Advertisement

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
Article

Abstract

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.

Keywords

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

Notes

Acknowledgments

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.

References

  1. 1.
    Agranovsky, Mark, Berenstein, Carlos, Kuchment, Peter: Approximation by spherical waves in \(L^p\)-spaces. J. Geom. Anal. 6(3), 365–383 (1997). 1996MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agranovsky, Mark, Quinto, Eric Todd: Injectivity sets for Radon transform over circles and complete systems of radial functions. J. Funct. Anal. 139, 383–414 (1996)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ambartsoumian, Gaik, Boman, Jan, Krishnan, Venkateswaran P., Quinto, Eric Todd: Microlocal analysis of an ultrasound transform with circular source and receiver trajectories. American Mathematical Society, Series Contemporary Mathematics 598, pp. 45–58 (2013)Google Scholar
  4. 4.
    Ambartsoumian, Gaik, Felea, Raluca, Krishnan, Venkateswaran P., Nolan, Clifford, Quinto, Eric Todd: A class of singular Fourier integral operators in synthetic aperture radar imaging. J. Funct. Anal. 264(1), 246–269 (2013)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ambartsoumian, Gaik, Gouia-Zarrad, Rim, Lewis, Matthew A.: Inversion of the circular Radon transform on an annulus. Inverse Probl. 26(10), 105015, 11 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ambartsoumian, Gaik, Krishnan, Venkateswaran P.: Inversion of a class of circular and elliptical Radon transforms. 2014. To appear in Contemporary Mathematics, Proceedings of the International Conference on Complex Analysis and Dynamical Systems VI, (2013)Google Scholar
  7. 7.
    Ambartsoumian, Gaik, Kuchment, Peter: On the injectivity of the circular Radon transform. Inverse Probl. 21(2), 473–485 (2005)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ambartsoumian, Gaik, Kunyansky, Leonid: Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Probl. Imaging 8(2), 339–359 (2014)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Andersson, Lars-Erik: On the determination of a function from spherical averages. SIAM J. Math. Anal. 19(1), 214–232 (1988)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Antipov, Yuri A., Estrada, Ricardo, Rubin, Boris: Method of analytic continuation for the inverse spherical mean transform in constant curvature spaces. J. Anal. Math. 118(2), 623–656 (2012)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Cormack, Allen M.: Representation of a function by its line integrals, with some radiological applications. J. Appl. Phys. 34(9), 2722–2727 (1963)MATHCrossRefGoogle Scholar
  12. 12.
    Denisjuk, Alexander: Integral geometry on the family of semi-spheres. Fract. Calc. Appl. Anal. 2(1), 31–46 (1999)MATHMathSciNetGoogle Scholar
  13. 13.
    Fawcett, John A.: Inversion of \(n\)-dimensional spherical averages. SIAM J. Appl. Math. 45(2), 336–341 (1985)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Finch, David, Haltmeier, Markus, Rakesh: Inversion of spherical means and the wave equation in even dimensions. SIAM J. Appl. Math. 68(2), 392–412 (2007)Google Scholar
  15. 15.
    Finch, David, Patch, Sarah K., Rakesh: Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal. 35(5), 1213–1240 (2004). (electronic)Google Scholar
  16. 16.
    Finch, David, Rakesh: The spherical mean value operator with centers on a sphere. Inverse Probl. 23(6), S37–S49 (2007)Google Scholar
  17. 17.
    Frikel, Jürgen, Quinto, Eric Todd: Artifacts in incomplete data tomography - with applications to photoacoustic tomography and sonar (2014). http://www.arxiv.org/abs/1407.3453
  18. 18.
    Greenleaf, Allan, Uhlmann, Gunther: Non-local inversion formulas for the X-ray transform. Duke Math. J. 58, 205–240 (1989)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Guillemin, Victor: Some remarks on integral geometry. Technical Report, MIT (1975)Google Scholar
