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Mathematics of Hybrid Imaging: A Brief Review

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The Mathematical Legacy of Leon Ehrenpreis

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 16))

Abstract

The article provides a brief survey of the mathematics of newly being developed so-called “hybrid” (also called “multi-physics” or “multi-wave”) imaging techniques.

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Notes

  1. 1.

    Besides the name AET, suggested in the original paper [148], other names are also used: Ultrasound Modulated EIT (UMEIT) [17], Impediography [10], and some others.

  2. 2.

    The reader should be aware that this classification of hybrid modalities into three classes, although being reasonable, is not commonly accepted and is used in this text only for the author’s convenience.

  3. 3.

    Some microlocal arguments in [6], although correct, were incomplete. The missing arguments are provided in [123].

  4. 4.

    Here instability = ill-posedness = super-algebraic decay of singular values of the direct mapping.

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Acknowledgements

This work was partially supported by the NSF DMS Grant 0604778, as well as by MSRI and IAMCS. The author expresses his gratitude to these institutions. Thanks also go to many colleagues with whom the author discussed the hybrid imaging methods (the author’s attempt to write specific names produced a very long, while still incomplete, list).

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  1. Texas A&M University, College Station, TX, 77843-3368, USA

    Peter Kuchment

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  1. Peter Kuchment
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Correspondence to Peter Kuchment .

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  1. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, Milano, 20133, Italy

    Irene Sabadini

  2. Schmid College of Science and Technology, Chapman University, University Dr. 1, Orange, 92868, California, USA

    Daniele C Struppa

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Dedicated to the memory of Professor Leon Ehrenpreis, a great mathematician, human being, and friend.

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Kuchment, P. (2012). Mathematics of Hybrid Imaging: A Brief Review. In: Sabadini, I., Struppa, D. (eds) The Mathematical Legacy of Leon Ehrenpreis. Springer Proceedings in Mathematics, vol 16. Springer, Milano. https://doi.org/10.1007/978-88-470-1947-8_12

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