Abstract

The sparse transforms currently used in the model-based reconstruction method for photoacoustic computed tomography (PACT) are predefined and they typically cannot capture the underlying features of the specific data sets adequately, thus limiting the high-quality recovery of photoacoustic images. In this work, we present an advanced reconstruction model using the K-VSD dictionary learning technique and present the in vivo results after adapting the model into the 3D PACT system. The in vivo experiments were performed on an IRB approved human hand and two rats. When compared to the traditional sparse transform, experimental results using our proposed method improved accuracy and contrast to noise ration of the reconstructed photoacoustic images, on average, by 3.7 and 1.8 times in the case of 50% sparse-sampling rate, respectively. We also compared the performance of our algorithm against other techniques, and imaging speed was 60% faster than other approaches. Our system would require sparse-transducer array and lower number of data acquisition hardware (DAQs) potentially reducing the cost of the system. Thus, our work provides a new way for reconstructing photoacoustic images, and it would enable the development of new high-speed low-cost 3D PACT for various biomedical applications.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (7)

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Z. Liu, L. Yu, and H. Sun, “Image restoration via Bayesian dictionary learning with nonlocal structured beta process,” J. Vis. Commun. Image Represent. 52, 159–169 (2018).
[Crossref]

C. Jiang, Q. Zhang, R. Fan, and Z. Hu, “Super-resolution CT Image Reconstruction Based on Dictionary Learning and Sparse Representation,” Sci. Rep. 8(1), 8799 (2018).
[Crossref] [PubMed]

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

2017 (2)

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

2016 (1)

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

2015 (2)

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

2014 (4)

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

Y. Wang and L. Ying, “Compressed sensing dynamic cardiac cine MRI using learned spatiotemporal dictionary,” IEEE Trans. Biomed. Eng. 61(4), 1109–1120 (2014).
[Crossref] [PubMed]

G. Paltauf and R. Nuster, “Artifact removal in photoacoustic section imaging by combining an integrating cylindrical detector with model-based reconstruction,” J. Biomed. Opt. 19(2), 026014 (2014).
[Crossref] [PubMed]

J. Xia and L. V. Wang, “Small-animal whole-body photoacoustic tomography: a review,” IEEE Trans. Biomed. Eng. 61(5), 1380–1389 (2014).
[Crossref] [PubMed]

2012 (3)

2011 (1)

L. Nie, Z. Guo, and L. V. Wang, “Photoacoustic tomography of monkey brain using virtual point ultrasonic transducers,” J. Biomed. Opt. 16(7), 076005 (2011).
[Crossref] [PubMed]

2010 (1)

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

2008 (1)

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

2007 (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

2003 (1)

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Adler, J.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Appleton, C. M.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Arabul, M. U.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Arridge, S.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Bai, T.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Beard, P.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Betcke, M.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Bitton, R.

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

Cao, N.

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

Cao, R.

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

Carney, P. R.

Chen, S.

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

Chen, Y.

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

Cox, B.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Engan, K.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Fan, R.

C. Jiang, Q. Zhang, R. Fan, and Z. Hu, “Super-resolution CT Image Reconstruction Based on Dictionary Learning and Sparse Representation,” Sci. Rep. 8(1), 8799 (2018).
[Crossref] [PubMed]

Govinahallisathyanarayana, S.

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

Guo, Z.

L. Nie, Z. Guo, and L. V. Wang, “Photoacoustic tomography of monkey brain using virtual point ultrasonic transducers,” J. Biomed. Opt. 16(7), 076005 (2011).
[Crossref] [PubMed]

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

Hauptmann, A.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Heres, H. M.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Hossack, J. A.

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

Hu, D.

L. Zhou, J. Wang, and D. Hu, “The application of dictionary based compressed sensing for photoacoustic image,” in International Conference on Machine Learning and Cybernetics (IEEE, 2015), pp. 98–102.

Hu, P.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Hu, S.

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

Hu, Z.

C. Jiang, Q. Zhang, R. Fan, and Z. Hu, “Super-resolution CT Image Reconstruction Based on Dictionary Learning and Sparse Representation,” Sci. Rep. 8(1), 8799 (2018).
[Crossref] [PubMed]

Huynh, N.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Jia, X.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Jiang, C.

