Abstract

The simplicity and computational efficiency of back-projection formulae have made them a popular choice in optoacoustic tomography. Nonetheless, exact back-projection formulae exist for only a small set of tomographic problems. This limitation is overcome by algebraic algorithms, but at the cost of higher numerical complexity. In this paper, we present a generic algebraic framework for calculating back-projection operators in optoacoustic tomography. We demonstrate our approach in a two-dimensional optoacoustic-tomography example and show that once the algebraic back-projection operator has been found, it achieves a comparable run time to that of the conventional back-projection algorithm, but with the superior image quality of algebraic methods.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (4)

L. Ding, X. L. Dean-Ben, and D. Razansky, “Efficient 3-D Model-Based Reconstruction Scheme for Arbitrary Optoacoustic Acquisition Geometries,” IEEE Trans. Med. Imaging 36(9), 1858–1867 (2017).
[Crossref] [PubMed]

J. Xiao, X. Luo, K. Peng, and B. Wang, “Improved back-projection method for circular-scanning-based photoacoustic tomography with improved tangential resolution,” Appl. Opt. 56(32), 8983–8990 (2017).
[Crossref] [PubMed]

G. Drozdov and A. Rosenthal, “Analysis of negatively focused ultrasound detectors in optoacoustic tomography,” IEEE Trans. Med. Imaging 36(1), 301–309 (2017).
[Crossref] [PubMed]

X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
[Crossref] [PubMed]

2015 (3)

H. Huang, G. Bustamante, R. Peterson, and J. Y. Ye, “An adaptive filtered back-projection for photoacoustic image reconstruction,” Med. Phys. 42(5), 2169–2178 (2015).
[Crossref] [PubMed]

Y. Zhao, X. Zhao, and P. Zhang, “An extended algebraic reconstruction technique (E-ART) for dual spectral CT,” IEEE Trans. Med. Imaging 34(3), 761–768 (2015).
[Crossref] [PubMed]

A. Taruttis and V. Ntziachristos, “Advances in real-time multispectral optoacoustic imaging and its applications,” Nat. Photonics 9(4), 219–227 (2015).
[Crossref]

2014 (5)

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Acoustic Inversion in optoacoustic tomography: a review,” Curr. Med. Imaging Rev. 9(4), 318–336 (2014).
[Crossref] [PubMed]

J. Turner, H. Estrada, M. Kneipp, and D. Razansky, “Improved optoacoustic microscopy through three-dimensional spatial impulse response synthetic aperture focusing technique,” Opt. Lett. 39(12), 3390–3393 (2014).
[Crossref] [PubMed]

M. Haltmeier, “Universal inversion formulas for recovering a function from spherical means,” SIAM J. Math. Anal. 46(1), 214–232 (2014).
[Crossref]

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
[Crossref] [PubMed]

J. Prakash, A. S. Raju, C. B. Shaw, M. Pramanik, and P. K. Yalavarthy, “Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography,” Biomed. Opt. Express 5(5), 1363–1377 (2014).
[Crossref] [PubMed]

2013 (5)

C. B. Shaw, J. Prakash, M. Pramanik, and P. K. Yalavarthy, “Least squares QR-based decomposition provides an efficient way of computing optimal regularization parameter in photoacoustic tomography,” J. Biomed. Opt. 18(8), 080501 (2013).
[Crossref] [PubMed]

X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
[Crossref] [PubMed]

M. Omar, J. Gateau, and V. Ntziachristos, “Raster-scan optoacoustic mesoscopy in the 25-125 MHz range,” Opt. Lett. 38(14), 2472–2474 (2013).
[Crossref] [PubMed]

C. Lutzweiler, X. L. Deán-Ben, and D. Razansky, “Expediting model-based optoacoustic reconstructions with tomographic symmetries,” Med. Phys. 41(1), 013302 (2013).
[Crossref] [PubMed]

J. Aguirre, A. Giannoula, T. Minagawa, L. Funk, P. Turon, and T. Durduran, “A low memory cost model based reconstruction algorithm exploiting translational symmetry for photoacustic microscopy,” Biomed. Opt. Express 4(12), 2813–2827 (2013).
[Crossref] [PubMed]

2012 (5)

K. Wang and M. A. Anastasio, “A simple Fourier transform-based reconstruction formula for photoacoustic computed tomography with a circular or spherical measurement geometry,” Phys. Med. Biol. 57(23), N493–N499 (2012).
[Crossref] [PubMed]

