Abstract

Coded aperture X-ray computed tomography (CT) has the potential to revolutionize X-ray tomography systems in medical imaging and air and rail transit security - both areas of global importance. It allows either a reduced set of measurements in X-ray CT without degradation in image reconstruction, or measure multiplexed X-rays to simplify the sensing geometry. Measurement reduction is of particular interest in medical imaging to reduce radiation, and airport security often imposes practical constraints leading to limited angle geometries. Coded aperture compressive X-ray CT places a coded aperture pattern in front of the X-ray source in order to obtain patterned projections onto a detector. Compressive sensing (CS) reconstruction algorithms are then used to recover the image. To date, the coded illumination patterns used in conventional CT systems have been random. This paper addresses the code optimization problem for general tomography imaging based on the point spread function (PSF) of the system, which is used as a measure of the sensing matrix quality which connects to the restricted isometry property (RIP) and coherence of the sensing matrix. The methods presented are general, simple to use, and can be easily extended to other imaging systems. Simulations are presented where the peak signal to noise ratios (PSNR) of the reconstructed images using optimized coded apertures exhibit significant gain over those attained by random coded apertures. Additionally, results using real X-ray tomography projections are presented.

© 2017 Optical Society of America

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References

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2017 (1)

A. Parada-Mayorga and G. R. Arce, “Colored coded aperture design in compressive spectral imaging via minimum coherence,” IEEE Trans. on Computational Imaging,  3(2), 202–216 (2017).
[Crossref]

2016 (2)

2015 (2)

2014 (2)

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

2013 (2)

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

J. S. Jorgensen, E. Y. Sidky, and X. Pan, “Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-Ray CT,” IEEE Trans. Med. Imag. 32(2), 460–473 (2013).
[Crossref]

2011 (1)

H. Rauhut, “Compressive sensing and structured random matrices,” Radon Series Comp. Appl. Math XX, 1–94 (2011).

2009 (2)

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7), 1395–1408 (2009).
[Crossref]

2008 (2)

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25, 2960–2970 (2008).
[Crossref]

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Magazine 25(2), 21–30 (2008).
[Crossref]

2007 (2)

M. Lustig, D. L Donoho, and J. M Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine 58, 1182–1195 (2007).
[Crossref]

M. Elad, “Optimized projections for compressed sensing,” in IEEE Trans. on Signal Processing 55(12), 5695–5702 (2007).
[Crossref]

2006 (1)

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

1985 (2)

Aarle, W. V.

Arce, G. R.

A. Parada-Mayorga and G. R. Arce, “Colored coded aperture design in compressive spectral imaging via minimum coherence,” IEEE Trans. on Computational Imaging,  3(2), 202–216 (2017).
[Crossref]

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23, 32788–32802 (2015).
[Crossref] [PubMed]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25, 2960–2970 (2008).
[Crossref]

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

A. Parada-Mayorga, A. P. Cuadros, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis via coherence analysis,” in Proceedings of IEEE International Symposium on Biomedical Imaging (2017).

X. Ma and G. R. Arce, “Computational lithography.” (John Wiley & Sons, Inc, 2010).
[Crossref]

Arguello, H.

Y. Mejia and H. Arguello, “Filtered gradient reconstruction algorithm for compressive spectral imaging,” Optical Engineering,  56(4), 041306 (2016).
[Crossref]

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23, 32788–32802 (2015).
[Crossref] [PubMed]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

E. Mojica, S. Pertuz, and H. Arguello, “High-resolution coded-aperture design for compressive X-ray tomography using low resolution detectors,” Opt. Commun. (posted 30 June 2017, in press).

A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

Barrett, H. H.

Batenburg, K. J.

Beenhouwer, J. D.

Bleichrodt, F.

Brady, D. J.

D. J. Brady, A. Mrozack, K. MacCabe, and P. Llull, “Compressive tomography,” Adv. Opt. Photon. 7, 756–813 (2015).
[Crossref]

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).

Buzug, T. M.

T. M. Buzug, Computed tomography: from photon statistics to modern cone-beam CT (Springer, 2008).

Candes, E.

E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Processing Magazine 25(2), 21–30 (2008).
[Crossref]

Cant, J.

Carin, L.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Choi, K.

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).

Correa, C.

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

Cuadros, A. P.

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23, 32788–32802 (2015).
[Crossref] [PubMed]

A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

A. Parada-Mayorga, A. P. Cuadros, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis via coherence analysis,” in Proceedings of IEEE International Symposium on Biomedical Imaging (2017).

