Abstract

Coded aperture X-ray computed tomography (CAXCT) is a novel X-ray imaging system capable of reconstructing high quality images from a reduced set of measurements. Coded apertures are placed in front of the X-ray source in CAXCT so as to obtain patterned projections onto a detector array. Then, compressive sensing (CS) reconstruction algorithms are used to reconstruct the linear attenuation coefficients. The coded aperture is an important factor that influences the point spread function (PSF), which in turn determines the capability to sample the linear attenuation coefficients of the object. A coded aperture optimization approach was recently proposed based on the coherence of the system matrix; however, this algorithm is memory intensive and it is not able to optimize the coded apertures for large image sizes required in many applications. This paper introduces a significantly more efficient approach for coded aperture optimization that reduces the memory requirements and the execution time by orders of magnitude. The features are defined as the inner product of the vectors representing the geometric paths of the X-rays with the sparse basis representation of the object; therefore, the algorithm aims to find a subset of features that minimizes the information loss compared to the complete set of projections. This subset corresponds to the unblocking elements in the optimized coded apertures. The proposed approach solves the memory and runtime limitations of the previously proposed algorithm and provides a significant gain in the reconstruction image quality compared to that attained by random coded apertures in both simulated datasets and real datasets.

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

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References

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2017 (3)

H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
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J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

A. Cuadros and G. R. Arce, “Coded aperture optimization in compressive X-ray tomography: a gradient descent approach,” Opt. Express 25(20), 23833–23849 (2017).
[Crossref] [PubMed]

2016 (5)

2015 (1)

2014 (4)

2013 (1)

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

2009 (3)

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

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

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

2008 (2)

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
[Crossref]

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

2007 (2)

R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag 24(4), 118–121 (2007).
[Crossref]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing 1(4), 586–597 (2007).
[Crossref]

2003 (1)

G. Y. Isabelle and A. Elisseeff, “An introduction to variable and feature selection,” Journal of Machine Learning Research 3, 1157–1182 (2003).

2001 (1)

F. Natterer, “Inversion of the attenuated Radon transform,” Inverse problems 17(1), 113 (2001).
[Crossref]

1996 (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

1993 (1)

F. T. Lin, Y. K. Cheng, and C. H. Ching, “Applying the genetic approach to simulated annealing in solving some NP-hard problems,” IEEE Trans. Systems, Man, and Cybernetics 23(6), 1752–1767 (1993)
[Crossref]

1991 (1)

T. L. Moore, “Using Euler’s formula to solve plane separation problems,” The College Mathematics Journal 22(2), 125–130 (1991).
[Crossref]

1985 (1)

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Medical Imaging 4(1), 14–25 (1985).
[Crossref] [PubMed]

1983 (1)

K. H. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM Journal on Applied Mathematics 43(3), 546–552 (1983).
[Crossref]

1982 (1)

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

1981 (1)

K. C. Tam and Victor Perez-Mendez, “Tomographical imaging with limited-angle input,” J. Opt. Soc. Am. A 71(5), 582–592 (1981).
[Crossref]

Aarle, W. V.

Arce, G. R.

A. Cuadros and G. R. Arce, “Coded aperture optimization in compressive X-ray tomography: a gradient descent approach,” Opt. Express 25(20), 23833–23849 (2017).
[Crossref] [PubMed]

H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
[Crossref]

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[Crossref]

A. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 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,” IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

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. Cuadros and G. R. Arce, “Coded aperture compressive X-ray spectral CT,” in Proceedings of IEEE Conference on Sampling Theory and Applications (IEEE2017), pp. 548–551.

Arguello, H.

A. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 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,” IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

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.

Baraniuk, R. G.

R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag 24(4), 118–121 (2007).
[Crossref]

Baron, D.

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[Crossref]

Batenburg, K. J.

Beenhouwer, J. D.

Bindman, R. S.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Bleichrodt, F.

Bluman, A. G.

A. G. Bluman, Elementary Statistics (McGraw Hill2013).

Brady, D.

J. Greenberg, K. Krishnamurthy, and D. Brady, “Compressive single-pixel snapshot x-ray diffraction imaging,” Opt. Lett. 39(1), 111–114 (2014).
[Crossref]

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
[Crossref]

Brady, D. J.