  20. 20.
    Guillemin, Victor, Sternberg, Shlomo: Geometric asymptotics. American Mathematical Society, Providence (1977). Mathematical Surveys, No. 14MATHCrossRefGoogle Scholar
  21. 21.
    Hansen, Per Christian: The truncated SVD as a method for regularization. BIT 27(4), 534–553 (1987)MATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Horn, Roger A., Johnson, Charles R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)MATHGoogle Scholar
  23. 23.
    Kuchment, Peter, Kunyansky, Leonid: Mathematics of thermoacoustic tomography. Eur. J. Appl. Math. 19(2), 191–224 (2008)MATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Lavrentiev, M.M., Romanov, V.G., Vasiliev, V.G.: Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Mathematics, vol. 167. Springer, Berlin (1970)Google Scholar
  25. 25.
    Mensah, Serge, Franceschini, Émilie: Near-field ultrasound tomography. J. Acoust. Soc. Am. 121(3), 1423–1433 (2007)CrossRefGoogle Scholar
  26. 26.
    Mensah, Serge, Franceschini, Émilie, Lefevre, Jean-Pierre: Mammographie ultrasonore en champ proche. Trait. Signal 23(3–4), 259–276 (2006)MATHGoogle Scholar
  27. 27.
    Mensah, Serge, Franceschini, Émilie, Pauzin, Marie-Christine: Ultrasound mammography. Nucl. Instrum. Methods Phys. Res. 571(3), 52–55 (2007)CrossRefGoogle Scholar
  28. 28.
    Moon, Sunghwan: On the determination of a function from an elliptical Radon transform. J. Math. Anal. Appl. 416(2), 724–734 (2014)MATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Nguyen, Linh V.: A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging 3(4), 649–675 (2009)MATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Nguyen, Linh V.: Spherical mean transform: a PDE approach. Inverse Probl. Imaging 7(1), 243–252 (2013)MATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Norton, Stephen J.: Reconstruction of a two-dimensional reflecting medium over a circular domain: exact solution. J. Acoust. Soc. Am. 67(4), 1266–1273 (1980)MATHMathSciNetCrossRefGoogle Scholar
  32. 32.
    Norton, Stephen J., Linzer, Melvin: Reconstructing spatially incoherent random sources in the nearfield: exact inversion formulas for circular and spherical arrays. J. Acoust. Soc. Am. 76(6), 1731–1736 (1984)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Plato, Robert: The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math. 36(2), 331–351 (2012)MATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Press, William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian P.: Numerical Recipes in C, 2nd edn. Cambridge University Press, Cambridge (1992). The art of scientific computingMATHGoogle Scholar
  35. 35.
    Quinto, Eric Todd: Singularities of the X-ray transform and limited data tomography in \({\mathbb{R}}^2\) and \({\mathbb{R}}^3\). SIAM J. Math. Anal. 24, 1215–1225 (1993)MATHMathSciNetCrossRefGoogle Scholar
  36. 36.
    Romanov, V.G.: An inversion formula in a problem of integral geometry on ellipsoids. Mat. Zametki 46(4), 124–126 (1989)MATHMathSciNetGoogle Scholar
  37. 37.
    Rubin, Boris: Inversion formulae for the spherical mean in odd dimensions and the Euler–Poisson–Darboux equation. Inverse Probl. 24(2), 025021, 10 (2008)CrossRefGoogle Scholar
  38. 38.
    Tricomi, F.G.: Integral Equations. Dover Publications Inc, New York (1985). Reprint of the 1957 originalGoogle Scholar
  39. 39.
    Volchkov, V.V.: Integral Geometry and Convolution Equations. Kluwer Academic Publishers, Dordrecht (2003)MATHCrossRefGoogle Scholar
  40. 40.
    Volterra, Vito: Theory of functionals and of integral and integro-differential equations. With a preface by G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by E. Whittaker. Dover Publications Inc, New York (1959)Google Scholar
  41. 41.
    Weiss, Richard: Product integration for the generalized Abel equation. Math. Comput. 26, 177–190 (1972)MATHCrossRefGoogle Scholar

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

Personalised recommendations