C. Jiang, Q. Zhang, R. Fan, and Z. Hu, “Super-resolution CT Image Reconstruction Based on Dictionary Learning and Sparse Representation,” Sci. Rep. 8(1), 8799 (2018).
[Crossref] [PubMed]

Jiang, H.

Jiang, M. S.

Jiang, S.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Jiang, Z.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

Kim, C.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

Kreutz-Delgado, K.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Lee, T. W.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Li, C.

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

Li, L.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Liang, D.

Lin, L.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Lin, R.

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

Liu, C.

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

Liu, H.

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

Liu, Z.

Z. Liu, L. Yu, and H. Sun, “Image restoration via Bayesian dictionary learning with nonlocal structured beta process,” J. Vis. Commun. Image Represent. 52, 159–169 (2018).
[Crossref]

Liu, Z. J.

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

Lopata, R. G. P.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Lucka, F.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Maslov, K.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

Meng, G. F.

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

Meng, J.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

J. Meng, L. V. Wang, D. Liang, and L. Song, “In vivo optical-resolution photoacoustic computed tomography with compressed sensing,” Opt. Lett. 37(22), 4573–4575 (2012).
[Crossref] [PubMed]

J. Meng, L. V. Wang, L. Ying, D. Liang, and L. Song, “Compressed-sensing photoacoustic computed tomography in vivo with partially known support,” Opt. Express 20(15), 16510–16523 (2012).
[Crossref]

Miyazaki, Y.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

Mou, X.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Murray, J. F.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Nie, L.

L. Nie, Z. Guo, and L. V. Wang, “Photoacoustic tomography of monkey brain using virtual point ultrasonic transducers,” J. Biomed. Opt. 16(7), 076005 (2011).
[Crossref] [PubMed]

Ning, B.

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

Nuster, R.

G. Paltauf and R. Nuster, “Artifact removal in photoacoustic section imaging by combining an integrating cylindrical detector with model-based reconstruction,” J. Biomed. Opt. 19(2), 026014 (2014).
[Crossref] [PubMed]

Onuma, Y.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

Ourselin, S.

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

Paltauf, G.

G. Paltauf and R. Nuster, “Artifact removal in photoacoustic section imaging by combining an integrating cylindrical detector with model-based reconstruction,” J. Biomed. Opt. 19(2), 026014 (2014).
[Crossref] [PubMed]

Pan, C. H.

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

Park, J.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Rao, B. D.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Rutten, M. C. M.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Sejnowski, T. J.

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Serruys, P. W.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

Shi, J.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Shi, P.

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

Shung, K. K.

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

Song, L.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

J. Meng, L. V. Wang, L. Ying, D. Liang, and L. Song, “Compressed-sensing photoacoustic computed tomography in vivo with partially known support,” Opt. Express 20(15), 16510–16523 (2012).
[Crossref]

J. Meng, L. V. Wang, D. Liang, and L. Song, “In vivo optical-resolution photoacoustic computed tomography with compressed sensing,” Opt. Lett. 37(22), 4573–4575 (2012).
[Crossref] [PubMed]

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

Sotomi, Y.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

Sun, H.

Z. Liu, L. Yu, and H. Sun, “Image restoration via Bayesian dictionary learning with nonlocal structured beta process,” J. Vis. Commun. Image Represent. 52, 159–169 (2018).
[Crossref]

Sun, M.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

Suwannasom, P.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

Tenekecioglu, E.

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

van de Vosse, F. N.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

van Sambeek, M. R. H. M.

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Wang, B.

Wang, G.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Wang, J.

L. Zhou, J. Wang, and D. Hu, “The application of dictionary based compressed sensing for photoacoustic image,” in International Conference on Machine Learning and Cybernetics (IEEE, 2015), pp. 98–102.

Wang, L. V.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

J. Xia and L. V. Wang, “Small-animal whole-body photoacoustic tomography: a review,” IEEE Trans. Biomed. Eng. 61(5), 1380–1389 (2014).
[Crossref] [PubMed]

J. Meng, L. V. Wang, D. Liang, and L. Song, “In vivo optical-resolution photoacoustic computed tomography with compressed sensing,” Opt. Lett. 37(22), 4573–4575 (2012).
[Crossref] [PubMed]

J. Meng, L. V. Wang, L. Ying, D. Liang, and L. Song, “Compressed-sensing photoacoustic computed tomography in vivo with partially known support,” Opt. Express 20(15), 16510–16523 (2012).
[Crossref]

L. Nie, Z. Guo, and L. V. Wang, “Photoacoustic tomography of monkey brain using virtual point ultrasonic transducers,” J. Biomed. Opt. 16(7), 076005 (2011).
[Crossref] [PubMed]

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

Wang, Y.

