Abstract

The model-based image reconstruction approaches in photoacoustic tomography have a distinct advantage compared to traditional analytical methods for cases where limited data is available. These methods typically deploy Tikhonov based regularization scheme to reconstruct the initial pressure from the boundary acoustic data. The model-resolution for these cases represents the blur induced by the regularization scheme. A method that utilizes this blurring model and performs the basis pursuit deconvolution to improve the quantitative accuracy of the reconstructed photoacoustic image is proposed and shown to be superior compared to other traditional methods via three numerical experiments. Moreover, this deconvolution including the building of an approximate blur matrix is achieved via the Lanczos bidagonalization (least-squares QR) making this approach attractive in real-time.

© 2014 Optical Society of America

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2014 (1)

J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron.20, 6800609 (2014).
[CrossRef]

2013 (5)

J. Chen, R. Lin, H. Wang, J. Meng, H. Zheng, and L. Song, “Blind-deconvolution optical-resolution photoacoustic microscopy in vivo,” Opt. Express21, 7316–7327 (2013).
[CrossRef] [PubMed]

N. A. Rejesh, H. Pullagurla, and M. Pramanik, “Deconvolution based deblurring of reconstructed images in photoacoustic/thermoacoustic tomography,” J. Opt. Soc. Am. A30, 1994–2001 (2013).
[CrossRef]

C. Huang, K. Wang, L. Nie, L. V. Wang, and M. A. Anastasio, “Full-Wave Iterative Image Reconstruction in Photoacoustic Tomography with Acoustically Inhomogeneous Media,” IEEE Trans. Med. Imag.32, 1097–1110 (2013).
[CrossRef]

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, 080501:1–3 (2013).

J. Prakash and P. K. Yalavarthy, “A LSQR-type method provides a computationally efficient automated optimal choice of regularization parameter in diffuse optical tomography,” Med. Phys.40, 033101 (2013).
[CrossRef] [PubMed]

2012 (4)

K. Wang, R. Su, A. A. Oraevsky, and M. A. Anastasio, “Investigation of iterative image reconstruction in three-dimensional optoacoustic tomography,” Phys. Med. Bio.57, 5399–5423 (2012).
[CrossRef]

X. L. Dean-Ben, V. Ntziachristos, and D. Razansky, “Acceleration of Optoacoustic Model-Based Reconstruction Using Angular Image Discretization,” IEEE Trans. Med. Imag.31, 1154–1162 (2012).
[CrossRef]

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, 1922–1928 (2012).
[CrossRef]

L. H. V. Wang and S. Hu, “Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs,” Science335, 1458–1462, (2012).
[CrossRef] [PubMed]

2011 (1)

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

2010 (3)

Z. Guo, C. H. Li, L. Song, and L. H. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt.15, 021311 (2010).
[CrossRef] [PubMed]

B. E. Treeby and B. T. Cox, “k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields,” J. Biomed. Opt.15, 021314 (2010).
[CrossRef] [PubMed]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast Image Recovery Using Variable Splitting and Constrained Optimization,” IEEE Trans. Image Process.19, 2345–2356 (2010).
[CrossRef] [PubMed]

2009 (2)

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

M. Pramanik and L. H. V. Wang, “Thermoacoustic and photoacoustic sensing of temperature,” J. Biomed. Opt.14, 054024 (2009).
[CrossRef] [PubMed]

2008 (8)

K. H. Song and L. H. V. Wang, “Noninvasive photoacoustic imaging of the thoracic cavity and the kidney in small and large animals,” Med. Phys.35, 4524–4529 (2008).
[CrossRef] [PubMed]

M. Pramanik, G. Ku, C. H. Li, and L. H. V. Wang, “Design and evaluation of a novel breast cancer detection system combining both thermoacoustic (TA) and photoacoustic (PA) tomography,” Med. Phys.35, 2218–2223 (2008).
[CrossRef] [PubMed]

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

P. Kuchment and L. Kunyansky, “Mathematics of thermoacoustic and photoacoustic tomography,” European J. App. Math.19, 191–224 (2008).