A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
[Crossref] [PubMed]

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imaging 31(10), 1922–1928 (2012).
[Crossref] [PubMed]

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(17), 5399–5423 (2012).
[Crossref] [PubMed]

M. Á. Araque Caballero, A. Rosenthal, J. Gateau, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic imaging using focused detector scanning,” Opt. Lett. 37(19), 4080–4082 (2012).
[Crossref] [PubMed]

2011 (4)

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

D. Razansky, A. Buehler, and V. Ntziachristos, “Volumetric real-time multispectral optoacoustic tomography of biomarkers,” Nat. Protoc. 6(8), 1121–1129 (2011).
[Crossref] [PubMed]

Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Model-based optoacoustic inversion with arbitrary-shape detectors,” Med. Phys. 38(7), 4285–4295 (2011).
[Crossref] [PubMed]

2010 (5)

D. Razansky, S. Kellnberger, and V. Ntziachristos, “Near-field radiofrequency thermoacoustic tomography with impulse excitation,” Med. Phys. 37(9), 4602–4607 (2010).
[Crossref] [PubMed]

C. Zhang, C. Li, and L. V. Wang, “Fast and robust deconvolution-based image reconstruction for photoacoustic tomography in circular geometry: experimental validation,” IEEE Photonics J. 2(1), 57–66 (2010).
[Crossref] [PubMed]

A. Rosenthal, D. Razansky, and V. Ntziachristos, “Fast semi-analytical model-based acoustic inversion for quantitative optoacoustic tomography,” IEEE Trans. Med. Imaging 29(6), 1275–1285 (2010).
[Crossref] [PubMed]

X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
[Crossref] [PubMed]

2009 (3)

J. Provost and F. Lesage, “The application of compressed sensing for photo-acoustic tomography,” IEEE Trans. Med. Imaging 28(4), 585–594 (2009).
[Crossref] [PubMed]

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inverse Probl. 25(12), 123009 (2009).
[Crossref] [PubMed]

C. Li, N. Duric, P. Littrup, and L. Huang, “In vivo breast sound-speed imaging with ultrasound tomography,” Ultrasound Med. Biol. 35(10), 1615–1628 (2009).
[Crossref] [PubMed]

2007 (3)

P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
[Crossref]

L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23(1), 373–383 (2007).
[Crossref]

P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046706 (2007).
[Crossref] [PubMed]

2006 (1)

W. A. Kalender, “X-ray computed tomography,” Phys. Med. Biol. 51(13), R29–R43 (2006).
[Crossref] [PubMed]

2005 (1)

M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016706 (2005).
[Crossref] [PubMed]

2003 (1)

R. M. Lewitt and S. Matej, “Overview of methods for image reconstruction from projections in emission computed tomography,” Proc. IEEE 91(10), 1588–1611 (2003).
[Crossref]

2002 (2)

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry,” IEEE Trans. Med. Imaging 21(7), 823–828 (2002).
[Crossref] [PubMed]

1986 (1)

T. Taylor and L. R. Lupton, “Resolution, artifacts and the design of computed tomography systems,” Nucl. Instrum. Methods Phys. Res. A 242(3), 603–609 (1986).
[Crossref]

1982 (1)

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8(1), 43–71 (1982).
[Crossref]

Aguirre, J.

Ahlgren, U.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Anastasio, M. A.

K. Wang and M. A. Anastasio, “A simple Fourier transform-based reconstruction formula for photoacoustic computed tomography with a circular or spherical measurement geometry,” Phys. Med. Biol. 57(23), N493–N499 (2012).
[Crossref] [PubMed]

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(17), 5399–5423 (2012).
[Crossref] [PubMed]

Araque Caballero, M. Á.

Baldock, R.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Bauer-Marschallinger, J.

P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
[Crossref]

Bouman, C. A.

Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
[Crossref] [PubMed]

Buehler, A.

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imaging 31(10), 1922–1928 (2012).
[Crossref] [PubMed]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

D. Razansky, A. Buehler, and V. Ntziachristos, “Volumetric real-time multispectral optoacoustic tomography of biomarkers,” Nat. Protoc. 6(8), 1121–1129 (2011).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
[Crossref] [PubMed]

Burgholzer, P.