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

Y. Wang and L. Ying, “Compressed sensing dynamic cardiac cine MRI using learned spatiotemporal dictionary,” IEEE Trans. Biomed. Eng. 61(4), 1109–1120 (2014).
[Crossref] [PubMed]

Xia, J.

J. Xia and L. V. Wang, “Small-animal whole-body photoacoustic tomography: a review,” IEEE Trans. Biomed. Eng. 61(5), 1380–1389 (2014).
[Crossref] [PubMed]

Xiang, L.

Xiang, S. M.

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

Yan, H.

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

Yang, J.

Ying, L.

Y. Wang and L. Ying, “Compressed sensing dynamic cardiac cine MRI using learned spatiotemporal dictionary,” IEEE Trans. Biomed. Eng. 61(4), 1109–1120 (2014).
[Crossref] [PubMed]

J. Meng, L. V. Wang, L. Ying, D. Liang, and L. Song, “Compressed-sensing photoacoustic computed tomography in vivo with partially known support,” Opt. Express 20(15), 16510–16523 (2012).
[Crossref]

Yu, L.

Z. Liu, L. Yu, and H. Sun, “Image restoration via Bayesian dictionary learning with nonlocal structured beta process,” J. Vis. Commun. Image Represent. 52, 159–169 (2018).
[Crossref]

Zhang, C. S.

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

Zhang, Q.

Zhang, R.

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Zhang, Y.

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

Zhang, Y. D.

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

Zheng, J.

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

Zhou, L.

L. Zhou, J. Wang, and D. Hu, “The application of dictionary based compressed sensing for photoacoustic image,” in International Conference on Machine Learning and Cybernetics (IEEE, 2015), pp. 98–102.

Biomed. Opt. Express (1)

Eur. Heart J. Cardiovasc. Imaging (1)

P. Suwannasom, Y. Sotomi, Y. Miyazaki, E. Tenekecioglu, Y. Onuma, and P. W. Serruys, “Multimodality imaging to detect vulnerable plaque in coronary arteries and its clinical application,” Eur. Heart J. Cardiovasc. Imaging 18(6), 613–620 (2018).
[PubMed]

IEEE Trans. Biomed. Eng. (2)

J. Xia and L. V. Wang, “Small-animal whole-body photoacoustic tomography: a review,” IEEE Trans. Biomed. Eng. 61(5), 1380–1389 (2014).
[Crossref] [PubMed]

Y. Wang and L. Ying, “Compressed sensing dynamic cardiac cine MRI using learned spatiotemporal dictionary,” IEEE Trans. Biomed. Eng. 61(4), 1109–1120 (2014).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (2)

T. Bai, H. Yan, X. Jia, S. Jiang, G. Wang, and X. Mou, “Z-Index parameterization for volumetric CT image reconstruction via 3-D dictionary learning,” IEEE Trans. Med. Imaging 36(12), 2466–2478 (2017).
[Crossref] [PubMed]

A. Hauptmann, F. Lucka, M. Betcke, N. Huynh, J. Adler, B. Cox, P. Beard, S. Ourselin, and S. Arridge, “Model based learning for accelerated, limited-view 3D photoacoustic tomography,” IEEE Trans. Med. Imaging 37(6), 1382–1393 (2018).
[Crossref] [PubMed]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

M. U. Arabul, H. M. Heres, M. C. M. Rutten, M. R. H. M. van Sambeek, F. N. van de Vosse, and R. G. P. Lopata, “Investigation on the effect of spatial compounding on photoacoustic images of carotid plaques in the in vivo available rotational range,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 65(3), 440–447 (2018).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

S. M. Xiang, G. F. Meng, Y. Wang, C. H. Pan, and C. S. Zhang, “Image deblurring with coupled dictionary learning,” Int. J. Comput. Vis. 114(2–3), 248–271 (2015).
[Crossref]