Y. Hristova, P. Kuchment, and L. V. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inv. Problems24, 055006 (2008).
[CrossRef]

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt.13, 054052 (2008).
[CrossRef] [PubMed]

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag.25, 21–30 (2008).
[CrossRef]

J. Romberg, “Imaging via compressive sampling,” IEEE Signal Process. Mag.25, 14–20 (2008).
[CrossRef]

2007 (1)

P. C. Hansen, “Regularization tools version 4.0 for Matlab 7.3,” Numerical Algorithms46, 189–194 (2007).
[CrossRef]

2006 (1)

H. F. Zhang, K. Maslov, G. Stoica, and L. H. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotech.24, 848–851 (2006).
[CrossRef]

2005 (2)

M. Xu and L. H. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Review E71, 016706 (2005).
[CrossRef]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Ku, and L. H. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag.24, 199–210 (2005).
[CrossRef]

2004 (2)

B. Yin, D. Xing, Y. Wang, Y. Zeng, Y. Tan, and Q. Chen, “Fast photoacoustic imaging system based on 320-element linear transducer array,” Phys. Med. Bio.49, 1339–1346 (2004).
[CrossRef]

Y. Wang, X. Y. Xie, X. D. Wang, G. Ku, K. L. Gill, D. P. O’Neal, G. Stoica, and L. H. V. Wang, “Photoacoustic Tomography of a Nanoshell Contrast Agent in the in Vivo Rat Brain,” Nano Lett.4, 1689–1692 (2004).
[CrossRef]

2003 (2)

A. A. Karabutov, E. V. Savateeva, and A. A. Oraevsky, “Optoacoustic tomography: New modality of laser diagnostic systems,” Laser Phys.13, 711–723 (2003).

K. P. Kostli and P. C. Beard, “Two-dimensional photoacoustic imaging by use of Fourier transform image reconstruction and a detector with an anisotropic response,” App. Opt.42, 1899–1908 (2003).
[CrossRef]

2002 (1)

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acous. Soc. Am.112, 1536–1544 (2002).
[CrossRef]

2001 (1)

M. E. Kilmer and D. P. OLeary, “Choosing regularization parameters in iterative methods for ill-posed problems,” SIAM J. Matrix Anal. Appl.22, 1204–1221 (2001).
[CrossRef]

2000 (1)

G. Ku and L. H. V. Wang, “Scanning thermoacoustic tomography in biological tissue,” Med. Phys.27, 1195–1202 (2000).
[CrossRef] [PubMed]

1996 (1)

J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized- likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process.5, 1346–1358 (1996).
[CrossRef]

1982 (1)

C. C. Paige and M. A. Saunders, “LSQR: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Software8, 43–71 (1982).
[CrossRef]

1973 (1)

B. R. Hunt, “The application of constrained least square estimation to image restoration by digital computer,” IEEE Trans. Comput.C-22, 805–812 (1973).
[CrossRef]

Afonso, M.

M. Figueiredo, J. Bioucas-Dias, and M. Afonso, “Fast Frame-Based Image Deconvolution Using Variable Splitting and Constrained Optimization,” IEEE Worskhop on Statistical Signal Processing, Cardiff, Wales, 2009.

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast Image Recovery Using Variable Splitting and Constrained Optimization,” IEEE Trans. Image Process.19, 2345–2356 (2010).
[CrossRef] [PubMed]

Anastasio, M. A.