P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
[Crossref]

P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046706 (2007).
[Crossref] [PubMed]

Bustamante, G.

H. Huang, G. Bustamante, R. Peterson, and J. Y. Ye, “An adaptive filtered back-projection for photoacoustic image reconstruction,” Med. Phys. 42(5), 2169–2178 (2015).
[Crossref] [PubMed]

Davidson, D.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Dean-Ben, X. L.

L. Ding, X. L. Dean-Ben, and D. Razansky, “Efficient 3-D Model-Based Reconstruction Scheme for Arbitrary Optoacoustic Acquisition Geometries,” IEEE Trans. Med. Imaging 36(9), 1858–1867 (2017).
[Crossref] [PubMed]

Deán-Ben, X. L.

A. Özbek, X. L. Deán-Ben, and D. Razansky, “Optoacoustic imaging at kilohertz volumetric frame rates,” Optica 5(7), 857 (2018).
[Crossref]

X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
[Crossref] [PubMed]

X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
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C. Lutzweiler, X. L. Deán-Ben, and D. Razansky, “Expediting model-based optoacoustic reconstructions with tomographic symmetries,” Med. Phys. 41(1), 013302 (2013).
[Crossref] [PubMed]

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imaging 31(10), 1922–1928 (2012).
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J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
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A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
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T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
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L. Ding, X. L. Dean-Ben, and D. Razansky, “Efficient 3-D Model-Based Reconstruction Scheme for Arbitrary Optoacoustic Acquisition Geometries,” IEEE Trans. Med. Imaging 36(9), 1858–1867 (2017).
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G. Drozdov and A. Rosenthal, “Analysis of negatively focused ultrasound detectors in optoacoustic tomography,” IEEE Trans. Med. Imaging 36(1), 301–309 (2017).
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Durduran, T.

Duric, N.

C. Li, N. Duric, P. Littrup, and L. Huang, “In vivo breast sound-speed imaging with ultrasound tomography,” Ultrasound Med. Biol. 35(10), 1615–1628 (2009).
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Englmeier, K.-H.

Estrada, H.

Fehm, T. F.

X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
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Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry,” IEEE Trans. Med. Imaging 21(7), 823–828 (2002).
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X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
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Funk, L.

Gateau, J.

Giannoula, A.

Gottschalk, S.

X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
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P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
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M. Haltmeier, “Universal inversion formulas for recovering a function from spherical means,” SIAM J. Math. Anal. 46(1), 214–232 (2014).
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P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
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P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046706 (2007).
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Hecksher-Sørensen, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Hill, B.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
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Hsieh, J.

Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
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H. Huang, G. Bustamante, R. Peterson, and J. Y. Ye, “An adaptive filtered back-projection for photoacoustic image reconstruction,” Med. Phys. 42(5), 2169–2178 (2015).
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C. Li, N. Duric, P. Littrup, and L. Huang, “In vivo breast sound-speed imaging with ultrasound tomography,” Ultrasound Med. Biol. 35(10), 1615–1628 (2009).
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Jetzfellner, T.

A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
[Crossref] [PubMed]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
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X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
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Jiang, S. B.

X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
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D. Razansky, S. Kellnberger, and V. Ntziachristos, “Near-field radiofrequency thermoacoustic tomography with impulse excitation,” Med. Phys. 37(9), 4602–4607 (2010).
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Kunyansky, L. A.

L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23(1), 373–383 (2007).
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J. Provost and F. Lesage, “The application of compressed sensing for photo-acoustic tomography,” IEEE Trans. Med. Imaging 28(4), 585–594 (2009).
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R. M. Lewitt and S. Matej, “Overview of methods for image reconstruction from projections in emission computed tomography,” Proc. IEEE 91(10), 1588–1611 (2003).
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C. Zhang, C. Li, and L. V. Wang, “Fast and robust deconvolution-based image reconstruction for photoacoustic tomography in circular geometry: experimental validation,” IEEE Photonics J. 2(1), 57–66 (2010).
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C. Li, N. Duric, P. Littrup, and L. Huang, “In vivo breast sound-speed imaging with ultrasound tomography,” Ultrasound Med. Biol. 35(10), 1615–1628 (2009).
[Crossref] [PubMed]

Li, R.

X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
[Crossref] [PubMed]

Littrup, P.

