Skip to main content
SpringerLink
Log in

Microlocal Analysis of FIOs with Singularities

  • Conference paper
  • First Online:
Microlocal Methods in Mathematical Physics and Global Analysis

Part of the book series: Trends in Mathematics ((RESPERSP))

Abstract

Overview In this talk we describe the composition calculus of Fourier Integral Operators (FIOs) with fold and cusp singularities. Such operators appear in many inverse scattering problems, where the composition calculus can be used as a tool for recovering images. In these problems, caustics occur and create artifacts which make the reconstruction more complicated and challenging. The goal is to understand these artifacts, find their strength and try to remove them.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Beylkin, G. Imaging of discontinuities in the inverse problem by inversion of a generalized Radon transform, Jour. Math. Phys. 28 (1985), 99–108.

    Article  MathSciNet  Google Scholar 

  2. Duistermaat, J.J., Guillemin, V., The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. math., 29 (1975), 39–79.

    Article  MathSciNet  MATH  Google Scholar 

  3. Felea, R. Composition calculus of Fourier integral operators with fold and blowdown singularities, Comm. P.D.E, 30 (13) (2005), 1717–1740.

    Google Scholar 

  4. Felea, R., Greanleaf, A., An FIO calculus for marine seismic imaging: folds and cross caps, Communications in PDEs, 33 (1), (2008), 45–77.

    Article  MATH  Google Scholar 

  5. Felea, R., Greanleaf, A., Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math Ress Lett, 17 (5) (2010), 867–886.

    MATH  Google Scholar 

  6. Felea, R., Greenleaf, A., Pramanik, M., An FIO calculus for marine seismic imaging, II: Sobolev estimates, Math. Annalen, (2011).

    Google Scholar 

  7. Givental, A. Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. (English) Func. Anal. Appl. 20 (3) (1986), 197–203.

    Google Scholar 

  8. Greenleaf, A., Uhlmann, G., Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Jour. Func. Anal., 89 (1990), 220–232.

    Article  MathSciNet  Google Scholar 

  9. Greenleaf, A., Uhlmann,G., Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier, Grenoble, 40 (1990), 443–466.

    Google Scholar 

  10. Guillemin, V., Uhlmann, G., Oscillatory integrals with singular symbols. Duke Math. J. 48 (1) (1981), 251–267.

    Article  MathSciNet  MATH  Google Scholar 

  11. Hörmander, L., Fourier integral operators, I. Acta mathematica, 127 (1971), 79–183.

    Article  MathSciNet  MATH  Google Scholar 

  12. Melrose, R.B., Uhlmann, G.A., Lagrangian intersection and the Cauchy problem. Comm. Pure Appl. Math., 32 (4) (1979), 483–519.

    Article  MathSciNet  MATH  Google Scholar 

  13. Nolan, C.J., Scattering in the presence of fold caustics. SIAM J. Appl. Math. 61 (2) (2000), 659–672.

    Article  MathSciNet  MATH  Google Scholar 

  14. Nolan, C.J., Symes, W.W., Global solutions of a linearized inverse problem for the acoustic wave equation, Comm. in PDE 22, (1997), 919–952.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rochester Institute of Technology, School of mathematical sciences, Rochester, NY, 14620, USA

    Raluca Felea

Authors
  1. Raluca Felea
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raluca Felea .

Editor information

Editors and Affiliations

  1. Ammerländer Heerstr. 114-118, Oldenburg, 26111, Germany

    Daniel Grieser

  2. Inst. Mathematik, Universität Tübingen, Auf der Morgenstelle 10, Tübingen, 72076, Germany

    Stefan Teufel

  3. , Department of Mathematics, Bldg. 380, Stanford University, 450, Serra Mall, Stanford, 94305-2125, California, USA

    Andras Vasy

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Basel

About this paper

Cite this paper

Felea, R. (2013). Microlocal Analysis of FIOs with Singularities. In: Grieser, D., Teufel, S., Vasy, A. (eds) Microlocal Methods in Mathematical Physics and Global Analysis. Trends in Mathematics(). Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0466-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation