk 11 22 … NN VcVc × × × Notes 1.the reconstruction may be larger than V; 2.Low computational cost; 3.Requires the pre-computation of R k Inversion algorithm Take as estimate of V the union of those  k such that Basic Inversion Method 15

| | +1 The Noise 16

Basic inversion algorithm revisited: 1.for any k compute s k 2.take as estimate of V the union of those  k such that s k =1 Complete inversion algorithm: 1.for any k compute s k 2.form the estimate V  defined as union of those  k such that s k >  sksk  Proper threshold… Inversion Method 17

- Insulating objects (size 20cm×20cm ×40cm) in a pipeline (diameter 200cm) filled by tap water (conductivity of mS cm -1 at 20°C); - The point-like electrodes are equally spaced along the circumferential direction (15 electrodes) and along the axial direction (5 electrodes spaced of 40 cm) for a total of 75 electrodes. F. Calvano, G. Rubinacci and A. Tamburrino, “Fast Methods for Shape Reconstruction in Electrical Resistance Tomography”, NDT & E International, vol. 46, pp , DOI information: /j.ndteint , March 2012 Comparison with others N.I. methods #1 18

Monotonicity Factorization Music 0.1% noise 1% noise Comparison with others N.I. methods #1 19

Probe Conducting specimen Anomaly Eddy Current Tomography 20

Parabolic PDE (time-harmonic operation) Mathematical model 21

Impedance Analizer PC (Inversion Algorithm) ECT coils V c V Impedance Analyzer (self and mutual impedances) Personal Computer (imaging algorithm) c Anomaly (  =  i ) i1i1 i2i2 iMiM Conductor (  =  b )  i >  b Problem Definition 22

Phase 1 VcVc # l # k Phase 2 V Time-harmonic operations (frequency is prescribed) Eddy Current Data Anomaly (  =  i ) Conductor (  =  b )  i >  b 23

Low frequency expansion 24

Phase 1 Phase 2 VcVc DαDα Phase 1 Phase 2 VcVc DβDβ A. Tamburrino and G. Rubinacci, “Fast Methods for Quantitative Eddy-Current Tomography of Conductive Materials”, IEEE Trans. Magn., vol. 42, no. 8, pp , Larger anomaly  smaller Ohmic losses prescribed coil currents) (small  ) Large skin-depth regime 25

 Specimen: 33mm  18mm  2mm  Material (Al):  b =5.74   m  Contrast:  i /  b =10  Frequencies: 250Hz, 800Hz, 1200Hz  skin-depth values: 7.63mm, 4.27mm, 3.48mm  Coils system: 23 in a closest-packed array  Lift-off: 0.5mm  Measurements noise: multiplicative (  1%)  2 nd layer discretization: 22  12 elements Numerical examples 26

Noisy measurements Pre-computed and stored) Sign index spatial distribution 27

Test defects 28

T map =52ms Real Time Imaging 29

T map =52ms, T opt =434s The 1/(1-x) transformation 30

(I) (II) (III) Indices distribution before the nonlinear transformation ? 31

(I) (II) (III) Indices distribution after the nonlinear transformation 32

Test defects Reconstructions 33

External Coil Internal diameter=5mm, external diameter=10.5mm, height=6.5mm, number of turns=700. Internal Coil internal diameter=1mm, external diameter=4mm, height=3mm, number of turns=180. The excitation frequency is 20kHz Test case: printed circuit board Experimental setup 34

Top Bottom A. Tamburrino, F. Calvano, S. Ventre, G. Rubinacci, “Non-iterative imaging method for experimental data inversion in eddy current tomography”, NDT&E International 47 (April, 2012) 26–34 The Specimen (PCB) 35

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DβDβ DαDα A. Tamburrino, S. Ventre, G. Rubinacci, “Recent developments of a Monotonicity Imaging Method for Magnetic Induction Tomography in the small skin-depth regime” Inverse Problems v. 26, , July By increasing the size of an anomaly the magnetic energy increases prescribed coil currents) (large  ) Small skin-depth regime 37

Free response and natural modes: The set of time constants is discrete The corresponding eigen-functions form a complete basis (Z. Zhou e A. Tamburrino, private communication). Conducting specimen Anomaly Natural modes in ECT 38

Phase 1 Phase 2 VcVc DβDβ Phase 1 Phase 2 VcVc DαDα DαDα A. Tamburrino, Z. Su, N. Lei, S. Paul, L. Udpa and S. Udpa, ‘’The Monotonicity Imaging Method For Time-domain (Pulsed) Eddy Current Imaging’’, presented at the 19th Int. Workshop on Electromagnetic Nondestructive Evaluation, China, June 25-28,  i >  b Larger anomaly  smaller time-constant Natural modes in ECT By increasing the electrical resistivity the natural modes decay faster 39