J. Biomed. Opt. (6)

J. Meng, Z. Jiang, L. V. Wang, J. Park, C. Kim, M. Sun, Y. Zhang, and L. Song, “High-speed, sparse-sampling three-dimensional photoacoustic computed tomography in vivo based on principal component analysis,” J. Biomed. Opt. 21(7), 076007 (2016).
[Crossref] [PubMed]

L. Song, K. Maslov, R. Bitton, K. K. Shung, and L. V. Wang, “Fast 3-D dark-field reflection-mode photoacoustic microscopy in vivo with a 30-MHz ultrasound linear array,” J. Biomed. Opt. 13(5), 054028 (2008).
[Crossref] [PubMed]

G. Paltauf and R. Nuster, “Artifact removal in photoacoustic section imaging by combining an integrating cylindrical detector with model-based reconstruction,” J. Biomed. Opt. 19(2), 026014 (2014).
[Crossref] [PubMed]

L. Nie, Z. Guo, and L. V. Wang, “Photoacoustic tomography of monkey brain using virtual point ultrasonic transducers,” J. Biomed. Opt. 16(7), 076005 (2011).
[Crossref] [PubMed]

Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010).
[Crossref] [PubMed]

J. Meng, C. Liu, J. Zheng, R. Lin, and L. Song, “Compressed sensing based virtual-detector photoacoustic microscopy in vivo,” J. Biomed. Opt. 19(3), 036003 (2014).
[Crossref] [PubMed]

J. Vis. Commun. Image Represent. (1)

Z. Liu, L. Yu, and H. Sun, “Image restoration via Bayesian dictionary learning with nonlocal structured beta process,” J. Vis. Commun. Image Represent. 52, 159–169 (2018).
[Crossref]

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007).
[Crossref] [PubMed]

Nat. Commun. (1)

L. Lin, P. Hu, J. Shi, C. M. Appleton, K. Maslov, L. Li, R. Zhang, and L. V. Wang, “Single-breath-hold photoacoustic computed tomography of the breast,” Nat. Commun. 9(1), 2352 (2018).
[Crossref] [PubMed]

Neural Comput. (1)

K. Kreutz-Delgado, J. F. Murray, B. D. Rao, K. Engan, T. W. Lee, and T. J. Sejnowski, “Dictionary learning algorithms for sparse representation,” Neural Comput. 15(2), 349–396 (2003).
[Crossref] [PubMed]

Neurocomputing (1)

Y. Wang, N. Cao, Z. J. Liu, and Y. D. Zhang, “Real-time dynamic MRI using parallel dictionary learning and dynamic total variation,” Neurocomputing 238, 410–419 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

S. Chen, H. Liu, P. Shi, and Y. Chen, “Sparse representation and dictionary learning penalized image reconstruction for positron emission tomography,” Phys. Med. Biol. 60(2), 807–823 (2015).
[Crossref] [PubMed]

Sci. Rep. (2)

S. Govinahallisathyanarayana, B. Ning, R. Cao, S. Hu, and J. A. Hossack, “Dictionary learning-based reverberation removal enables depth-resolved photoacoustic microscopy of cortical microvasculature in the mouse brain,” Sci. Rep. 8(1), 985 (2018).
[Crossref] [PubMed]

C. Jiang, Q. Zhang, R. Fan, and Z. Hu, “Super-resolution CT Image Reconstruction Based on Dictionary Learning and Sparse Representation,” Sci. Rep. 8(1), 8799 (2018).
[Crossref] [PubMed]

Other (5)

T. M. Quantm and W. K. Jeong, “Compressed sensing reconstruction of dynamic contrast enhanced MRI using GPU-accelerated convolutional sparse coding,” in 2016 IEEE 13th International Symposium on Biomedical Imaging (IEEE, 2016), pp. 518–521.

S. Ravishankar and Y. Bresler, “Sparsifying transform learning for compressed sensing MRI,” in 2013 IEEE 10th International Symposium on Biomedical Imaging (IEEE, 2013), pp. 17–20.
[Crossref]

L. Zhou, J. Wang, and D. Hu, “The application of dictionary based compressed sensing for photoacoustic image,” in International Conference on Machine Learning and Cybernetics (IEEE, 2015), pp. 98–102.

The Laser Institute of America, American National Standard for Safe Use of Lasers, (ANSI Z136.1–2000), The Laser Institute of America, (2000)

D. Karimi and R. K. Ward, ” A novel structured dictionary for fast processing of 3D medical images, with application to computed tomography restoration and denoising,” in Medical Imaging 2016: Image Processing (SPIE., 2016), 9784: 97840N.

