A Weighted Fresnel Approximation for the Delays Used in Focused Beamforming

  • Andrea Trueco
  • Vittorio Murino
Part of the Acoustical Imaging book series (ACIM, volume 23)


Beamforming is a linear technique aimed at processing array signals in order to enhance incoming signals from a selected steering direction and to abate incoming signals from any other direction1. Thanks to its flexibility, beamforming can be successfully employed in many application fields with different objectives. In any case, if the array works under far-field conditions, the delays required by the beamforming operation can be easily computed in an exact way, whereas, if it works under near-field conditions, the focalization of beamforming is required to take into account the curvature of waves. In the latter case, a fast computation of the exact delays is often prohibitive, then an approximate version is preferred: generally, the Fresnel approximation (obtained by the expansion of the time-independent free-space Green’s function2,3) is adopted1. Moreover, the Fresnel approximation makes it possible to apply the Fast Fourier Transform (FFT) in the implementation of beamforming even when focalization is necessary4, thus resulting in a great computational profit. Despite its simplicity and advantages, the Fresnel approximation has a well defined region of validity2 that forces potential steering directions to be contained inside a narrow scanning region. This constraint is heavy in applications (e.g., acoustic imaging) that require a wide region of view, for both medical and underwater investigations. To avoid this drawback, some imaging techniques that do not need the Fresnel approximation have been devised5,6 but, unfortunately, they increase the computational load and/or the system complexity.


Fast Fourier Transform Focalization Distance Validity Region Process Array Signal Scanning Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1. R.O. Nielsen, Sonar Signal Processing, Artech House, Boston, 1991.Google Scholar
  2. 2. L.J. Ziomek, “Three Necessary Conditions for the Validity of the Fresnel Phase Approximation for the Near-Field Beam Pattern of an Aperture,” IEEE Journ. Ocean. Engin., vol. 18, pp. 73–75, January 1993.CrossRefGoogle Scholar
  3. 3. J.W. Goodman, Introduction to Fourier Optic, McGraw-Hill, New York, 1968.Google Scholar
  4. 4. V. Murino, A. Trucco, “Dynamic Focusing by FFT Beamforming for Underwater 3D Imaging,” Acoustics Letters, vol. 17, no. 9, pp. 169–172, March 1994.Google Scholar
  5. 5. M. Soumekh, “Array Imaging with Beam-Steered Data,” IEEE Transactions on Image Processing, vol. 1, pp. 379–390, July 1992.ADSCrossRefGoogle Scholar
  6. 6. R.K. Hansen, P.A. andersen, “3D Acoustic Camera for Underwater Imaging,” Acoustical Imaging, vol. 20, pp. 723–727, 1993.CrossRefGoogle Scholar
  7. 7. C.R. Rao, S.K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley, New York, 1971.Google Scholar
  8. 8. V. Murino, A. Trucco, “Underwater 3D Imaging by FFT Dynamic Focused Beamforming,” 1st IEEE Intern. Conf. Image Processing, Austin, Texas (USA), vol. I, pp. 890–894, November 1994.Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Andrea Trueco
    • 1
  • Vittorio Murino
    • 2
  1. 1.Dpt. of Biophysical and Electronic Engineering (DIBEUniversity of GenoaGenoaItaly
  2. 2.Dpt. of Mathematics and Computer Science (DIMIUniversity of UdineUdineItaly

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