Test Case 40

Difference of time constants (sorted in descending order): Structure configurations Void size (L×W×H) 2×2×1(left) 4×4×1(right) 4×4×1(left) 6×6×1(right) 2×2×1(left) 2×2×2(right) By increasing the size of an anomaly the time constants decrease. Evidence of monotonicity 41

Difference of time constants (sorted in descending order): Structure configurations Void size (L×W×H) 4×4×1(left) 4×4×2(right) 4×4×2(left) 6×6×1(right) 2×2×2(left) 4×4×1(right) If anomalies do not include each other, some time constants increase and some decrease. Evidence of monotonicity 42

Induced current density: Induced currents concentrating on the surface (corresponding to eigenvalues 1, 2, 3) Induced currents concentrating in the center (corresponding to eigenvalues 6, 7, 8) Natural modes (eigenvectors) Spatial localization of the modes 43

Noise level 0.1%Real defectReconstruction Surface defect Inner defect Numerical Example (3D) 44

Phase 1 Phase 2 VcVc Microwave Hyperbolic PDE Microwave Tomography 45

Transmission Eigenvalues 46 The transmission eigenvalue problem is related to nonscattering incident fields

Phase 1 Phase 2 VcVc DβDβ Phase 1 Phase 2 VcVc DβDβ DαDα F. Cakoni, H. Haddar, “Transmission Eigenvalues in Inverse Scattering Theory”, inside Out II, MSRI Publications, Vol. 60, 2012 n i <n b Monotonicity 47

Numerical Example (2D) 48

Elliptic PDEsParabolic PDEsHyperbolic PDEs OkLF ok, HF okOk (imaging alg.?) Elliptic PDEsParabolic PDEsHyperbolic PDEs N/AOk? Frequency domain f Static Quasi-static Wave prop. Time domain Overview 49

Conclusions Monotonicity can be exploited for effective real time imaging algorithms in several different physical setting (Elliptic, Parabolic, and Hyperbolic PDEs) The theoretical frame is valid without any approximation for a finite number of measurements, differently from other imaging approaches requiring an infinite and continuous set of measurements. Open questions Parabolic and Hyperbolic case Multiple anomalies Conclusions and future developments 50

S. Ventre DIEI, Università di Cassino e del Lazio Meridionale, 03043, Cassino, Italy (Electrical Resistance Tomography, Eddy Current Testing in the frequency domain) Z. Su, Z. Zhou L. Udpa and S. Udpa Nondestructive Evaluation Laboratory, Michigan State University, East Lansing, Michigan, 48824, USA (Eddy current testing in the time-domain) D. Colton, F. Cakoni, P. Monk Department of Mathematical Sciences, University of Delaware, Newark, Delaware , USA (Microwave tomography) Work supported in part by the European Commission, grant agreement no , FP7. Acknowledgements 51

 F. Cakoni, D. Colton and P. Monk, The linear Sampling Method in Inverse Electromagnetic Scattering, SIAM, CBMS-NSF Regional Conference Series in Applied Mathematics,  A. Kirsch, Characterization of the shape of a scattering obstacle using the spectral data of the far field operator, Inverse Problems, 14 (1998), pp  A. J. Devaney, Super-resolution processing of multi-static data using time-reversal and MUSIC, preprint in  A. Tamburrino, G. Rubinacci, “A new non-iterative inversion method for Electrical Resistance Tomography”, Inverse Problems, vol. 18, pp , December  A. Tamburrino, G. Rubinacci “Fast methods for quantitative eddy current tomography of conductive materials”, IEEE Trans. on Magnetics, vol. 42, no. 8, pp , August 2006  A. Tamburrino, S. Ventre, G. Rubinacci, “Recent developments of a Monotonicity Imaging Method for Magnetic Induction Tomography in the small skin-depth regime” Inverse Problems v. 26, (21pp), July 2010  A. Tamburrino, F. Calvano, S. Ventre, G. Rubinacci, “Non-iterative imaging method for experimental data inversion in eddy current tomography”, NDT&E International 47 (April, 2012) 26–34, doi: /j.ndteint  G. Rubinacci, A. Tamburrino, S. Ventre, “Numerical optimization and regularization of a fast eddy current imaging method”, IEEE Trans. on Magnetics, vol. 42, no. 4, pp ,  A. Tamburrino, “Monotonicity based imaging methods for elliptic and parabolic inverse problems”, Jour. of Inverse and Ill-posed Problems, vol. 14, n. 6, pp , September 2006  M. Soleimani and A. Tamburrino, “Shape reconstruction in magnetic induction tomography using multifrequency data”, Int. Jour. of Information and System Sciences, vol. 2, no. 3, pp , 2006  M. de Magistris, M. Morozov, G. Rubinacci, A. Tamburrino, S. Ventre, “A monotonicity based approach for electromagnetic inspection of concrete rebars ”, COMPEL, vol. 26, no. 2, pp , May 2007  G. Rubinacci, A. Tamburrino, S. Ventre, “Concrete rebars inspection by eddy current testing”, International Journal of Applied Electromagnetic and Mechanics, vol. 25, nos. 1-4, pp , 2007  G. Rubinacci, A. Tamburrino, S. Ventre, “Eddy current imaging of surface breaking defects by using monotonicity based methods”, ACES Journal, vol. 23, no. 1, pp , March 2008  F. Calvano, G. Rubinacci and A. Tamburrino, “Fast Methods for Shape Reconstruction in Electrical Resistance Tomography”, NDT & E International, vol. 46, pp , March  A. Tamburrino, Z. Su, N. Lei, L. Udpa and S. Udpa, ‘’The Monotonicity Imaging Method For Time-domain (Pulsed) Eddy Current Imaging’’, presented at the 19th Int. Workshop on Electromagnetic Nondestructive Evaluation, China, June 25-28, References 52