Dabravolski, A.

Davenport, M. A.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications, (Cambridge University, 2011).

Dhawan, A. P.

Donoho, D. L

M. Lustig, D. L Donoho, and J. M Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine 58, 1182–1195 (2007).
[Crossref]

Duarte, M. F.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications, (Cambridge University, 2011).

Duarte-Carvajalino, J. M.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7), 1395–1408 (2009).
[Crossref]

Elad, M.

M. Elad, “Optimized projections for compressed sensing,” in IEEE Trans. on Signal Processing 55(12), 5695–5702 (2007).
[Crossref]

Eldar, Y. C.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications, (Cambridge University, 2011).

Glick, S.

I. Reiser and S. Glick, Tomosynthesis Imaging (Taylor and Francis, 2014).

Gordon, R.

Hämäläinen, K.

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

Holmgren, A.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Hou, W.

W. Hou and C. Zhang, “Analysis of compressed sensing based CT reconstruction with low radiation,” in Proceedings of International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS), (2014), pp. 291–296.

Janssens, E.

Jeon, H.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Jorgensen, J. S.

J. S. Jorgensen, E. Y. Sidky, and X. Pan, “Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-Ray CT,” IEEE Trans. Med. Imag. 32(2), 460–473 (2013).
[Crossref]

Kaganovsky, Y.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, “Principles of computerized tomographic imaging(Society for Industrial and Applied Mathematics, 2001).
[Crossref]

Kallonen, A.

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

Kao, C. M.

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Kittle, D. S.

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

Kolehmainen, V.

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

Kutyniok, G.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications, (Cambridge University, 2011).

Lassas, M.

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

Li, D.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Llull, P.

Lustig, M.

M. Lustig, D. L Donoho, and J. M Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine 58, 1182–1195 (2007).
[Crossref]

Ma, X.

MacCabe, K.

D. J. Brady, A. Mrozack, K. MacCabe, and P. Llull, “Compressive tomography,” Adv. Opt. Photon. 7, 756–813 (2015).
[Crossref]

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Mejia, Y.

Y. Mejia and H. Arguello, “Filtered gradient reconstruction algorithm for compressive spectral imaging,” Optical Engineering,  56(4), 041306 (2016).
[Crossref]

Mojica, E.

E. Mojica, S. Pertuz, and H. Arguello, “High-resolution coded-aperture design for compressive X-ray tomography using low resolution detectors,” Opt. Commun. (posted 30 June 2017, in press).

Mrozack, A.

Natterer, F.

F. Natterer, The mathematics of computerized tomography (Vieweg Teubner Verlag, 1986).

Niinimäki, K.

K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

O’Sullivan, J.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Palenstijn, W. J.

Pan, X.

J. S. Jorgensen, E. Y. Sidky, and X. Pan, “Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-Ray CT,” IEEE Trans. Med. Imag. 32(2), 460–473 (2013).
[Crossref]

Pan, X. H.

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Parada-Mayorga, A.

A. Parada-Mayorga and G. R. Arce, “Colored coded aperture design in compressive spectral imaging via minimum coherence,” IEEE Trans. on Computational Imaging,  3(2), 202–216 (2017).
[Crossref]

A. Parada-Mayorga, A. P. Cuadros, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis via coherence analysis,” in Proceedings of IEEE International Symposium on Biomedical Imaging (2017).

Pauly, J. M

M. Lustig, D. L Donoho, and J. M Pauly, “MRI Sparse: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine 58, 1182–1195 (2007).
[Crossref]

Paxman, R. G.

Peitsch, C.

A. P. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23, 32788–32802 (2015).
[Crossref] [PubMed]

A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

Pertuz, S.

E. Mojica, S. Pertuz, and H. Arguello, “High-resolution coded-aperture design for compressive X-ray tomography using low resolution detectors,” Opt. Commun. (posted 30 June 2017, in press).

Politte, D.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. MacCabe, D. Politte, J. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” in J. Opt. Soc. Am. A 31, 1369–1394 (2014).
[Crossref]

Ramirez, A.

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

Rangayyan, R.

Rauhut, H.

H. Rauhut, “Compressive sensing and structured random matrices,” Radon Series Comp. Appl. Math XX, 1–94 (2011).

Reiser, I.

I. Reiser and S. Glick, Tomosynthesis Imaging (Taylor and Francis, 2014).

Rueda, H.