M. Hassan, J. A. Greenberg, and D. J. Brady, “Snapshot fan beam coded aperture coherent scatter tomography,” Opt. Express 24(16), 18277–18289 (2016).
[Crossref] [PubMed]

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. P. MacCabe, D. G. Politte, J. A. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” J. Opt. Soc. Am. A 31(7), 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,” IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

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

Candes, E. J.

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

Cant, J.

Carin, L.

Y. Kaganovsky, D. Li, A. Holmgren, H. Jeon, K. P. MacCabe, D. G. Politte, J. A. O’Sullivan, L. Carin, and D. J. Brady, “Compressed sampling strategies for tomography,” J. Opt. Soc. Am. A 31(7), 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,” IEEE Signal Process. Mag 31(1), 105–115 (2014).
[Crossref]

Chen, Q.

Cheng, K. W.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Cheng, Y. K.

F. T. Lin, Y. K. Cheng, and C. H. Ching, “Applying the genetic approach to simulated annealing in solving some NP-hard problems,” IEEE Trans. Systems, Man, and Cybernetics 23(6), 1752–1767 (1993)
[Crossref]

Ching, C. H.

F. T. Lin, Y. K. Cheng, and C. H. Ching, “Applying the genetic approach to simulated annealing in solving some NP-hard problems,” IEEE Trans. Systems, Man, and Cybernetics 23(6), 1752–1767 (1993)
[Crossref]

Choi, K.

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

Cuadros, A.

Cuadros, A. P.

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.

Dabravolski, A.

Dai, H.

De Gonzalez, A. B.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Elisseeff, A.

G. Y. Isabelle and A. Elisseeff, “An introduction to variable and feature selection,” Journal of Machine Learning Research 3, 1157–1182 (2003).

Figueiredo, M. A. T.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing 1(4), 586–597 (2007).
[Crossref]

Foucart, S.

S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing (Springer2013).
[Crossref]

Fu, C.

H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
[Crossref]

Gould, R.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Greenberg, J.

Greenberg, J. A.

Gu, G.

Hall, M. A.

M. A. Hall, “Correlation-based Feature Selection for Machine Learning,” https://www.cs.waikato.ac.nz/~mhall/thesis.pdf

Hassan, M.

He, W.

Herman, G. T.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

Holmgren, A.

Hsieh, J.

J. Hsieh, Computed tomography: principles, Design, Artifacts, and Recent Advances (SPIE2015).

Isabelle, G. Y.

G. Y. Isabelle and A. Elisseeff, “An introduction to variable and feature selection,” Journal of Machine Learning Research 3, 1157–1182 (2003).

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

Janssens, E.

Jeon, H.

John, R.

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
[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. Medical Imaging 32(2), 460–473 (2013).
[Crossref]

Kaganovsky, Y.

Khadivi, K. O.

K. O. Khadivi, “Computed tomography: fundamentals, system technology, image quality, applications,” Medical Physics 33(8), 3076 (2016).
[Crossref]

Kim, K. P.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Kisner, S. J.

S. J. Kisner, “Image Reconstruction for X-ray Computed Tomography in Security Screening Applications,” Purdue University (2013).

Kittle, D. S.

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

Kouris, K.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

Krishnamurthy, K.

Lau, D. L.

H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
[Crossref]

Lent, A.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

Lewitt, R. M.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

Li, D.

Li, J. D.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Li, L.

F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

Liao, F.

Lin, F. T.

F. T. Lin, Y. K. Cheng, and C. H. Ching, “Applying the genetic approach to simulated annealing in solving some NP-hard problems,” IEEE Trans. Systems, Man, and Cybernetics 23(6), 1752–1767 (1993)
[Crossref]

Lipson, J.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Liu, H. A.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Liu, X.

Lv, X.

F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

Ma, Y.

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[Crossref]

MacCabe, K. P.

Mahesh, M.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Mao, T.

Marcus, R.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Miglioretti, D. L.

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

Moore, T. L.

T. L. Moore, “Using Euler’s formula to solve plane separation problems,” The College Mathematics Journal 22(2), 125–130 (1991).
[Crossref]

Morstatter, F.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Natterer, F.

F. Natterer, “Inversion of the attenuated Radon transform,” Inverse problems 17(1), 113 (2001).
[Crossref]

Nowak, R. D.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing 1(4), 586–597 (2007).
[Crossref]

O’Sullivan, J. A.