C. Huang, K. Wang, L. Nie, L. V. Wang, and M. A. Anastasio, “Full-Wave Iterative Image Reconstruction in Photoacoustic Tomography with Acoustically Inhomogeneous Media,” IEEE Trans. Med. Imag.32, 1097–1110 (2013).
[CrossRef]

K. Wang, R. Su, A. A. Oraevsky, and M. A. Anastasio, “Investigation of iterative image reconstruction in three-dimensional optoacoustic tomography,” Phys. Med. Bio.57, 5399–5423 (2012).
[CrossRef]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Ku, and L. H. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag.24, 199–210 (2005).
[CrossRef]

Beard, P. C.

K. P. Kostli and P. C. Beard, “Two-dimensional photoacoustic imaging by use of Fourier transform image reconstruction and a detector with an anisotropic response,” App. Opt.42, 1899–1908 (2003).
[CrossRef]

Bioucas-Dias, J.

M. Figueiredo, J. Bioucas-Dias, and M. Afonso, “Fast Frame-Based Image Deconvolution Using Variable Splitting and Constrained Optimization,” IEEE Worskhop on Statistical Signal Processing, Cardiff, Wales, 2009.

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast Image Recovery Using Variable Splitting and Constrained Optimization,” IEEE Trans. Image Process.19, 2345–2356 (2010).
[CrossRef] [PubMed]

Bodapati, S.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

Buehler, A.

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, 1922–1928 (2012).
[CrossRef]

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

Candes, E.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag.25, 21–30 (2008).
[CrossRef]

Carson, J. J. L.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt.13, 054052 (2008).
[CrossRef] [PubMed]

Chen, J.

Chen, Q.

B. Yin, D. Xing, Y. Wang, Y. Zeng, Y. Tan, and Q. Chen, “Fast photoacoustic imaging system based on 320-element linear transducer array,” Phys. Med. Bio.49, 1339–1346 (2004).
[CrossRef]

Chen, X. Y.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

Cheng, Z.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

Cox, B. T.

B. E. Treeby and B. T. Cox, “k-Wave: MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields,” J. Biomed. Opt.15, 021314 (2010).
[CrossRef] [PubMed]

Dai, H. J.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

De la Zerda, A.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

Dean-Ben, X. L.

X. L. Dean-Ben, V. Ntziachristos, and D. Razansky, “Acceleration of Optoacoustic Model-Based Reconstruction Using Angular Image Discretization,” IEEE Trans. Med. Imag.31, 1154–1162 (2012).
[CrossRef]

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, 1922–1928 (2012).
[CrossRef]

Dehghani, H.

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. Imag. 2014 (in press).
[CrossRef]

Dima, A.

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

Ephrat, P.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt.13, 054052 (2008).
[CrossRef] [PubMed]

Fessler, J. A.

J. A. Fessler and W. L. Rogers, “Spatial resolution properties of penalized- likelihood image reconstruction methods: Space-invariant tomographs,” IEEE Trans. Image Process.5, 1346–1358 (1996).
[CrossRef]

Figueiredo, M.

M. Figueiredo, J. Bioucas-Dias, and M. Afonso, “Fast Frame-Based Image Deconvolution Using Variable Splitting and Constrained Optimization,” IEEE Worskhop on Statistical Signal Processing, Cardiff, Wales, 2009.

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast Image Recovery Using Variable Splitting and Constrained Optimization,” IEEE Trans. Image Process.19, 2345–2356 (2010).
[CrossRef] [PubMed]

Gambhir, S. S.

A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
[CrossRef]

Gill, K. L.

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P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt.13, 054052 (2008).
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A. De la Zerda, C. Zavaleta, S. Keren, S. Vaithilingam, S. Bodapati, Z. Liu, J. Levi, B. R. Smith, T. J. Ma, O. Oralkan, Z. Cheng, X. Y. Chen, H. J. Dai, B. T. Khuri-Yakub, and S. S. Gambhir, “Carbon nanotubes as photoacoustic molecular imaging agents in living mice,” Nat. Nanotech.3, 557–562 (2008).
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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. Bio.57, 5399–5423 (2012).
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L. H. V. Wang and S. Hu, “Photoacoustic Tomography: In Vivo Imaging from Organelles to Organs,” Science335, 1458–1462, (2012).
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Z. Guo, C. H. Li, L. Song, and L. H. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt.15, 021311 (2010).
[CrossRef] [PubMed]

