Journal of Mathematical Imaging and Vision

, Volume 53, Issue 1, pp 78–91 | Cite as

An Efficient Numerical Algorithm for the Inversion of an Integral Transform Arising in Ultrasound Imaging

  • Souvik Roy
  • Venkateswaran P. Krishnan
  • Praveen Chandrashekar
  • A. S. Vasudeva Murthy


We present an efficient and novel numerical algorithm for inversion of transforms arising in imaging modalities such as ultrasound imaging, thermoacoustic and photoacoustic tomography, intravascular imaging, non-destructive testing, and radar imaging with circular acquisition geometry. Our algorithm is based on recently discovered explicit inversion formulas for circular and elliptical Radon transforms with radially partial data derived by Ambartsoumian, Gouia-Zarrad, Lewis and by Ambartsoumian and Krishnan. These inversion formulas hold when the support of the function lies on the inside (relevant in ultrasound imaging, thermoacoustic and photoacoustic tomography, non-destructive testing), outside (relevant in intravascular imaging), both inside and outside (relevant in radar imaging) of the acquisition circle. Given the importance of such inversion formulas in several new and emerging imaging modalities, an efficient numerical inversion algorithm is of tremendous topical interest. The novelty of our non-iterative numerical inversion approach is that the entire scheme can be pre-processed and used repeatedly in image reconstruction, leading to a very fast algorithm. Several numerical simulations are presented showing the robustness of our algorithm.


Circular Radon transform Elliptical Radon transform Volterra integral equations Truncated singular value decomposition Theromoacoustic tomography Photoacoustic tomography Ultrasound reflectivity imaging Intravascular imaging Radar imaging 



VPK would like to thank Rishu Saxena for her valuable input and discussions during the initial stages of this work. SR and VPK would like to express their gratitude to Gaik Ambartsoumian and Eric Todd Quinto for several fruitful discussions and important suggestions. VPK was partially supported by NSF grant DMS 1109417. All authors benefited from the support of the Airbus Group Corporate Foundation Chair “Mathematics of Complex Systems” established at TIFR Centre for Applicable Mathematics and TIFR International Centre for Theoretical Sciences, Bangalore, India.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Souvik Roy
    • 1
  • Venkateswaran P. Krishnan
    • 1
  • Praveen Chandrashekar
    • 1
  • A. S. Vasudeva Murthy
    • 1
  1. 1.TIFR Centre for Applicable MathematicsBangaloreIndia

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