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. Medical Imaging 32(2), 460–473 (2013).
[Crossref]

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

Peitsch, C.

A. Cuadros, C. Peitsch, H. Arguello, and G. R. Arce, “Coded aperture optimization for compressive X-ray tomosynthesis,” Opt. Express 23(25), 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.

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K. C. Tam and Victor Perez-Mendez, “Tomographical imaging with limited-angle input,” J. Opt. Soc. Am. A 71(5), 582–592 (1981).
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S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing (Springer2013).
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H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
[Crossref]

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[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. Medical Imaging 32(2), 460–473 (2013).
[Crossref]

Sidky, E.Y.

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

Sijbers, J.

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B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Medical Imaging 4(1), 14–25 (1985).
[Crossref] [PubMed]

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A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

Tam, K. C.

K. C. Tam and Victor Perez-Mendez, “Tomographical imaging with limited-angle input,” J. Opt. Soc. Am. A 71(5), 582–592 (1981).
[Crossref]

Tan, J.

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[Crossref]

Tang, J. L.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Trevino, R. P.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Tuy, H.

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
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K. H. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM Journal on Applied Mathematics 43(3), 546–552 (1983).
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Vannier, M.

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

Wagadarikar, A.

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
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E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag 25(2), 21–30 (2008).
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F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

Wang, K.

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.

Wang, S. H.

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Willett, R.

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
[Crossref]

Wright, S. J.

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing 1(4), 586–597 (2007).
[Crossref]

Xu, J.

F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

Yang, Y.

F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

Ye, L.

Zhuang, J.

ACM Computing Surveys (CSUR) (1)

J. D. Li, K. W. Cheng, S. H. Wang, F. Morstatter, R. P. Trevino, J. L. Tang, and H. A. Liu, “Feature selection: A data perspective,” ACM Computing Surveys (CSUR) 6(50), 94 (2017).

Appl. Opt. (2)

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 41(10), B44–B51 (2008).
[Crossref]

H. Dai, G. Gu, W. He, F. Liao, J. Zhuang, X. Liu, and Q. Chen, “Adaptive compressed sampling based on extended wavelet trees,” Appl. Opt. 53(29), 6619–6628 (2014).
[Crossref] [PubMed]

Archives of internal medicine (1)

R. S. Bindman, J. Lipson, R. Marcus, K. P. Kim, M. Mahesh, R. Gould, A. B. De Gonzalez, and D. L. Miglioretti, “Radiation dose associated with common computed tomography examinations and the associated lifetime attributable risk of cancer,” Archives of internal medicine 169(22), 2078–2086 (2009)
[Crossref]

IEEE Journal of Selected Topics in Signal Processing (3)

H. Rueda, C. Fu, D. L. Lau, and G. R. Arce, “Single Aperture Spectral+ ToF Compressive Camera: Toward Hyperspectral+ Depth Imagery,” IEEE Journal of Selected Topics in Signal Processing 11(7), 992–1003 (2017).
[Crossref]

M. A. T. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems,” IEEE Journal of Selected Topics in Signal Processing 1(4), 586–597 (2007).
[Crossref]

J. Tan, Y. Ma, H. Rueda, D. Baron, and G. R. Arce, “Compressive Hyperspectral Imaging via Approximate Message Passing,” IEEE Journal of Selected Topics in Signal Processing,  10(2), 389–401 (2016).
[Crossref]

IEEE Signal Process. Mag (3)

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

R. G. Baraniuk, “Compressive sensing [lecture notes],” IEEE Signal Process. Mag 24(4), 118–121 (2007).
[Crossref]

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

IEEE Trans. Medical Imaging (3)

K. Kouris, H. Tuy, A. Lent, G. T. Herman, and R. M. Lewitt, “Reconstruction from sparsely sampled data by art with interpolated rays,” IEEE Trans. Medical Imaging 1(3), 161–167 (1982).
[Crossref] [PubMed]

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

B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Medical Imaging 4(1), 14–25 (1985).
[Crossref] [PubMed]

IEEE Trans. Systems, Man, and Cybernetics (1)

F. T. Lin, Y. K. Cheng, and C. H. Ching, “Applying the genetic approach to simulated annealing in solving some NP-hard problems,” IEEE Trans. Systems, Man, and Cybernetics 23(6), 1752–1767 (1993)
[Crossref]

Inverse problems (2)

F. Natterer, “Inversion of the attenuated Radon transform,” Inverse problems 17(1), 113 (2001).
[Crossref]

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

J. Opt. Soc. Am. A (2)

Journal of Machine Learning Research (1)

G. Y. Isabelle and A. Elisseeff, “An introduction to variable and feature selection,” Journal of Machine Learning Research 3, 1157–1182 (2003).