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Figures (9)

Fig. 1
Fig. 1 Flowchart of our proposed algorithm based on learning dictionary
Fig. 2
Fig. 2 Training samples and the corresponding dictionaries. (A-C), MAP images of a human hand and two rats, reconstructed by BP with the data from all transducer elements; (1) – (8), 8 full-sampling photoacoustic images reconstructed by BP.
Fig. 3
Fig. 3 Reconstructed photoacoustic images and the errors of a human hand. (A1) – (A4), MAP images reconstructed by BP, Wavelet-TV, DL and DL-TV with 2/3 SR; (B1) – (B4), Results reconstructed by the four methods with 1/2 SR; (C1) – (C4), Results reconstructed by the four methods with 1/3 SR. x is the lateral direction of the transducer array, y is the mechanical scanning direction, colorbar represents the grayscale range of the error images.
Fig. 4
Fig. 4 Reconstructed results and errors of one B-scan (indicated by the yellow line in Fig. 3 (A1)) of the human hand. (a1) – (a4), B-scan images reconstructed by BP, Wavelet-TV, DL, and DL-TV with 2/3 SR; (b1) – (b4), Results reconstructed with 1/2 SR; (c1) – (c4), Results reconstructed with 1/3 SR; z represents the depth direction.
Fig. 5
Fig. 5 MAP photoacoustic images and the corresponding errors of Rat-A and Rat-B. (A1) – (A3) and (D1) – (D3), Results of Rat-A and Rat-B reconstructed by BP, Wavelet-TV and DL-TV with 2/3 SR, respectively; (B1) – (B3) and (E1) – (E3), Results of Rat-A and Rat-B reconstructed by the three methods with 1/2 SR, respectively; (C1) – (C3) and (F1) – (F3), Results of Rat-A and Rat-B reconstructed by the three methods with 1/3 SR, respectively.
Fig. 6
Fig. 6 Reconstructed results and errors of two B-scans from Rat-A and Rat-B (indicated by the vertical dashed lines shown in Figs. 5(A1) and (D1)). (a1) – (a3) and (d1) – (d3), B-scan images reconstructed by BP, Wavelet-TV, and DL-TV with 2/3 SR, respectively; (b1) – (b4) and (e1) – (e3), Results reconstructed with 1/2 SR of the two B-scans, respectively; (c1) – (c4) and (f1) – (f3), Results reconstructed with 1/3 SR of the two B-scans, respectively.
Fig. 7
Fig. 7 Photoacoustic amplitudes (relative optical absorption) along the chosen dashed lines in B-scan images of (a) Human hand (Ref. in Fig. 4), (b) Rat-A and (c) Rat-B (Refs. in Fig. 6).
Fig. 8
Fig. 8 MSE Curves of the 166-frame photoacoustic images reconstructed by different methods with 1/2 SR. (a), Curves of the human hand data set; (b), Curves of the Rat-A data set; (c), Curves of the Rat-B data set; (d) – (f), the bar graphs of maximum MSE, minimum MSE and average MSE of the three data sets.
Fig. 9
Fig. 9 Convergence curves with iteration number of the B-scans (a) one B-scan from the human hand, (b) one B-scan from the Rat-A, and (c) one B-scan from the Rat-B.

Tables (3)

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Table 1 Comparisons of the imaging speed of the different reconstruction methods

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Table 2 Algorithm 1. Optimization algorithm of DL-based reconstruction for one B-scan

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Table 2 Values of some key parameters used in our experiments

Equations (8)

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X
min D , α i = 1 S y i D α i 2 2 + μ i = 1 S α i 0
Ψ
min X F = K X - Y 2 2 + λ 1 Ψ X 1 + λ 2 T V ( X )
min X , { α j } F = 1 2 K X - Y 2 2 + λ 1 2 j R j X D α j 2 2 + μ j α j 0 + λ 2 T V ( X )
F ( X ) = K Τ ( K X - Y ) + λ 1 j R j Τ ( R j X - D α j ) + λ 2 ( T V ( X ) )
( T V ( X ) ) = ( T r ' ( W r ) - 1 T r X + X ( T c ( W c ) - 1 T c ' ) )
M S E ( X ^ ) = X ^ X 2 2 / N

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