C. Li, N. Duric, P. Littrup, and L. Huang, “In vivo breast sound-speed imaging with ultrasound tomography,” Ultrasound Med. Biol. 35(10), 1615–1628 (2009).
[Crossref] [PubMed]

Lou, Y.

X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
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C. Lutzweiler, X. L. Deán-Ben, and D. Razansky, “Expediting model-based optoacoustic reconstructions with tomographic symmetries,” Med. Phys. 41(1), 013302 (2013).
[Crossref] [PubMed]

Ma, R.

X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
[Crossref] [PubMed]

Matej, S.

R. M. Lewitt and S. Matej, “Overview of methods for image reconstruction from projections in emission computed tomography,” Proc. IEEE 91(10), 1588–1611 (2003).
[Crossref]

Matt, G. J.

P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046706 (2007).
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Minagawa, T.

Ntziachristos, V.

A. Taruttis and V. Ntziachristos, “Advances in real-time multispectral optoacoustic imaging and its applications,” Nat. Photonics 9(4), 219–227 (2015).
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A. Rosenthal, V. Ntziachristos, and D. Razansky, “Acoustic Inversion in optoacoustic tomography: a review,” Curr. Med. Imaging Rev. 9(4), 318–336 (2014).
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M. Omar, J. Gateau, and V. Ntziachristos, “Raster-scan optoacoustic mesoscopy in the 25-125 MHz range,” Opt. Lett. 38(14), 2472–2474 (2013).
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X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
[Crossref] [PubMed]

M. Á. Araque Caballero, A. Rosenthal, J. Gateau, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic imaging using focused detector scanning,” Opt. Lett. 37(19), 4080–4082 (2012).
[Crossref] [PubMed]

A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
[Crossref] [PubMed]

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imaging 31(10), 1922–1928 (2012).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Model-based optoacoustic inversion with arbitrary-shape detectors,” Med. Phys. 38(7), 4285–4295 (2011).
[Crossref] [PubMed]

D. Razansky, A. Buehler, and V. Ntziachristos, “Volumetric real-time multispectral optoacoustic tomography of biomarkers,” Nat. Protoc. 6(8), 1121–1129 (2011).
[Crossref] [PubMed]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
[Crossref] [PubMed]

A. Rosenthal, D. Razansky, and V. Ntziachristos, “Fast semi-analytical model-based acoustic inversion for quantitative optoacoustic tomography,” IEEE Trans. Med. Imaging 29(6), 1275–1285 (2010).
[Crossref] [PubMed]

D. Razansky, S. Kellnberger, and V. Ntziachristos, “Near-field radiofrequency thermoacoustic tomography with impulse excitation,” Med. Phys. 37(9), 4602–4607 (2010).
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Oraevsky, A. A.

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(17), 5399–5423 (2012).
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Özbek, A.

Paige, C. C.

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8(1), 43–71 (1982).
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Paltauf, G.

P. Burgholzer, G. J. Matt, M. Haltmeier, and G. Paltauf, “Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(4), 046706 (2007).
[Crossref] [PubMed]

P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
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X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inverse Probl. 25(12), 123009 (2009).
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Perry, P.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Peterson, R.

H. Huang, G. Bustamante, R. Peterson, and J. Y. Ye, “An adaptive filtered back-projection for photoacoustic image reconstruction,” Med. Phys. 42(5), 2169–2178 (2015).
[Crossref] [PubMed]

Pogue, B. W.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
[Crossref] [PubMed]

Prakash, J.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
[Crossref] [PubMed]

J. Prakash, A. S. Raju, C. B. Shaw, M. Pramanik, and P. K. Yalavarthy, “Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography,” Biomed. Opt. Express 5(5), 1363–1377 (2014).
[Crossref] [PubMed]

C. B. Shaw, J. Prakash, M. Pramanik, and P. K. Yalavarthy, “Least squares QR-based decomposition provides an efficient way of computing optimal regularization parameter in photoacoustic tomography,” J. Biomed. Opt. 18(8), 080501 (2013).
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Pramanik, M.

J. Prakash, A. S. Raju, C. B. Shaw, M. Pramanik, and P. K. Yalavarthy, “Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography,” Biomed. Opt. Express 5(5), 1363–1377 (2014).
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C. B. Shaw, J. Prakash, M. Pramanik, and P. K. Yalavarthy, “Least squares QR-based decomposition provides an efficient way of computing optimal regularization parameter in photoacoustic tomography,” J. Biomed. Opt. 18(8), 080501 (2013).
[Crossref] [PubMed]

Provost, J.