M. Pramanik and L. H. V. Wang, “Thermoacoustic and photoacoustic sensing of temperature,” J. Biomed. Opt.14, 054024 (2009).
[CrossRef] [PubMed]

K. H. Song and L. H. V. Wang, “Noninvasive photoacoustic imaging of the thoracic cavity and the kidney in small and large animals,” Med. Phys.35, 4524–4529 (2008).
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M. Pramanik, G. Ku, C. H. Li, and L. H. V. Wang, “Design and evaluation of a novel breast cancer detection system combining both thermoacoustic (TA) and photoacoustic (PA) tomography,” Med. Phys.35, 2218–2223 (2008).
[CrossRef] [PubMed]

H. F. Zhang, K. Maslov, G. Stoica, and L. H. V. Wang, “Functional photoacoustic microscopy for high-resolution and noninvasive in vivo imaging,” Nat. Biotech.24, 848–851 (2006).
[CrossRef]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Ku, and L. H. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag.24, 199–210 (2005).
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Y. Wang, X. Y. Xie, X. D. Wang, G. Ku, K. L. Gill, D. P. O’Neal, G. Stoica, and L. H. V. Wang, “Photoacoustic Tomography of a Nanoshell Contrast Agent in the in Vivo Rat Brain,” Nano Lett.4, 1689–1692 (2004).
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[CrossRef] [PubMed]

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C. Huang, K. Wang, L. Nie, L. V. Wang, and M. A. Anastasio, “Full-Wave Iterative Image Reconstruction in Photoacoustic Tomography with Acoustically Inhomogeneous Media,” IEEE Trans. Med. Imag.32, 1097–1110 (2013).
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[CrossRef]

B. Yin, D. Xing, Y. Wang, Y. Zeng, Y. Tan, and Q. Chen, “Fast photoacoustic imaging system based on 320-element linear transducer array,” Phys. Med. Bio.49, 1339–1346 (2004).
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Y. Wang, X. Y. Xie, X. D. Wang, G. Ku, K. L. Gill, D. P. O’Neal, G. Stoica, and L. H. V. Wang, “Photoacoustic Tomography of a Nanoshell Contrast Agent in the in Vivo Rat Brain,” Nano Lett.4, 1689–1692 (2004).
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B. Yin, D. Xing, Y. Wang, Y. Zeng, Y. Tan, and Q. Chen, “Fast photoacoustic imaging system based on 320-element linear transducer array,” Phys. Med. Bio.49, 1339–1346 (2004).
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J. Prakash, C. B. Shaw, R. Manjappa, R. Kanhirodan, and P. K. Yalavarthy, “Sparse Recovery Methods Hold Promise for Diffuse Optical Tomographic Image Reconstruction,” IEEE J. Sel. Top. Quantum Electron.20, 6800609 (2014).
[CrossRef]

J. Prakash and P. K. Yalavarthy, “A LSQR-type method provides a computationally efficient automated optimal choice of regularization parameter in diffuse optical tomography,” Med. Phys.40, 033101 (2013).
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Figures (6)

Fig. 1
Fig. 1

Schematic of Photoacoustic data acquisition set-up with shaded square region indicating the imaging domain.

Fig. 2
Fig. 2

Flowchart showing the major steps used in the proposed method.

Fig. 3
Fig. 3

(a) Derenzo phantom that is used for the study of resolution characteristics (dimensions are in millimeters). (b–h) Reconstructed photoacoustic images using k-wave interpolated, LSQR with optimal choice of λ, LSQR with heuristic choice of λ, Basis Pursuit Deconvolution (BPD) with optimal choice of λ in LSQR framework, BPD with heuristic choice of λ and 40 dB noise in LSQR framework, BPD with heuristic choice of λ and 30 dB noise in LSQR framework, BPD with heuristic choice of λ and 20 dB noise in LSQR framework, respectively.