 A. Tamburrino, Z. Su, N. Lei, L. Udpa and S. Udpa, “The Monotonicity Imaging Method for PECT”, accepted for publication in: Electromagnetic Nondestructive Evaluation (XVIII), AMSTERDAM, (NETHERLANDS) IOS PRESS,  Z. Su, A. Tamburrino, S. Ventre, L. Udpa, and S. Udpa, “Time Domain Monotonicity Based Inversion Method for Eddy Current Tomography”, 31st International Review of Progress in Applied Computational Electromagnetics (ACES 2015), Williamsburg (Virginia, USA), pp , March 22-26,  A. Tamburrino, L. Barbato, D. Colton, P. Monk, “Imaging of Dielectric Objects Via Monotonicity of the Transmission Eigenvalues”, accepted for presentation at “The 12th International Conference on Mathematical and Numerical Aspects of Wave Propagation”, (Karlsruhe, Germany) July 20-24,  Zhiyi Su, Antonello Tamburrino, Salvatore Ventre, Lalita Udpa and Satish Udpa, “Monotonicity of time-constants and real-time imaging in eddy current testing”, accepted for presentation at the 42st Annual Review of Progress in Quantitative Nondestructive Evaluation Conference, Minneapolis, (MN, USA), July 25-31,  Zhiyi Su, Antonello Tamburrino, Salvatore Ventre, Lalita Udpa, and Satish Udpa, “Time-domain monotonicity based inversion method for ECT”, presented at the 2015 Inverse Problems Symposium, East Lansing (MI, USA), May 31st-June 2nd,

Music methodMonotonicity methodFactorization method 0.1% noise 1% noise Void in Aluminum background Comparison with others N.I. methods #2 54

Integral formulation: ECT data: ( is the voltage induced in the k-th coil by the eddy currents only) Numerical modeling 55

NNNN NMNM Sparse Discretization: J=  T, Only the conductive domain needs to be discretized. (N k s are edge-element based shape functions) Galerkin’s method:  Numerical modeling 56

Proof:  Key property of 57

EFIE Problem definition 58

E0E0 EsEs   + + Proper conditions at infinity E=E 0 +E s Problem definition 59

for “small”  Nature of the low-frequency breakdown

Measurements in time-domain: Measurements in frequency-domain: Time constant corresponds to a pole for the Laplace transform: Measure of the time constants 61

where charges solenoidal currents Loop-star representation

Invertible matrices ! Numerical model

Scaling for “small”  Invertible matrices

Loop-star shape functions: requirements  

Loop-star shape functions: construction Region 1 Free space

Loop-star shape functions: uniqueness

Find linearly independent loop shape functions; find linearly independent star shape functions that are also linearly independent w.r.t. the loop shape functions. loop shape functions: well assessed fully automated procedure star shape functions: new fully automated procedure 

Loop-star shape functions: uniqueness The maximal number of linearly independent star shape functions: F-1 The number of boundary co-tree edges: (Euler formula: N-E+F=2-2p) E-N+1=F-1+2p Automatic procedure based on the Gauss elimination to detect the set of F-1 linearly independent star shape functions