Experimental validation of a fast non-iterative imaging algorithm

for eddy current tomography 

Flavio Calvano1, Guglielmo Rubinacci1, Antonello Tamburrino2 and Salvatore Ventre2

 1 Ass. EURATOM/ENEA/CREATE, DIEL, Università di Napoli Federico II, Italy

2 Ass. EURATOM/ENEA/CREATE, DAEIMI, Università di Cassino, Italy

Eddy Current TomographyEddy Current Tomography

Probe

Conducting specimen

Anomaly

H0

Hr J

pick-up coil

excitation & pick-up

coil

a)

ImpedanceAnalizer

PC(InversionAlgorithm)

ECT coils

Vc

V

ImpedanceAnalyzer

(auto and mutualimpedances)

PersonalComputer

(imaging algorithm)

c

Anomaly (=i)

i1

i2

iM

Conductor (=b)

i>b

Problem DefinitionProblem Definition

Phase 1 Vc

coillI

coilkv

# l

# k

Phase 2

V

lj

Icoill

coilk

klcoilcoiljI

vZ

0

Eddy Current DataEddy Current Data

Key quantity for the inversion method

Low frequency expansionsLow frequency expansions

Matrix of the mutual impedances between coils

5)3(3

0

4)2(20

5)5(5)4(4)3(3)2(200

0with

O

O

Ojjj

coil

coil

coil

PLX

PRR

PPPPLRZ

Phase 1

Phase 2

Vc

D

Phase 1Phase 2 Vc

D

22 PP cVDD

MonotonicityMonotonicity

A. Tamburrino and G. Rubinacci, “Fast Methods for Quantitative Eddy-Current Tomography of Conductive Materials”, IEEE Trans. Magn., vol. 42, no. 8, pp. 2017-2028, 2006.

definite-semi positive is 2222 PPPP

DD false 22 PP

... for 22 PP VD

Phase 1Phase 2 Vc

V

Vkk false 22 PP

Phase 1

Phase 2Vc

k

22 and ... kkD PP

Inversion: underlying ideaInversion: underlying idea

Vkk false 22 PP

V

Inversion: underlying ideaInversion: underlying idea

Basic inversion algorithm:

Take as estimate of V the union ofthose k such that 0PP )2()2(

k

free error not 2P

corrupted areof seigenvalue The ~~ 2)2( PP k

The test for k is no longer valid !

The NoiseThe Noise

The Sign IndexThe Sign Index

1,1ˆ

1,

1,

M

jjk

M

jjk

ks

At each k we associate

jk , is the j-th eigenvalue of 22 ~~ PP k

Experimental setupExperimental setup

External CoilInternal diameter=5mm, external diameter=10.5mm, height=6.5mm, number of turns=700.

Internal Coilinternal diameter=1mm, external diameter=4mm, height=3mm, number of turns=180.

The excitation frequency is 20kHz

Benckmark: printed circuit board

ResultsResultsReconstructed Map

Region under test measurements

Test domain measurements 2kP 2P

ResultsResults

Reconstructed Map

ResultsResults

Reconstructed Map with top test domains Reconstructed Map with the bottom test domains

Top view Bottom view (scanned from the top view)

2.4252 0.2171-0.2098- 22.3422

kP

Top

Bottom

2.399 0.104-0.100- 21.5852

kP

Estimated Noise level : 50 m

ResultsResultsTop view Bottom view (scanned from the top view)

Reconstructed Map with top test domains Reconstructed Map with the bottom test domains

CONCLUSIONSCONCLUSIONS

•A fast inversion method for inverting eddy-current testing data has been applied to the identification of the shape of inclusions in a conductor by eddy current tomography.

•The eddy-current data consists of the variation of the impedance matrix using an a-priori designed with numerical simulation array of coils to scan the specimen under test.

•The second-order moment P(2) accounts for the resistive contribution to the changes of the impedance matrix occurring at relatively low frequencies.

•A direct imaging algorithm based on monotonicity principle is available that allows real-time imaging on directly measured experimental data.

REFERENCESREFERENCES

•A. Tamburrino and G. Rubinacci, “A new non-iterative inversion method for electrical impedance tomography”, Inverse Problems, pp. 1809–1829, 2002.

•A. Tamburrino and G. Rubinacci, “Fast Methods for Quantitative Eddy-Current Tomography of Conductive Materials”, IEEE Trans. Magn., vol. 42, no. 8, pp. 2017-2028, 2006.

•A. Tamburrino, S. Ventre, G. Rubinacci, “Recent developments of a Monotonicity Imaging Method for Magnetic Induction Tomography” accepted for publication on Inverse Problems.

•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. 46-52, 2008.