Abstract

The development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography. In this paper we address this issue for photoacoustic computed tomography in circular geometry. We investigate the Galerkin least squares method for that purpose. For approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function. This includes many systems that have previously been employed in photoacoustic tomography, such as generalized Kaiser--Bessel basis functions or the natural pixel basis. By exploiting an isometry property of the forward problem we are able to efficiently set up the Galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution. We establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm.

Keywords

  1. photoacoustic imaging
  2. computed tomography
  3. Galerkin least squares method
  4. Kaiser--Bessel functions
  5. Radon transform
  6. least squares approach

MSC codes

  1. 65R32
  2. 45Q05
  3. 92C55

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References

1.
M. Agranovsky and P. Kuchment, Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed, Inverse Problems, 23 (2007), pp. 2089--2102.
2.
M. Ansorg, F. Filbir, W. R. Madych, and R. Seyfried, Summability kernels for circular and spherical mean data, Inverse Problems, 29 (2013), 015002.
3.
S. R. Arridge, M. M. Betcke, B. T. Cox, F. Lucka, and B. E. Treeby, On the adjoint operator in photoacoustic tomography, Inverse Problems, 32 (2016), 115012.
4.
P. Beard, Biomedical photoacoustic imaging, Interface Focus, 1 (2011), pp. 602--631.
5.
Z. Belhachmi, T. Glatz, and O. Scherzer, A direct method for photoacoustic tomography with inhomogeneous sound speed, Inverse Problems, 32 (2016), 045005.
6.
T. Blu and M. Unser, Approximation error for quasi-interpolators and (multi-)wavelet expansions, Appl. Comput. Harmon. Anal., 6 (1999), pp. 219--251.
7.
P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors, Inverse Problems, 23 (2007), pp. S65--S80.
8.
P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Rev. E (3), 75 (2007), 046706.
9.
X. L. Dean-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography, IEEE Trans. Med. Imag., 31 (2012), pp. 1922--1928.
10.
G. J. Diebold, T. Sun, and M. I. Khan, Photoacoustic monopole radiation in one, two, and three dimensions, Phys. Rev. Lett., 67 (1991), pp. 3384--3387.
11.
D. Finch, M. Haltmeier, and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math., 68 (2007), pp. 392--412, https://doi.org/10.1137/070682137.
12.
D. Finch, S. K. Patch, and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), pp. 1213--1240, https://doi.org/10.1137/S0036141002417814.
13.
R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, J. Theoret. Biol., 29 (1970), pp. 471--481.
14.
M. Haltmeier, A mollification approach for inverting the spherical mean Radon transform, SIAM J. Appl. Math., 71 (2011), pp. 1637--1652, https://doi.org/10.1137/110821561.
15.
M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, Comput. Math. Appl., 65 (2013), pp. 1025--1036.
16.
M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, SIAM J. Math. Anal., 46 (2014), pp. 214--232, https://doi.org/10.1137/120881270.
17.
M. Haltmeier and L. V. Nguyen, Analysis of iterative methods in photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 10 (2017), pp. 751--781, https://doi.org/10.1137/16M1104822.
18.
M. Haltmeier and S. Pereverzyev, Jr., Recovering a function from circular means or wave data on the boundary of parabolic domains, SIAM J. Imaging Sci., 8 (2015), pp. 592--610, https://doi.org/10.1137/140960219.
19.
M. Haltmeier and S. Pereverzyev, Jr., The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces, J. Math. Anal. Appl., 429 (2015), pp. 366--382.
20.
M. Haltmeier, O. Scherzer, P. Burgholzer, R. Nuster, and G. Paltauf, Thermoacoustic tomography and the circular Radon transform: Exact inversion formula, Math. Models Methods Appl. Sci., 17 (2007), pp. 635--655.
21.
M. Haltmeier, T. Schuster, and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry, Math. Methods Appl. Sci., 28 (2005), pp. 1919--1937.
22.
M. Haltmeier and G. Zangerl, Spatial resolution in photoacoustic tomography: Effects of detector size and detector bandwidth, Inverse Problems, 26 (2010), 125002.
23.
G. T. Herman, Basis functions in image reconstruction from projections: A tutorial introduction, Sens. Imaging, 16 (2015), pp. 1--21.
24.
Y. Hristova, P. Kuchment, and L. Nguyen, Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006.
25.
A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, Classics Appl. Math. 33, SIAM, Philadelphia, 2001, https://doi.org/10.1137/1.9780898719277.
26.
R. Kress, Linear Integral Equations, 2nd ed., Springer-Verlag, Berlin, 1999.
27.
R. A. Kruger, W. L. Kiser, D. R. Reinecke, G. A. Kruger, and K. D. Miller, Thermoacoustic molecular imaging of small animals, Mol. Imaging, 2 (2003), pp. 113--123.
28.
P. Kuchment, The Radon Transform and Medical Imaging, CBMS-NSF Regional Conf. Ser. in Appl. Math. 85, SIAM, Philadelphia, 2014, https://doi.org/10.1137/1.9781611973297.
29.
P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, in Handbook of Mathematical Methods in Imaging, Springer, New York, 2011, pp. 817--865.
30.
L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), pp. 373--383.
31.
L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems, 23 (2007), pp. S11--S20.