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

Sapiro, G.

J. M. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18(7), 1395–1408 (2009).
[Crossref]

Sidky, E. Y.

J. S. Jorgensen, E. Y. Sidky, and X. Pan, “Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-Ray CT,” IEEE Trans. Med. Imag. 32(2), 460–473 (2013).
[Crossref]

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Sijbers, J.

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K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

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A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

Zhang, C.

W. Hou and C. Zhang, “Analysis of compressed sensing based CT reconstruction with low radiation,” in Proceedings of International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS), (2014), pp. 291–296.

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J. S. Jorgensen, E. Y. Sidky, and X. Pan, “Quantifying admissible undersampling for sparsity-exploiting iterative image reconstruction in X-Ray CT,” IEEE Trans. Med. Imag. 32(2), 460–473 (2013).
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IEEE Trans. on Computational Imaging (1)

A. Parada-Mayorga and G. R. Arce, “Colored coded aperture design in compressive spectral imaging via minimum coherence,” IEEE Trans. on Computational Imaging,  3(2), 202–216 (2017).
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in IEEE Signal Process. Mag (1)

G. R. Arce, D. J. Brady, L. Carin, H. Arguello, and D. S. Kittle, “Compressive coded aperture spectral imaging: An Introduction,” in IEEE Signal Process. Mag 31(1), 105–115 (2014).
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J. X-Ray Sci. Technol. (1)

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

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K. Hämäläinen, A. Kallonen, V. Kolehmainen, M. Lassas, K. Niinimäki, and S. Siltanen, “Sparse tomography,” Computational Methods in Science and Engineering, SIAM 35, B644–B665,(2013).

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T. M. Buzug, Computed tomography: from photon statistics to modern cone-beam CT (Springer, 2008).

T. A. Bubba, A. Hauptmann, S. Huotari, J. Rimpelainen, and S. Siltanen, “Tomographic x-ray data of a lotus root filled with attenuating objects,” https://arxiv.org/abs/1609.07299 .

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[Crossref]

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications, (Cambridge University, 2011).

W. Hou and C. Zhang, “Analysis of compressed sensing based CT reconstruction with low radiation,” in Proceedings of International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS), (2014), pp. 291–296.

A. Parada-Mayorga, A. P. Cuadros, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis via coherence analysis,” in Proceedings of IEEE International Symposium on Biomedical Imaging (2017).

K. Hämäläinen, A. Harhanen, A. Kallonen, A. Kujanpää, E. Niemi, and S. Siltanen, “Tomographic X-ray data of a walnut,” http://arxiv.org/abs/1502.04064 .

G. R. Arce, A. Ramirez, H. Rueda, H. Arguello, and C. Correa, “Compressive Spectral Imaging,” in Wiley Encyclopedia of Electrical and Electronics Engineering, (John Wiley & Sons, Inc., 2016).

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A. P. Cuadros, K. Wang, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture design for compressive X-ray tomosynthesis,” in Imaging and Applied Optics 2015, OSA Technical Digest, Optical Society of America, paper CW2F.2.

E. Mojica, S. Pertuz, and H. Arguello, “High-resolution coded-aperture design for compressive X-ray tomography using low resolution detectors,” Opt. Commun. (posted 30 June 2017, in press).

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Figures (10)