Fig. 4
Fig. 4

(a) Numerical blood vessel phantom that is used for the study (dimensions are in millimeters). (b–f) Reconstructed photoacoustic images using k-wave interpolated, LSQR with heuristic choice of λ, LSQR with optimal choice of λ, Basis pursuit deconvolution (BPD) with heuristic choice of λ in LSQR framework, BPD with optimal choice of λ in LSQR framework, respectively. (g) One-dimensional cross-sectional plot for the results presented in (a),(c),(d),(e), and (f) along the dotted line shown in (a).

Fig. 5
Fig. 5

(a) Numerical PAT phantom that is used for the study (dimensions are in millimeters). (b–f) Reconstructed photoacoustic images using k-wave interpolated, LSQR with heuristic choice of λ, LSQR with optimal choice of λ, Basis pursuit deconvolution (BPD) with heuristic choice of λ in LSQR framework, BPD with optimal choice of λ in LSQR framework, respectively. (g) One-dimensional cross-sectional plot for the results presented in (a),(c),(d),(e), and (f) along the dotted line shown in (a).

Fig. 6
Fig. 6

Plot of the variation of the Residual error and the number of Lanczos iterations for numerical blood vessel phantom (Fig. 4).

Tables (3)

Tables Icon

Algorithm 1: Spilt augmented lagrangian shrinkage algorithm (SALSA) [41]

Tables Icon

Table 1 The Pearson Correlation (PC) and Contrast to Noise Ratio (CNR) of the reconstructed initial pressure distribution for the results presented in this work.

Tables Icon

Table 2 Typical computational time (in seconds) for reconstructing the initial pressure distribution for the various methods presented in this work. Note that the time taken for building the system matrix (375 seconds) was excluded from this.

Equations (24)

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p 2 t 2 = c 2 ( r ) Δ r p , t 0 , r R 2 p ( r , 0 ) = f ( r ) , p t ( r , 0 ) = 0 , p ( y , t ) = g ( y , t ) , for y B , t 0
A x = b
x b p = A T b
Ω = A x b 2 2 + λ x 2 2
x Tikh = ( A T A + λ I ) 1 A T b
U k + 1 ( β 0 e 1 ) = b
A V k = U k + 1 B k
A T U k + 1 = V k B k T + α k + 1 v k + 1 e k + 1 T
U k = [ u 1 , u 2 , , u k ] ; V k = [ v 1 , v 2 , , v k ]
b A x = U k + 1 ( β 0 e 1 B k x ( k ) ) ; x = V k x ( k ) ,
Ω ˜ = β 0 e 1 B k x ( k ) 2 2 + λ x ( k ) 2 2
x est ( k ) = ( B k T B k + λ I ) 1 β 0 B k T e 1 ; x LSQR = V k x est ( k )
x est ( k ) = ( B k T B k + λ I ) 1 B k T U k + 1 T b
x est ( k ) = ( B k T B k + λ I ) 1 B k T U k + 1 T A x
x est ( k ) = ( B k T B k + λ I ) 1 B k T U k + 1 T A V k x ( k )
U k + 1 T A V k = B k
x est ( k ) = ( B k T B k + λ I ) 1 B k T B k x ( k )
x est ( k ) = M x ( k )
M = ( B k T B k + λ I ) 1 B k T B k
Ω ˜ = M x ( k ) x est ( k ) 2 2 + λ l 1 x ( k ) 1
x ˜ LSQR = V k x d ( k )
PC ( x , x recon ) = COV ( x , x recon ) σ ( x ) σ ( x recon )
CNR = μ roi μ back ( σ roi 2 a roi + σ back 2 a back ) 1 2
x ( k ) x d ( k ) 2 < x ( k ) x est ( k ) 2

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