J. Provost and F. Lesage, “The application of compressed sensing for photo-acoustic tomography,” IEEE Trans. Med. Imaging 28(4), 585–594 (2009).
[Crossref] [PubMed]

Raju, A. S.

Razansky, D.

A. Özbek, X. L. Deán-Ben, and D. Razansky, “Optoacoustic imaging at kilohertz volumetric frame rates,” Optica 5(7), 857 (2018).
[Crossref]

L. Ding, X. L. Dean-Ben, and D. Razansky, “Efficient 3-D Model-Based Reconstruction Scheme for Arbitrary Optoacoustic Acquisition Geometries,” IEEE Trans. Med. Imaging 36(9), 1858–1867 (2017).
[Crossref] [PubMed]

X. L. Deán-Ben, T. F. Fehm, S. J. Ford, S. Gottschalk, and D. Razansky, “Spiral volumetric optoacoustic tomography visualizes multi-scale dynamics in mice,” Light Sci. Appl. 6(4), e16247 (2017).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Acoustic Inversion in optoacoustic tomography: a review,” Curr. Med. Imaging Rev. 9(4), 318–336 (2014).
[Crossref] [PubMed]

J. Turner, H. Estrada, M. Kneipp, and D. Razansky, “Improved optoacoustic microscopy through three-dimensional spatial impulse response synthetic aperture focusing technique,” Opt. Lett. 39(12), 3390–3393 (2014).
[Crossref] [PubMed]

X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
[Crossref] [PubMed]

C. Lutzweiler, X. L. Deán-Ben, and D. Razansky, “Expediting model-based optoacoustic reconstructions with tomographic symmetries,” Med. Phys. 41(1), 013302 (2013).
[Crossref] [PubMed]

A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
[Crossref] [PubMed]

M. Á. Araque Caballero, A. Rosenthal, J. Gateau, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic imaging using focused detector scanning,” Opt. Lett. 37(19), 4080–4082 (2012).
[Crossref] [PubMed]

X. L. Deán-Ben, A. Buehler, V. Ntziachristos, and D. Razansky, “Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imaging 31(10), 1922–1928 (2012).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Model-based optoacoustic inversion with arbitrary-shape detectors,” Med. Phys. 38(7), 4285–4295 (2011).
[Crossref] [PubMed]

D. Razansky, A. Buehler, and V. Ntziachristos, “Volumetric real-time multispectral optoacoustic tomography of biomarkers,” Nat. Protoc. 6(8), 1121–1129 (2011).
[Crossref] [PubMed]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
[Crossref] [PubMed]

A. Rosenthal, D. Razansky, and V. Ntziachristos, “Fast semi-analytical model-based acoustic inversion for quantitative optoacoustic tomography,” IEEE Trans. Med. Imaging 29(6), 1275–1285 (2010).
[Crossref] [PubMed]

D. Razansky, S. Kellnberger, and V. Ntziachristos, “Near-field radiofrequency thermoacoustic tomography with impulse excitation,” Med. Phys. 37(9), 4602–4607 (2010).
[Crossref] [PubMed]

Rosenthal, A.

G. Drozdov and A. Rosenthal, “Analysis of negatively focused ultrasound detectors in optoacoustic tomography,” IEEE Trans. Med. Imaging 36(1), 301–309 (2017).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Acoustic Inversion in optoacoustic tomography: a review,” Curr. Med. Imaging Rev. 9(4), 318–336 (2014).
[Crossref] [PubMed]

X. L. Deán-Ben, R. Ma, A. Rosenthal, V. Ntziachristos, and D. Razansky, “Weighted model-based optoacoustic reconstruction in acoustic scattering media,” Phys. Med. Biol. 58(16), 5555–5566 (2013).
[Crossref] [PubMed]

A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
[Crossref] [PubMed]

M. Á. Araque Caballero, A. Rosenthal, J. Gateau, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic imaging using focused detector scanning,” Opt. Lett. 37(19), 4080–4082 (2012).
[Crossref] [PubMed]

A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
[Crossref] [PubMed]

A. Rosenthal, V. Ntziachristos, and D. Razansky, “Model-based optoacoustic inversion with arbitrary-shape detectors,” Med. Phys. 38(7), 4285–4295 (2011).
[Crossref] [PubMed]

