Abstract
In this paper, 3-D vector field tomography (3D-VFT) is employed to reconstruct three-dimensional, irrotational fields in a bounded cubic domain. A sampling process along the scanning lines that further assigns the derived points to preordained finite reconstruction points accomplishes data redundancy, lacking when the problem is formed in the continuous domain, and results in the formulation of an over-determined system of linear equations. The only precondition to the system solution, that corresponds to a discretized inversion of the Ray transform, is the known location and values of a limited number of boundary points. The method is accompanied by a theoretical analysis on the regularization achieved and the errors introduced. The effectiveness and robustness of the method are demonstrated by means of simulations of electric fields, a series of perturbation tests, and a comparison with two alternative baseline methodologies.
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References
Aben, H.K.: Kerr effect tomography for general axisymmetric field. Appl. Opt. 26(14), 2921–2924 (1987)
Braun, H., Hauck, A.: Tomographic reconstruction of vector fields. IEEE Trans. Signal Process. 39(2), 464–471 (1991)
Defrise, M., Gullberg, G.T.: 3D reconstruction of tensors and vectors (2005)
Efremov, N., Poluektov, N., Kharchenko, V.: Tomography of ion and atom velocities in plasmas. J. Quant. Spectrosc. Radiative Transf. 53(6), 723–728 (1995)
Friedman, A.: Partial Differential Equations. Courier Dover Publications, New York (2011)
Fulton, S.R., Ciesielski, P.E., Schubert, W.H.: Multigrid methods for elliptic problems: a review. Monthly Weather Rev. 114(5), 943–959 (1986)
Giannakidis, A., Kotoulas, L., Petrou, M.: Improved 2-d vector field reconstruction using virtual sensors and the radon transform. In: Proceedings of the 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2008. EMBS 2008, pp. 2725–2728. IEEE (2008)
Giannakidis, A., Kotoulas, L., Petrou, M.: Virtual sensors for 2D vector field tomography. JOSA A 27(6), 1331–1341 (2010)
Giannakidis, A., Petrou, M.: Sampling bounds for 2-D vector field tomography. J. Math. Imaging Vis. 37(2), 151–165 (2010)
Giannakidis, A., Petrou, M.: Improved 2D vector field estimation using probabilistic weights. JOSA A 28(8), 1620–1635 (2011)
Herman, G.T.: Fundamentals of Computerized Tomography: Image Reconstruction from Projections. Springer, Berlin (2009)
Hertz, H.M.: Kerr effect tomography for nonintrusive spatially resolved measurements of asymmetric electric field distributions. Appl. Opt. 25(6), 914–921 (1986)
Jovanovic, I.: Inverse problems in acoustic tomography: theory and applications. Ph.D. Thesis, EPFL, Lausanne (2008)
Juhlin, P.: Principles of doppler tomography. Technical Report, Lund University (Sweden). Department of Mathematics (1992)
Juhlin, S.P.: Doppler tomography. In: Proceedings of the 15th Annual International Conference of the IEEE, Engineering in Medicine and Biology Society, 1993, pp. 212–213. IEEE (1993)
Kaczmarz, S.: Angenäherte auflösung von systemen linearer gleichungen. Bulletin International de lAcademie Polonaise des Sciences et des Lettres 35, 355–357 (1937)
Koulouri, A., Petrou, M.: Vector field tomography: reconstruction of an irrotational field in the discrete domain. In: Proceedings of the SPPR (2012)
Louis, A.K., Maass, P.: A mollifier method for linear operator equations of the first kind. Inverse Probl. 6(3), 427 (1990)
Norton, S.J.: Tomographic reconstruction of 2-D vector fields: application to flow imaging. Geophys. J. Int. 97(1), 161–168 (1989)
Norton, S.J.: Unique tomographic reconstruction of vector fields using boundary data. IEEE Trans. Image Process. 1(3), 406–412 (1992)
Norton, S.J., Linzer, M.: Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach. Ultrason. Imaging 4(3), 201–233 (1982)
Osman, N.F., Prince, J.L.: Reconstruction of vector fields in bounded domain vector tomography. In: Proceedings of the International Conference on Image Processing, 1997, vol. 1, pp. 476–479. IEEE (1997)
Papadaniil, C.D., Hadjileontiadis, L.J.: Towards an overall 3-D vector field reconstruction via discretization and a linear equations system. In: Proceedings of the IEEE 13th International Conference on Bioinformatics and Bioengineering (BIBE), 2013, pp. 1–4. IEEE (2013)
Petrou, M., Giannakidis, A.: Full tomographic reconstruction of 2D vector fields using discrete integral data. Comput. J. 54(9), 1491–1504 (2011)
Prince, J.L.: Tomographic reconstruction of 3-D vector fields using inner product probes. IEEE Trans. Image Process. 3(2), 216–219 (1994)
Radon, J.: Über die bestimmungen von funktionen durch ihre integralwerte längs gewisser mannigfaltkeiten. Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math-Nat. 69(16), 262–277 (1917)
Rieder, A., Schuster, T.: The approximate inverse in action iii: 3D-doppler tomography. Numerische Mathematik 97(2), 353–378 (2004)
Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)
Schuster, T.: The 3D doppler transform: elementary properties and computation of reconstruction kernels. Inverse Probl. 16(3), 701 (2000)
Schuster, T.: An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation. Inverse Probl. 17(4), 739 (2001)
Segre, S.: The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave. Plasma Phys. 20(4), 295 (1978)
Sparr, G., Strahlen, K., Lindstrom, K., Persson, H.: Doppler tomography for vector fields. Inverse Probl. 11(5), 1051 (1995)
Tao, Y.K., Davis, A.M., Izatt, J.A.: Single-pass volumetric bidirectional blood flow imaging spectral domain optical coherence tomography using a modified hilbert transform. Opt. Express 16(16), 12350–12361 (2008)
Valk, P.E.: Positron Emission Tomography: Basic Sciences. Springer, New York (2003)
Wernsdörfer, A.: Complete reconstruction of three-dimensional vector fields. In: Proceedings of the ECAPT’93 (Karlsruhe) (1993)
West, D.B., et al.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (2001)
Yuhai, C., Youquan, J., Wei, S.: Utilizing phase retard integration method of flow birefringence to analyse the flow with symmetric plane. Acta Mechanica Sinica 6(4), 374–381 (1990)
Zahn, M.: Transform relationship between kerr-effect optical phase shift and nonuniform electric field distributions. IEEE Trans. Dielectr. Electr. Insulation, 1(2), 235–246 (1994)
Acknowledgments
This work is inspired by and dedicated to the memory of the late Professor Maria Petrou. The authors would like to thank Dr. Vasiliki Kosmidou and Ms. Alexandra Koulouri for their valuable contribution to this work. The latter was carried out as part of the GSRT Research Excellent Grant ARISTEIA, within the 4th Strategic Objective of the operational program “Education and Lifelong Learning” entitled ‘Supporting the Human Capital in order to Promote Research and Innovation’, under Grant agreement 440, Project CBP: Cognitive Brain signal Processing lab, coordinated by the Information Technologies Institute—Centre for Research & Technology—Hellas.
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Papadaniil, C.D., Hadjileontiadis, L.J. Tomographic Reconstruction of 3-D Irrotational Vector Fields via a Discretized Ray Transform. J Math Imaging Vis 52, 285–302 (2015). https://doi.org/10.1007/s10851-015-0559-y
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DOI: https://doi.org/10.1007/s10851-015-0559-y