×

Regularized inversion of a distributed point source model for the reconstruction of defects in eddy current imaging. (English) Zbl 1360.65163

Summary: Purpose{ } - The inverse problem in the eddy current (EC) imaging of metallic parts is an ill-posed problem. The purpose of the paper is to compare the performances of regularized algorithms to estimate the 3D geometry of a surface breaking defect.{ }Design/methodology/approach{ } - The forward problem is solved using a mesh-free semi-analytical model, the distributed point source method, which allows EC data to be simulated according to the shape of the considered defect. The inverse problem is solved using two regularization methods, namely the Tikhonov (l2) and the 3D total variation (tv) methods, implemented with first- and second-order algorithms. The inversion performances were evaluated in terms of both mean square error (MSE) and computation time, while considering additive white and colored noise, respectively, standing for acquisition errors and model errors.{ }Findings{ } - In presence of colored noise, the authors found out that first- and second-order methods provide approximately the same result according to the SEs obtained while estimating the defect voxels. Nevertheless, in comparison with (l2), the (tv) regularization was proved to decrease the MSE by 10 voxels, at the cost of less than twice the computational effort.{ }Originality/value{ } - In this paper, an easy to implement mesh-free model, based on virtual defect current sources, was used to generated EC data relative to a defect positioned at the surface of a metallic part. A 3D total variation regularization approach was used in combination with the proposed model, which appears to be well suited to the reconstruction of volumic defects.

MSC:

65J22 Numerical solution to inverse problems in abstract spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aujol, J.-F. (2009), ”Some first-order algorithms for total variation based image restoration”, Journal Math Imaging Vision, Vol. 34, pp. 307-27. , · Zbl 1287.94012 · doi:10.1108/03321641111168093
[2] Bect, J., Blanc-Féraud, L., Aubert, G. and Chambolle, A. (2004), ”A l1-unified variational framework for image restoration”, Proceedings of the 8th European Conference on Computer Vision, Prague, Czech Republic, Part IV (Lecture Notes in Computer Science 3024). · Zbl 1098.68728
[3] Boyd, S. and Vandenberghe, L. (2004), Convex Optimization, Cambridge University Press, Cambridge. · Zbl 1058.90049
[4] Chambolle, A. (2004), ”An algorithm for total variation minimization and applications”, Journal of Mathematical Imaging and Vision, Vol. 20 No. 1, pp. 89-97. · Zbl 1366.94048 · doi:10.1108/03321641111168093
[5] Ciarlet, P.G. (1990), ”Introduction analyse numérique matricielle et optimisation”, Collection Mathématique Appliquée Pour la Maitrise, Masson, Paris.
[6] Combettes, P.L., DŨng, D. and VŨ, B.C. (2010), ”Dualization of signal recovery problems”, Set-valued and Variational Analysis, Vol. 18, pp. 373-404. · Zbl 1229.90123 · doi:10.1108/03321641111168093
[7] Hansen, P.C. (1994), ”Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems”, Numerical Algorithms, Vol. 6, pp. 1-35. · Zbl 0789.65029 · doi:10.1108/03321641111168093
[8] Idier, J. (2001), Approche Bayésienne Pour les Problèmes Inverses, Traité IC2, Série Traitement du Signal et de l’image, Hermès, Paris.
[9] Nikolova, M. and Mohammad-Djafari, A. (1996), ”Eddy current tomography using a binary Markov model”, Signal Processing, Vol. 9, pp. 119-32. , · Zbl 0875.94062 · doi:10.1108/03321641111168093
[10] Placko, D. and Kundu, T. (2007), DPSM for Modeling Engineering Problems, Wiley, Hoboken, NJ.
[11] Thomas, V., Joubert, P.-Y. and Vourch, E. (2009), ”Comparative study of sensing elements for the design of an eddy current probe dedicated to the imaging of aeronautical fastener holes”, Sensor Letters, Vol. 7 No. 3, pp. 460-5. , · Zbl 1360.65163 · doi:10.1108/03321641111168093
[12] Thomas, V., Joubert, P.-Y., Vourch, E. and Placko, D. (2010a), ”A novel modeling of surface breaking defects for eddy current quantitative imaging”, Proceedings of IEEE Sensors Applications Symposium 2010, Limerick, Ireland.
[13] Thomas, V., Joubert, P.-Y., Vourch, E. and Placko, D. (2010b), ”Distributed point source modeling for the multi-frequency eddy current imaging of surface breaking defects”, Proceedings of the 14th International IGTE Symposium, Graz, Austria.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.