A. Rosenthal, D. Razansky, and V. Ntziachristos, “Fast semi-analytical model-based acoustic inversion for quantitative optoacoustic tomography,” IEEE Trans. Med. Imaging 29(6), 1275–1285 (2010).
[Crossref] [PubMed]

T. Jetzfellner, A. Rosenthal, A. Buehler, A. Dima, K.-H. Englmeier, V. Ntziachristos, and D. Razansky, “Optoacoustic tomography with varying illumination and non-uniform detection patterns,” J. Opt. Soc. Am. A 27(11), 2488–2495 (2010).
[Crossref] [PubMed]

Ross, A.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Sauer, K. D.

Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
[Crossref] [PubMed]

Saunders, M. A.

C. C. Paige and M. A. Saunders, “LSQR: an algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8(1), 43–71 (1982).
[Crossref]

Sharpe, J.

J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002).
[Crossref] [PubMed]

Shaw, C. B.

J. Prakash, A. S. Raju, C. B. Shaw, M. Pramanik, and P. K. Yalavarthy, “Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography,” Biomed. Opt. Express 5(5), 1363–1377 (2014).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Song, W. Y.

X. Jia, Y. Lou, R. Li, W. Y. Song, and S. B. Jiang, “GPU-based fast cone beam CT reconstruction from undersampled and noisy projection data via total variation,” Med. Phys. 37(4), 1757–1760 (2010).
[Crossref] [PubMed]

Su, R.

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(17), 5399–5423 (2012).
[Crossref] [PubMed]

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Turner, J.

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X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inverse Probl. 25(12), 123009 (2009).
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Wang, K.

K. Wang and M. A. Anastasio, “A simple Fourier transform-based reconstruction formula for photoacoustic computed tomography with a circular or spherical measurement geometry,” Phys. Med. Biol. 57(23), N493–N499 (2012).
[Crossref] [PubMed]

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(17), 5399–5423 (2012).
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Wang, L. V.

C. Zhang, C. Li, and L. V. Wang, “Fast and robust deconvolution-based image reconstruction for photoacoustic tomography in circular geometry: experimental validation,” IEEE Photonics J. 2(1), 57–66 (2010).
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M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(1), 016706 (2005).
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Xu, Y.

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry,” IEEE Trans. Med. Imaging 21(7), 823–828 (2002).
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Yalavarthy, P. K.

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
[Crossref] [PubMed]

J. Prakash, A. S. Raju, C. B. Shaw, M. Pramanik, and P. K. Yalavarthy, “Basis pursuit deconvolution for improving model-based reconstructed images in photoacoustic tomography,” Biomed. Opt. Express 5(5), 1363–1377 (2014).
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C. B. Shaw, J. Prakash, M. Pramanik, and P. K. Yalavarthy, “Least squares QR-based decomposition provides an efficient way of computing optimal regularization parameter in photoacoustic tomography,” J. Biomed. Opt. 18(8), 080501 (2013).
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Ye, J. Y.

H. Huang, G. Bustamante, R. Peterson, and J. Y. Ye, “An adaptive filtered back-projection for photoacoustic image reconstruction,” Med. Phys. 42(5), 2169–2178 (2015).
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Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
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C. Zhang, C. Li, and L. V. Wang, “Fast and robust deconvolution-based image reconstruction for photoacoustic tomography in circular geometry: experimental validation,” IEEE Photonics J. 2(1), 57–66 (2010).
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Zhang, P.

Y. Zhao, X. Zhao, and P. Zhang, “An extended algebraic reconstruction technique (E-ART) for dual spectral CT,” IEEE Trans. Med. Imaging 34(3), 761–768 (2015).
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Y. Zhao, X. Zhao, and P. Zhang, “An extended algebraic reconstruction technique (E-ART) for dual spectral CT,” IEEE Trans. Med. Imaging 34(3), 761–768 (2015).
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Zhao, Y.