Medical Physics (1)

K. O. Khadivi, “Computed tomography: fundamentals, system technology, image quality, applications,” Medical Physics 33(8), 3076 (2016).
[Crossref]

Nature Medicine (1)

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nature Medicine 2(4), 473–475 (1996).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (1)

Proc. SPIE (1)

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

SIAM Journal on Applied Mathematics (1)

K. H. Tuy, “An inversion formula for cone-beam reconstruction,” SIAM Journal on Applied Mathematics 43(3), 546–552 (1983).
[Crossref]

The College Mathematics Journal (1)

T. L. Moore, “Using Euler’s formula to solve plane separation problems,” The College Mathematics Journal 22(2), 125–130 (1991).
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M. A. Hall, “Correlation-based Feature Selection for Machine Learning,” https://www.cs.waikato.ac.nz/~mhall/thesis.pdf

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

Y. Adam, C. Thrampoulidis, and G. Wornell, “Analysis and Optimization of Aperture Design in Computational Imaging,” https://arxiv.org/pdf/1712.04541.pdf

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 .

A. G. Bluman, Elementary Statistics (McGraw Hill2013).

S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing (Springer2013).
[Crossref]

F. Wang, Y. Yang, X. Lv, J. Xu, and L. Li, “Feature selection using feature ranking, correlation analysis and chaotic binary particle swarm optimization,” in Proceedings of IEEE Conference on Software Engineering and Service Science (IEEE2014), pp. 305–309.

S. J. Kisner, “Image Reconstruction for X-ray Computed Tomography in Security Screening Applications,” Purdue University (2013).

J. Hsieh, Computed tomography: principles, Design, Artifacts, and Recent Advances (SPIE2015).

R. Stempert and D. Boye, “Volumetric Radiography of Watermarks,” http://digitome.davidson.edu/wp-content/uploads/2018/01/Volumetric-Radiography-of-Watermarks.pdf .

D. Boye, R. Garner, and R. Kozlowski, “Examining paintings on wood or canvas using 3D X-ray imaging with Digitome,” http://digitome.davidson.edu/wp-content/uploads/2018/01/Examining-paintings-using-3DX-ray.pdf .

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. Cuadros and G. R. Arce, “Coded aperture compressive X-ray spectral CT,” in Proceedings of IEEE Conference on Sampling Theory and Applications (IEEE2017), pp. 548–551.

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Figures (12)

