Nonlinear Fourier analysis for discontinuous conductivities: computational results. (English) Zbl 1349.78047
Summary: Two reconstruction methods of Electrical Impedance Tomography (EIT) are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics (CGO) solutions to D-bar equations involving the global uniqueness proofs for Calderón problem exposed in [A. I. Nachman, Ann. Math. (2) 143, No. 1, 71–96 (1996; Zbl 0857.35135); the first two author, Ann. Math. (2) 163, No. 1, 265–299 (2006; Zbl 1111.35004)]: the Astala-Päivärinta theory-based low-pass transport matrix method implemented in [the first author et al., Inverse Probl. Imaging 5, No. 3, 531–549 (2011; Zbl 1237.78014)] and the shortcut method which considers ingredients of both theories. The latter method is formally similar to the Nachman theory-based regularized EIT reconstruction algorithm studied in [K. Knudsen et al., Inverse Probl. Imaging 3, No. 4, 599–624 (2009; Zbl 1184.35314)] and several references from there. New numerical results are presented using parallel computation with size parameters larger than ever, leading mainly to two conclusions as follows. First, both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases, and there seems to be a Gibbs-like phenomenon producing ringing artifacts. Second, the transport matrix method loses accuracy away from a (freely chosen) pivot point located outside of the object to be studied, whereas the shortcut method produces reconstructions with more uniform quality.
MSC:
| 78A48 | Composite media; random media in optics and electromagnetic theory |
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 92C55 | Biomedical imaging and signal processing |
Keywords:
inverse problem; Beltrami equation; conductivity equation; inverse conductivity problem; complex geometrical optics solution; nonlinear Fourier transform; scattering transform; electrical impedance tomographyReferences:
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