Fig. 1
Fig. 1 (a) Coded aperture compressive fan beam X-ray CT system. Different coded apertures are placed in front of the rotating source for each location sp. Each row in the system matrix H describes the sensing for a particular detector element. The unblocked coded aperture elements select the rows of the matrix H associated with the detector elements that correspond to the unblocked coded aperture elements. (b) Sensing matrix Q D   × N 2, of a system with N2 = 9, P = 4, D = 6 and M = 3. Q is formed by the rows selected by the unblocked elements of the coded apertures. Each column of the coded aperture matrix C is associated with a detector element. If the coded aperture element associated with the ith detector element is 0 then ci = 0. Otherwise, the vector ci is a natural basis in D.
Fig. 2
Fig. 2 (a) Sparse structures of the system matrix Q, the coded aperture matrix C, and the complete CT system matrix H for a fan beam coded aperture compressive X-ray CT system with P = 16 view angles, M = 16 detectors per view angle and a N × N = 8 × 8 image. The coded apertures have 12.5% transmittance, that is D = 32. Non-zero entries are depicted as black pixels for visualization purposes. (b) Under-sampled sinogram of 64 × 64 Shepp-Logan Phantom with P = 128, M = 128 detectors per view angle and 50% under-sampling. A zoom of a particular row of the coded sinogram is depicted. The black elements denote the blocking elements of the coded aperture for that particular view angle, while the other detectors reflect the information of the corresponding spatial position of the complete uncoded sinogram depicted in (c).
Fig. 3
Fig. 3 (a) 128 × 128 Walnut phantom [23] and reconstructions for different subsampling scenarios using: (b) Optimized coded apertures obtained using the proposed algorithm, and (c) UD, and (d) UV strategies [8]. Measurements were simulated for N = 128, M = P = 256 and D = 16429. A zoom of the highlighted region is depicted to emphasize the error in the reconstructions. Figure (b) was reconstructed using the gradient projection for sparse reconstruction (GPSR) algorithm, Figs. (c) and (d) were reconstructed using a TV regularization algorithm.
Fig. 4
Fig. 4 (a) Quadratic penalty cost function encouraging solution in bi ∈ {0, 1} since the minimum error is obtained for binary values. (b) Transmittance and uniformity penalty cost function for M = P = 2N = 16 and μ = 2, i.e. approximately 50 % transmittance. Note the minimum error is achieved for the desired average number of rays.
Fig. 5
Fig. 5 (a) PSNR of the reconstructions obtained using random and optimal coded apertures with different subsampling rates. For the random coded apertures, 10 realizations were computed and the graph depicts the mean and the standard error as a solid line and shaded area respectively. (b) Γ error for optimal and random coded apertures. The subsampling rate corresponds to the percentage of the full set of detectors used in each case.
Fig. 6
Fig. 6 Gradient descent iterations of the optimization function and regularization terms for 47.3% (a) and 20.3% (b) subsampling rates. Gradient descent iterations of the mutual coherence for 47.3% (c) and 20.3% (d) subsampling rates. The subsampling rate corresponds to the percentage of the full set of detectors used in each case. The parameters for the simulation were M = P = 2N = 128.
Fig. 7
Fig. 7 (a) 64 × 64 Walnut phantom [23]. Reconstructions using: (b) optimized and (c) random coded apertures with subsampling rate 20.3%. Absolute error images for the (d) optimized and (e) random cases show more artifacts for random projections.(f) SVD of the CT matrix with optimized codes and random codes. The condition number of both the random κr and optimized κo cases are specified. The maximum σmax and minimum σmin values are specified for the random and optimal cases. Optimized projections result in a less ill-conditioned system.
Fig. 8
Fig. 8 Normalized absolute error plots for the reconstructions of the 64 × 64 Shepp-Logan phantom for subsampling rates of 32.47% using a Haar wavelet (a) and 32.4% using a Symlet wavelet (b) and M = P = 128.
Fig. 9
Fig. 9 Reconstructions of a slice of a lotus root from real X-ray tomographic projections, using: (a) Landweber iterations without using coded apertures [32] and (b) optimized and (c) random coded apertures with subsampling rate 20.6%. Absolute error images for the (d) optimized and (e) random cases are depicted.(f) SVD of the CT matrix with optimized codes and random codes. The condition number of both the random κr and optimized κo cases are specified. The maximum σmax and minimum σmin values are specified for the random and optimal cases.
Fig. 10
Fig. 10 Intensity graphs of the number of rays that measure certain pixel for D = 7742, and D = 3327 i.e. 47.3% and 20.3% subsampling rates respectively, with the corresponding standard deviation, mean and maximum values. Hot Spots, with radiation higher than the 95% of the maximum intensity, are located in the random case for both subsampling rates and are shown as red dots in the figure.

Equations (27)

