# American Institute of Mathematical Sciences

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

## Incomplete data problems in wave equation imaging

 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.
Citation: Frank Natterer. Incomplete data problems in wave equation imaging. Inverse Problems & 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, (1970). Google Scholar [2] N. Bleistein, J. K. Cohen and J. W. Stockwell (Jr.), "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion,", Springer, (2001). Google Scholar [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. doi: doi:10.1016/0161-7346(92)90073-5. Google Scholar [4] J. Claerbout, "Fundamentals of Geophysical Data Processing,", McGraw-Hill, (). Google Scholar [5] N. Duric et al., Development of ultraound tomography for breast imaging: Technical assessment,, Medical Physics, 32 (2005), 1375. doi: doi:10.1118/1.1897463. Google Scholar [6] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001). Google Scholar [7] F. Natterer, "The Attenuated Radon Transform for Complex Attenuation,", Technical Report, (2007). Google Scholar [8] F. Natterer, "Ultrasonic Image Reconstruction via Plane Wave Stacking,", Technical Report, (2004). Google Scholar [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), (2007). Google Scholar [10] F. Natterer, Reflectors in wave equation imaging,, Wave Motion, 45 (2008), 776. doi: doi:10.1016/j.wavemoti.2008.01.001. Google Scholar [11] R. G. Novikov, An inversion formula for the attenuated $X$-ray transform,, Ark. Mat., 40 (2002), 145. doi: doi:10.1007/BF02384507. Google Scholar [12] J. You, Attenuated Radon transform with complex coefficients,, Inverse Problems, 23 (2007), 1963. doi: doi:10.1088/0266-5611/23/5/010. Google Scholar

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##### References:
 [1] M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions,", Dover Publications, (1970). Google Scholar [2] N. Bleistein, J. K. Cohen and J. W. Stockwell (Jr.), "Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion,", Springer, (2001). Google Scholar [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. doi: doi:10.1016/0161-7346(92)90073-5. Google Scholar [4] J. Claerbout, "Fundamentals of Geophysical Data Processing,", McGraw-Hill, (). Google Scholar [5] N. Duric et al., Development of ultraound tomography for breast imaging: Technical assessment,, Medical Physics, 32 (2005), 1375. doi: doi:10.1118/1.1897463. Google Scholar [6] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM, (2001). Google Scholar [7] F. Natterer, "The Attenuated Radon Transform for Complex Attenuation,", Technical Report, (2007). Google Scholar [8] F. Natterer, "Ultrasonic Image Reconstruction via Plane Wave Stacking,", Technical Report, (2004). Google Scholar [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), (2007). Google Scholar [10] F. Natterer, Reflectors in wave equation imaging,, Wave Motion, 45 (2008), 776. doi: doi:10.1016/j.wavemoti.2008.01.001. Google Scholar [11] R. G. Novikov, An inversion formula for the attenuated $X$-ray transform,, Ark. Mat., 40 (2002), 145. doi: doi:10.1007/BF02384507. Google Scholar [12] J. You, Attenuated Radon transform with complex coefficients,, Inverse Problems, 23 (2007), 1963. doi: doi:10.1088/0266-5611/23/5/010. Google Scholar
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