Y. Zhao, X. Zhao, and P. Zhang, “An extended algebraic reconstruction technique (E-ART) for dual spectral CT,” IEEE Trans. Med. Imaging 34(3), 761–768 (2015).
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IEEE Photonics J. (1)

C. Zhang, C. Li, and L. V. Wang, “Fast and robust deconvolution-based image reconstruction for photoacoustic tomography in circular geometry: experimental validation,” IEEE Photonics J. 2(1), 57–66 (2010).
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IEEE Trans. Image Process. (1)

Z. Yu, J. B. Thibault, C. A. Bouman, K. D. Sauer, and J. Hsieh, “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization,” IEEE Trans. Image Process. 20(1), 161–175 (2011).
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L. Ding, X. L. Dean-Ben, and D. Razansky, “Efficient 3-D Model-Based Reconstruction Scheme for Arbitrary Optoacoustic Acquisition Geometries,” IEEE Trans. Med. Imaging 36(9), 1858–1867 (2017).
[Crossref] [PubMed]

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry,” IEEE Trans. Med. Imaging 21(7), 823–828 (2002).
[Crossref] [PubMed]

J. Prakash, H. Dehghani, B. W. Pogue, and P. K. Yalavarthy, “Model-resolution-based basis pursuit deconvolution improves diffuse optical tomographic imaging,” IEEE Trans. Med. Imaging 33(4), 891–901 (2014).
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A. Rosenthal, T. Jetzfellner, D. Razansky, and V. Ntziachristos, “Efficient framework for model-based tomographic image reconstruction using wavelet packets,” IEEE Trans. Med. Imaging 31(7), 1346–1357 (2012).
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P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier, and G. Paltauf, “Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors,” Inverse Probl. 23(6), S65–S80 (2007).
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L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23(1), 373–383 (2007).
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C. B. Shaw, J. Prakash, M. Pramanik, and P. K. Yalavarthy, “Least squares QR-based decomposition provides an efficient way of computing optimal regularization parameter in photoacoustic tomography,” J. Biomed. Opt. 18(8), 080501 (2013).
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A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky, and V. Ntziachristos, “Model-based optoacoustic inversions with incomplete projection data,” Med. Phys. 38(3), 1694–1704 (2011).
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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(17), 5399–5423 (2012).
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K. Wang and M. A. Anastasio, “A simple Fourier transform-based reconstruction formula for photoacoustic computed tomography with a circular or spherical measurement geometry,” Phys. Med. Biol. 57(23), N493–N499 (2012).
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Figures (8)

Fig. 1
Fig. 1 (a) The grid of the image and detector locations used for calculating the model matrixM. The image is divided into N x × N y square pixels with a pixel area of Δ x Δ y and the acoustic signals are sampled at N p positions over a line with a distance of Δ x between them. (b) The image grid on which the projection operator K is calculated. Here, only a single back-projection is calculated, and the number of pixels in the x directions is increased to N K = N x + N p 1.
Fig. 2
Fig. 2 An illustration of how the back-projection operator is applied on data from different detectors. The back-projected images are shown by the arcs and their span in the x direction is N K pixels. The back-projection operation is the same for all the detector locations, but the position of the projected image is shifted so it is always centered on the position of the detector. All the back-projected images overlap with the region of interest in which the image is to be reconstructed (shown in the gray square), where the final reconstruction is a sum of all the back-projected images.
Fig. 3
Fig. 3 (a-c) Images from the back-projection operatorKfor different time instants. The results were obtained with 120 iterations of the LSQR algorithm applied on Eq. (A9). (d) A 1D slice (solid curve) taken from the back-projection operator Kshown in Fig. 3(b) (t = 11 µs) at x = 0. The slice, obtained with 120 iterations, is compared to the one obtained with a single iteration (dashed curve).
Fig. 4
Fig. 4 (a-c) The input images from which the projection data originated and the corresponding reconstructions obtained via (d-f) the back-projection (BP) formula in Eq. (2); (g-l) the algebraic approach in which Eq. (5) was solved by using the LSQR algorithm with (g-i) a single iteration and (j-l) 120 iterations; and (m-r) the proposed algebraic back-projection (ABP) algorithm in which Eq. (A9) was solved by using the LSQR algorithm with (m-o) a single iteration and (p-r) 120 iterations.
Fig. 5
Fig. 5 The reconstruction of the images of Fig. 4(a)-4(c) from noisy data. The reconstructions were performed using (a-c) the back-projection formula (BP) in Eq. (2), (d-f) algebraic solution to Eq. (5) with 120 iterations and (g-i) the number of iterations that minimized the RMSE, and (j-l) algebraic back-projection (ABP) with 120 iterations in the solution of Eq. (A9).
Fig. 6
Fig. 6 The 1D profiles of the reconstructions shown in 5(a), 5(d), 5(g) and 5(j) over the horizontal line y = 0.
Fig. 7
Fig. 7 (a-c) The root-mean-square error (RMSE) of the algebraic and ABP reconstructions as a function of iteration for the images in Figs. 4(a)-4(c), respectively. While the algebraic reconstructions exhibit a lower RMSE, their RMSE is more susceptible to noise. The RMSE was calculated from the difference between the reconstructions and originating images. (d-f) The signal-to-noise ratio (SNR) of the algebraic and ABP reconstructions as a function of iteration for the images in Figs. 4(a)-4(c), respectively. The results are compared to the ones achieved by the back-projection (BP) formula. The SNR was calculated for each image dividing its maximum value with the standard deviation in a 10 × 10 pixel area on its top left. For all images, ABP obtained the highest SNR, while the BP formula obtained the lowest.
Fig. 8
Fig. 8 (a-c) Reconstruction of a microsphere with a diameter of 0.3 mm from experimental data using (a) BP (b) algebraic inversion with 5 iterations and (c) ABP operator calculated with 120 iterations. For the algebraic inversion, no further improvement in the peak value or width of the microsphere reconstruction was obtained with more iterations. (d) a 1D plot of the reconstruction through the horizontal line y = 3.72 mm, showing the 1D profile of the microsphere reconstructions.