32.
L. A. Kunyansky, Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform, Inverse Problems, 31 (2015), 095001.
33.
R. M. Lewitt, Multidimensional digital image representations using generalized Kaiser--Bessel window functions, J. Opt. Soc. Amer. A, 7 (1990), pp. 1834--1846.
34.
R. M. Lewitt, Alternatives to voxels for image representation in iterative reconstruction algorithms, Phys. Med. Biol., 37 (1992), pp. 705--716.
35.
A. K. Louis, Approximate inverse for linear and some nonlinear problems, Inverse Problems, 12 (1996), pp. 175--190.
36.
A. K. Louis and P. Maass, A mollifier method for linear operator equations of the first kind, Inverse Problems, 6 (1990), pp. 427--440.
37.
A. K. Louis and T. Schuster, A novel filter design technique in 2D computerized tomography, Inverse Problems, 12 (1996), pp. 685--696.
38.
S. Mallat, A Wavelet Tour of Signal Processing: The Sparse Way, 3rd ed., Elsevier/Academic Press, Amsterdam, 2009.
39.
S. Matej and R. M. Lewitt, Practical considerations for 3-D image reconstruction using spherically symmetric volume elements, IEEE Trans. Med. Imag., 15 (1996), pp. 68--78.
40.
F. Natterer, The Mathematics of Computerized Tomography, Classics Appl. Math. 32, SIAM, Philadelphia, 2001, https://doi.org/10.1137/1.9780898719284.
41.
F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging, 6 (2012), pp. 315--320.
42.
L. V. Nguyen, A family of inversion formulas for thermoacoustic tomography, Inverse Problems, 3 (2009), pp. 649--675.
43.
L. V. Nguyen and L. A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), pp. 748--769, https://doi.org/10.1137/15M1049683.
44.
M. Nilchian, J. P. Ward, C. Vonesch, and M. Unser, Optimized Kaiser-Bessel window functions for computed tomography, IEEE Trans. Image Process., 24 (2015), pp. 3826--3833.
45.
V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, Looking and listening to light: The evolution of whole-body photonic imaging, Nat. Biotechnol., 23 (2005), pp. 313--320.
46.
V. P. Palamodov, A uniform reconstruction formula in integral geometry, Inverse Problems, 28 (2012), 065014.
47.
G. Paltauf, R. Nuster, M. Haltmeier, and P. Burgholzer, Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors, Inverse Problems, 23 (2007), pp. S81--S94.
48.
G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, Iterative reconstruction algorithm for optoacoustic imaging, J. Opt. Soc. Amer., 112 (2002), pp. 1536--1544.
49.
A. Rieder and T. Schuster, The approximate inverse in action with an application to computerized tomography, SIAM J. Numer. Anal., 37 (2000), pp. 1909--1929, https://doi.org/10.1137/S0036142998347619.
50.
A. Rieder and T. Schuster, The approximate inverse in action III: 3D-Doppler tomography, Numer. Math., 97 (2004), pp. 353--378.
51.
H. Roitner, M. Haltmeier, R. Nuster, D. P. O'Leary, T. Berer, G. Paltauf, H. Grün, and P. Burgholzer, Deblurring algorithms accounting for the finite detector size in photoacoustic tomography, J. Biomed. Opt., 19 (2014), 056011.
52.
A. Rosenthal, V. Ntziachristos, and D. Razansky, Acoustic inversion in optoacoustic tomography: A review, Curr. Med. Imaging Rev., 9 (2013), pp. 318--336.
53.
Y. Salman, An inversion formula for the spherical mean transform with data on an ellipsoid in two and three dimensions, J. Math. Anal. Appl., 420 (2014), pp. 612--620.
54.
B. E. Treeby and B. T. Cox, k-wave: Matlab toolbox for the simulation and reconstruction of photoacoustic wave-fields, J. Biomed. Opt., 15 (2010), 021314.
55.
K. Wang, S. A. Ermilov, R. Su, H. Brecht, A. A. Oraevsky, and M. A. Anastasio, An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography, IEEE Trans. Med. Imag., 30 (2011), pp. 203--214.
56.
K. Wang, R. W. Schoonover, R. Su, A. Oraevsky, and M. A. Anastasio, Discrete imaging models for three-dimensional optoacoustic tomography using radially symmetric expansion functions, IEEE Trans. Med. Imag., 33 (2014), pp. 1180--1193.
57.
K. Wang, R. Su, A. A. Oraevsky, and M. A. Anastasio, Investigation of iterative image reconstruction in three-dimensional optoacoustic tomography, Phys. Med. Biol., 57 (2012), pp. 5399--5423.
58.
L. V. Wang and S. Hu, Photoacoustic tomography: In vivo imaging from organelles to organs, Science, 335 (2012), pp. 1458--1462.
59.
M. Xu and L. V. Wang, Time-domain reconstruction for thermoacoustic tomography in a spherical geometry, IEEE Trans. Med. Imag., 21 (2002), pp. 814--822.
60.
M. Xu and L. V. Wang, Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction, Phys. Rev. E (3), 67 (2003), 056605.
61.
M. Xu and L. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E (3), 71 (2005), 016706.
62.
J. Zhang, M. A. Anastasio, P. J. La Rivière, and L. V. Wang, Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography, IEEE Trans. Med. Imag., 28 (2009), pp. 1781--1790.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 160 - 184
ISSN (online): 1095-7170

History

Submitted: 22 December 2016
Accepted: 19 October 2017
Published online: 9 January 2018

Keywords

  1. photoacoustic imaging
  2. computed tomography
  3. Galerkin least squares method
  4. Kaiser--Bessel functions
  5. Radon transform
  6. least squares approach

MSC codes

  1. 65R32
  2. 45Q05
  3. 92C55

Authors

Affiliations

Sergiy Pereverzyev, Jr.

Funding Information

Tyrolean Science Fund

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