Fig. 1
Fig. 1 Optical setup of coded aperture X-ray CT. (a) The object is illuminated by coded fan-beam X-ray sources at P positions [S1, S2, · · · , SP], and the projections are captured by a flat detector. Part of the X-ray radiation is blocked by the blocking elements on the coded apertures and the correspondent pixels on the flat detector are discarded. (b) CA represents coded aperture, and the white and black squares represent unblocking and blocking elements, respectively.
Fig. 2
Fig. 2 The structure matrix W, the coded aperture matrix C, the vectorized object f = Ψz, where Ψ is the sparse basis matrix and z is the vectorized sparse representation, and the vectorized measurements y for a fan beam CAXCT system with P = 8 view angles, M = 8 detectors per view angle and a N × N = 4 × 4 image. The CT system matrix is H = . The coded apertures have 50% transmittance, that is, the number of unblocking elements on the coded apertures D = 32.
Fig. 3
Fig. 3 Division of CT system matrix H with 12 rows into 3 submatrices, S1, S2 and S3. (a) H is divided randomly. The 1 st , 4 th , 7 th and 10 th rows are combined in S1, the 3 rd , 5 th , 6 th and 9 th rows are combined in S2, and the 2 nd , 8 th , 11 th and 12 th rows are combined in S3. (b) H is divided such thay 1 st , 2 nd , 3 rd and 4 th rows are combined in S1, the 5 nd , 6 th , 7 th and 8 th rows are combined in S2, and the 9 th , 10 th , 11 th and 12 th rows are combined in S3. (c) H is divided uniformly. The 1 st , 4 th , 7 th and 10 th rows are combined in S1, the 2 nd , 5 th , 8 th and 11 th rows are combined in S2, and the 3 rd , 6 th , 9 th and 12 th rows are combined in S3.
Fig. 4
Fig. 4 (a) “Walnut phantom” and reconstructed images using (b) proposed approach, (c) gradient descent approach [20] and (d) random coded apertures. Absolute error images for (e) proposed approach, (f) gradient descent approach and (g) random coded apertures. Note that more artifacts are present in the reconstructions from random X-ray paths.
Fig. 5
Fig. 5 SVD of the sensing matrix using random and optimal coded apertures. The highest and lowest singular values are highlighted in each case.
Fig. 6
Fig. 6 The reconstructed 128 × 128 image of the “Walnut phantom” with optimized coded apertures at (a) 25%, (b) 50% and (c) 75% subsampling rate; with random coded aperture at (d) 25%, (e) 50% and (f) 75% subsampling rate.
Fig. 7
Fig. 7 The plots of singular values as a function of component numbers at 25%, 50% and 75% subsampling rate for optimized and random coded apertures of 128 × 128 images, respectively. The highest and lowest singular values are highlighted in each case.
Fig. 8
Fig. 8 The reconstructed 256×256 image of the “Walnut phantom” with two-stage optimized coded apertures at (a) 25%, (b) 50% and (c) 75% subsampling rate; “separated division” optimized coded apertures at (d) 25%, (e) 50% and (f) 75% subsampling rate; with random coded aperture at (g) 25%, (h) 50% and (i) 75% subsampling rate.
Fig. 9
Fig. 9 256 × 256 Lotus root reconstructed images with full available projections using (a) GPSR and (b) FPB algorithm.
Fig. 10
Fig. 10 128 × 128 Lotus root reconstructed images using (a) optimized coded apertures, (b) random coded apertures and (c) conventional CT where no coded apertures are used. All reconstructions shown in this figure use the same number of X-ray measurements, 50% subsampling rate.
Fig. 11
Fig. 11 256 × 256 Lotus root reconstructed images using (a) random coded apertures, (b) optimized coded apertures with “separated division” and (c) optimized coded apertures without “separated division”. Absolute error images for (d) random coded apertures, (e) optimized coded apertures with “separated division” and (f) optimized coded apertures without “separated division”. Note that more artifacts are present in the reconstructions from random X-ray paths.
Fig. 12
Fig. 12 The geometric structures of the moments (a), (b) and (c). The red lines, blue lines and green lines represent the X-ray paths in the object, the distances between the origin and the X-ray paths and the critical length, 2 2 N , respectively.

Tables (2)

Tables Icon

Algorithm 1 Local optimization of coded apertures

Tables Icon

Table 1 PSNR and runtime for the proposed approach and the gradient descent approach

Equations (10)

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

y ( θ , η ) = α ( θ , η ) I ( θ , η , E ) exp ( μ ( x , y , E ) d x d y ) d E ,
y = CWf ,
z ^ = arg min z y Az 2 2 + λ z 1 ,
A = CH ,
J = AA T 2 2 = CH ( CH ) T 2 2 = CHH T C 2 2 = N 2 CQC 2 2 = N 2 i = 1 MP j = 1 MP ( C i i Q i j C j j ) 2 .
arg min J = arg min i j = Q i j 2 Subject to C i i = 1 and C j j = 1 i , j
PSNR = 20 log 10 ( f max MSE ) ,
corrcoef 1 = 0 ( a N 2 ) ( b N 2 ) ( a N 2 ) ( a N 2 ) 2 ( b N 2 ) ( b N 2 ) 2 ,
corrcoef 2 = 1 N 2 ( a N 2 ) ( b N 2 ) ( a N 2 ) ( a N 2 ) 2 ( b N 2 ) ( b N 2 ) 2 .
Δ = ( corrcoef 2 ) 2 ( corrcoef 1 ) 2 = N 2 2 a b N 2 ( ( a N 2 ) ( a N 2 ) 2 ) ( ( b N 2 ) ( b N 2 ) 2 ) .

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