These lecture notes provide a mathematical treatment of time-reversal experiments with a special emphasis on telecommunication. A direct link is established between time-reversal experiments and the adjoint imaging method. Based on this relationship, several iterative schemes are proposed for optimizing MIMO (multiple-input multiple-output) time-reversal systems in underwater acoustic and in wireless communication systems. Whereas in typical imaging applications these iterative schemes require the repeated solution of forward problems in a computer, the analogue in timereversal communication schemes consists of a small number of physical time-reversal experiments and does not require exact knowledge of the environment in which the communication system operates. The discussion is put in the general framework of wave propagation by symmetric hyperbolic systems, with detailed discussions of the linear acoustic system for underwater communication and of the time-dependent system of Maxwell's equations for telecommunication. Moreover, in its general form as treated here, the theory will also apply for several other models of wave-propagation, such as for example linear elastic waves.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Abramowitz and I.A. Stegun (1970): Handbook of Mathematical Functions. Dover.
G. Beylkin (1995): ‘On the fast Fourier transform of functions with singularities’, Appl. Comp. Harm. Anal. 2, 363-381.
Y. Censor (1981): ‘Row–action methods for huge and sparce systems and their applications’, SIAM Review 23, 444-466.
R. Clack and M. Defrise (1994): ‘A Cone–Beam Reconstruction Algorithm Using Shift–Variant Filtering and Cone–Beam Backprojection’, IEEE Trans. Med. Imag. 13, 186-195.
P.E. Danielsson, P. Edholm, J. Eriksson, M. Magnusson Seger, and H. Turbell (1999): The original π-method for helical cone-beam CT, Proc. Int. Meeting on Fully 3D-reconstruction, Egmond aan Zee, June 3-6.
L.A. Feldkamp, L.C. Davis, and J.W. Kress (1984): ‘Practical cone–beam algorithm’, J. Opt. Soc. Amer. A 6, 612-619.
K. Fourmont (1999): ‘Schnelle Fourier–Transformation bei nicht–äquidistanten Gittern und tomographische Anwendungen’, Dissertation Fachbereich Mathematik und Informatik der Universität Münster, Münster, Germany.
I.M. Gel‘fand, M.I. Graev, N.Y. Vilenkin (1965): Generalized Functions, Vol. 5: Integral Geometry and Representation Theory. Academic Press.
P. Grangeat (1991): ‘Mathematical framework of cone–beam reconstruction via the first derivative of the Radon transform’ in: G.T. Herman, A.K. Louis and F. Natterer (eds.): Lecture Notes in Mathematics 1497, 66-97.
C. Hamaker, K.T. Smith, D.C. Solmon and Wagner, S.L. (1980): ‘The divergent beam X-ray transform’, Rocky Mountain J. Math. 10, 253-283.
C. Hamaker and D.C. Solmon (1978): ‘The angles between the null spaces of X-rays’, J. Math. Anal. Appl. 62, 1-23.
S. Helgason (1999): The Radon Transform. Second Edition. Birkhäuser, Boston.
G.T. Herman (1980): Image Reconstruction from Projection. The Fundamentals of Computerized Tomography. Academic Press.
G.T. Herman and L. Meyer (1993): ‘Algebraic reconstruction techniques can be made computationally efficient’, IEEE Trans. Med. Imag. 12, 600-609.
H.M. Hudson and R.S. Larkin (1994): ‘Accelerated EM reconstruction using ordered subsets of projection data’, IEEE Trans. Med. Imag. 13, 601-609.
A.C. Kak and M. Slaney (1987): Principles of Computerized Tomography Imaging. IEEE Press, New York.
A. Katsevich (2002): ‘Theoretically exact filtered backprojection-type inversion algorithm for spiral CT’, SIAM J. Appl. Math. 62, 2012-2026.
H. Kudo and T. Saito (1994): ‘Derivation and implementation of a cone–beam reconstruction algorithm for nonplanar orbits’, IEEE Trans. Med. Imag. 13, 196-211.
F. Natterer (1986): The Mathematics of Computerized Tomography. John Wiley & Sons and B.G. Teubner. Reprint SIAM 2001.
F. Natterer and F. Wübbeling (2001): Mathematical Methods in Image Reconstruction. SIAM, Philadelphia.
F. Natterer and E.L. Ritman (2002): ‘Past and Future Directions in X-Ray Computed Tomography (CT)’, to appear in Int. J. Imaging Systems & Technology.
R.G. Novikov (2000): ‘An inversion formula for the attenuated X-ray transform’, Preprint, Departement de Mathématique, Université de Nantes.
V.A. Sharafutdinov (1994): Integral geometry of Tensor Fields. VSP, Utrecht.
G. Steidl (1998): ‘A note on fast Fourier transforms for nonequispaced grids’, Advances in Computational Mathematics 9, 337-352.
S. Webb (1990): From the Watching of Shadows. Adam Hilger.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Dorn, O. (2008). Time-Reversal and the Adjoint Imaging Method with an Application in Telecommunication. In: Bonilla, L.L. (eds) Inverse Problems and Imaging. Lecture Notes in Mathematics, vol 1943. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78547-7_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-78547-7_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78545-3
Online ISBN: 978-3-540-78547-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)