Equations (24)

Equations on this page are rendered with MathJax. Learn more.

p(r,t)= Γ 4πc t |rr'|=ct H(r') |rr'| dr',
H(r)= 2 Γ |rr'|=ct [ p(r',t)t t p(r',t) ] d Ω r (r') Ω 0 ,
d Ω r (r')= dS' |r'r | 2 [ n(r') r'r |r'r| ],
p=Mh,
h rec =argmin||pMh| | 2 ,
h rec = M p
p=( p 1 p 2 p N p 1 p N p ).
K=( K 1 K 2 K N K 1 K N K ),
h rec = K ^ p
K ^ =( K N p K N p 1 K 2 K 1 K N p +1 K N p K 3 K 2 K N p + N x 1 K N p + N x 2 K N x +1 K N x ),
p rec =M K ^ p.
M K ^ = I N p N t ,
K ν,1 =( K N p K N p +1 K N p + N x 1 ); K ν,2 =( K N p 1 K N p K N p + N x 2 ); K ν, N p 1 =( K 2 K 3 K N x +1 ); K ν, N p =( K 1 K 2 K N x ),
I ν,1 =( I N t 0 0 0 ); I ν,2 =( 0 I N t 0 0 ) I ν, N p 1 =( 0 0 I N t 0 ); I ν, N p =( 0 0 0 I N t ),
( M 0 0 0 0 M 0 0 0 0 M 0 0 0 0 M )( K ν,1 K ν,2 K ν, N p 1 K ν, N p )=( I ν,1 I ν,2 I ν, N p 1 I ν, N p ).
K ν =( 0 N x N y × N y ( N K N X ) I N x N y 0 N x N y × N y ( N K N X 1) I N x N y 0 N x N y × N y 0 N x N y × N y ( N K N X 2) I N x N y 0 N x N y ×2 N y 0 N x N y ×2 N y I N x N y 0 N x N y × N y ( N K N X 2) 0 N x N y × N y I N x N y 0 N x N y × N y ( N K N X 1) I N x N y 0 N x N y × N y ( N K N X ) )K.
M S K= I ν ,
M s =( 0 N t N p × N y ( N K N X ) M 0 N t N p × N y ( N K N X 1) M 0 N t N p × N y 0 N t N p × N y ( N K N X 2) M 0 N t N p ×2 N y 0 N t N p ×2 N y M 0 N t N p × N y ( N K N X 2) 0 N t N p × N y M 0 N t N p × N y ( N K N X 1) M 0 N t N p × N y ( N K N X ) ).
K=( K c,1 K c,2 K c, N t 1 K c, N t ),
I v =( I c,1 I c,2 I c, N t 1 I c, N t ),
M s K c,n = I c,n {n=1 N t }.
min M s K c,n I c,n 2 {n=1 N t },
p mat =( p 1 p 2 p N p 1 p N p ),
h rec = n=0 N p 1 h mat,n ( N y N p n N y : N y N k n N y ),

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