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z ^ = arg min z y Az 2 2 + λ z 1 ,
( 1 δ s ) x 2 2 A ˜ x 2 2 ( 1 + δ s ) x 2 2 ,
δ s = max S [ N 2 ] , | S | s A ˜ S * A ˜ S I 2
δ s A ˜ * A ˜ I F .
C ^ = arg min C A ˜ * A ˜ I F = arg min C Γ .
[ A ˜ T A ˜ ] m , n = i , j M P c i , c j w i ( m ) w j ( n ) ( i , j M P c i , c j w i ( m ) w j ( m ) ) 1 / 2 ( i , j M P c i , c j w i ( n ) w j ( n ) ) 1 / 2 .
[ A ˜ T A ˜ ] m , n = i = 1 M P b i w i ( m ) w i ( n ) ( i = 1 M P b i w i ( m ) w i ( m ) ) 1 / 2 ( i = 1 M P b i w i ( n ) w i ( n ) ) 1 / 2 ,
Γ = m = 1 N 2 n = 1 N 2 | i = 1 M P b i w i ( m ) w i ( n ) ( i = 1 M P b i w i ( m ) w i ( m ) ) 1 / 2 ( i = 1 M P b i w i ( n ) w i ( n ) ) 1 / 2 I m , n | 2 .
b ^ = arg min b Γ = arg min b m = 1 N 2 n = 1 N 2 | [ A ˜ T A ˜ ] m , n I m , n | 2
b ^ = arg min b m < n ( d ( m , n ) ( d ( m , m ) ) 1 / 2 ( d ( n , n ) ) 1 / 2 ) 2 = arg min b F ( b ) ,
F ( θ ) = m < n ( i = 1 M P cos θ i + 1 2 R i m , n ( i = 1 M P cos θ i + 1 2 R i m , m ) 1 / 2 ( i = 1 M P cos θ i + 1 2 R i n , n ) 1 / 2 ) 2 .
F ( θ ) = m < n ( d θ ( m , n ) ( d θ ( m , m ) ) 1 / 2 ( d θ ( n , n ) ) 1 / 2 ) 2 .
F ( θ ) = ( m < n d θ ( m , n ) R ¯ m , n d θ ( m , m ) d θ ( n , n ) + d θ ( m , n ) 2 R ¯ m , m 2 d θ ( m , m ) 2 d θ ( n , n ) + d θ ( m , n ) 2 R ¯ n , n 2 d θ ( m , m ) d θ ( n , n ) 2 ) sin θ ,
θ i k + 1 = θ i k α [ F ( θ i k ) + r ] ,
b ^ = arg min b F ( b ) + γ 1 V 1 ( b ) + γ 2 V 2 ( b ) ,
V 2 ( b ) = 1 N C k Ω C [ ( i = 1 M P b i H i , k ) μ ] 2 ,
V 2 ( b ) = 1 N C [ η ¯ T H Ω C μ ] [ H Ω C T ] [ η ¯ T ] 1 ,
J ( b ) = F ( b ) + γ 1 V 1 ( b ) + γ 2 V 2 ( b ) .
b ^ = arg min b m = 1 N 2 n = 1 N 2 | [ A ˜ T A ˜ ] m , n I m , n | 2 .
b ^ = arg min b m n ( [ A ˜ T A ˜ ] m , n 0 ) 2 + m = n ( [ A ˜ T A ˜ ] m , m 1 ) 2 .
b ^ = arg min b m n ( [ A ˜ T A ˜ ] m , n ) 2 = arg min b m < n ( [ A ˜ T A ˜ ] m , n ) 2 .
F ( θ ) θ i =   m < n [ d θ ( m , n ) θ i ] 2 d θ ( m , n ) d θ ( m , m ) d θ ( n , n ) d θ ( m , n ) 2 [ d θ ( m , m ) θ i d θ ( n , n ) + d θ ( n , n ) θ i d θ ( m , m ) ] ( d θ ( m , m ) ) 2 d θ ( n , n ) 2 .
F ( θ ) = ( m < n d θ ( m , n ) R ¯ m , n d θ ( m , m ) d θ ( n , n ) + d θ ( m , n ) 2 R ¯ m , m 2 d θ ( m , m ) 2 d θ ( n , n ) + d θ ( m , n ) 2 R ¯ n , n 2 d θ ( m , m ) d θ ( n , n ) 2 ) sin θ .
V 2 ( b ) = 1 N C k = 1 2 ( ( b 1 H k , 1 + b 2 H k , 2 ) μ ) 2 .
V 2 ( b ) b 1 = 1 N C { 2 [ ( b 1 H 1 , 1 + b 2 H 1 , 2 ) μ ] 1 2 b 1 H 1 , 1 + 2 [ ( b 1 H 2 , 1 + b 2 H 2 , 2 ) μ ] 1 2 b 1 H 2 , 1 }
V 2 ( b ) b 1 = 1 N C [ b T H Ω C μ ] [ H Ω C 1 ] T 1 b 1
V 2 ( b ) = 1 N C [ b T H Ω C μ ] [ H Ω C ] T [ b T ] 1

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