American Institute of Mathematical Sciences

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November  2010, 4(4): 685-691. doi: 10.3934/ipi.2010.4.685

Incomplete data problems in wave equation imaging

Frank Natterer 1,

1. 

University of Münster, Department of Mathematics and Computer Science, Einsteinstrasse 72, 48159 Münster, Germany

Received  December 2008 Published  September 2010

We study reflection imaging as an incomplete data problem in frequency domain. It turns out that this amounts to inverting the Fourier transform using only frequencies outside some set. By numerical simulations we show the effect of this incompleteness on concrete reconstruction problems. We try to complete the data by analytic continuation. An explicit formula is obtained by an inversion formula for the exponential Radon transform. We discuss the application to medical ultrasound tomography and to seismic imaging. We describe an alternative method based on the presence of reflectors.
Keywords: Seismic Imaging, Ultrasound Imaging, Tomography..
Mathematics Subject Classification: Primary: 35Q86; Secondary: 44A12, 65L09, 92C55, 15A2.
Citation: Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems and Imaging, 2010, 4 (4) : 685-691. doi: 10.3934/ipi.2010.4.685
References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," Dover Publications, Inc., New York, 1970.

[2]

N. Bleistein, J. K. Cohen and J. W. Stockwell (Jr.), "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer, 2001.

[3]

D. T. Borup, S. A. Johnson, W. W. Kim and M. J. Berggren, Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation, Ultrasonic Imaging, 14 (1992), 69-85. doi: doi:10.1016/0161-7346(92)90073-5.

[4]

J. Claerbout, "Fundamentals of Geophysical Data Processing," McGraw-Hill ,1976.

[5]

N. Duric et al., Development of ultraound tomography for breast imaging: Technical assessment, Medical Physics, 32 (2005), 1375-1386. doi: doi:10.1118/1.1897463.

[6]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001.

[7]

F. Natterer, "The Attenuated Radon Transform for Complex Attenuation," Technical Report, Fachbereich Mathematik und Informatik, Universität Münster, 2007.

[8]

F. Natterer, "Ultrasonic Image Reconstruction via Plane Wave Stacking," Technical Report, Fachbereich Mathematik und Informatik, Universität Münster, 2004.

[9]

F. Natterer, "Ultrasound Tomography with Fixed Linear Arrays of Transducers," Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Pisa, Italy, October 2007.

[10]

F. Natterer, Reflectors in wave equation imaging, Wave Motion, 45 (2008), 776-784. doi: doi:10.1016/j.wavemoti.2008.01.001.

[11]

R. G. Novikov, An inversion formula for the attenuated $X$-ray transform, Ark. Mat., 40 (2002), 145-167. doi: doi:10.1007/BF02384507.

[12]

J. You, Attenuated Radon transform with complex coefficients, Inverse Problems, 23 (2007), 1963-1971. doi: doi:10.1088/0266-5611/23/5/010.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions," Dover Publications, Inc., New York, 1970.

[2]

N. Bleistein, J. K. Cohen and J. W. Stockwell (Jr.), "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion," Springer, 2001.

[3]

D. T. Borup, S. A. Johnson, W. W. Kim and M. J. Berggren, Nonperturbative diffraction tomography via Gauss-Newton iteration applied to the scattering integral equation, Ultrasonic Imaging, 14 (1992), 69-85. doi: doi:10.1016/0161-7346(92)90073-5.

[4]

J. Claerbout, "Fundamentals of Geophysical Data Processing," McGraw-Hill ,1976.

[5]

N. Duric et al., Development of ultraound tomography for breast imaging: Technical assessment, Medical Physics, 32 (2005), 1375-1386. doi: doi:10.1118/1.1897463.

[6]

F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM, Philadelphia, 2001.

[7]

F. Natterer, "The Attenuated Radon Transform for Complex Attenuation," Technical Report, Fachbereich Mathematik und Informatik, Universität Münster, 2007.

[8]

F. Natterer, "Ultrasonic Image Reconstruction via Plane Wave Stacking," Technical Report, Fachbereich Mathematik und Informatik, Universität Münster, 2004.

[9]

F. Natterer, "Ultrasound Tomography with Fixed Linear Arrays of Transducers," Proceedings of the Interdisciplinary Workshop on Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT), Pisa, Italy, October 2007.

[10]

F. Natterer, Reflectors in wave equation imaging, Wave Motion, 45 (2008), 776-784. doi: doi:10.1016/j.wavemoti.2008.01.001.

[11]

R. G. Novikov, An inversion formula for the attenuated $X$-ray transform, Ark. Mat., 40 (2002), 145-167. doi: doi:10.1007/BF02384507.

[12]

J. You, Attenuated Radon transform with complex coefficients, Inverse Problems, 23 (2007), 1963-1971. doi: doi:10.1088/0266-5611/23/5/010.

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