Abstract

Spectral computed tomography (CT) relies on the spectral dependence of X-ray attenuation coefficients to separate projection measurements into more than two energy bins. Such data can be used to unveil tomographic material characterization — key in national security and medical imaging. This paper explores a radical departure from conventional methods used in spectral imaging. It relies on K-edge coded apertures to create spatially and spectrally coded, lower-dose, X-ray bundles that interrogate specific voxels of the object. The new approach referred to as compressive spectral X-ray imaging (CSXI) uses low-cost standard X-ray integrating detectors and acquires compressive measurements, which enable the reconstruction of energy binned images from fewer measurements. Various spectral and spatial coding strategies for structured illumination are explored. Subsampling in CSXI is accomplished by either view angle spectral subsampling, spatial subsampling enabled by block-unblock coded apertures placed at the source or detector side, or both. The careful design of subsampling strategies, spectral filters, coded apertures, and their placement, are shown to be critical for the quality of tomographic image reconstruction. The forward imaging model of CSXI, which is a non-linear ill-posed problem, is analyzed and a multi-stage algorithm is developed to address the estimation of the energy binned sinograms from the integrating detector measurements. Then, an Alternating Direction Method of Multipliers (ADMM) is used to solve a joint sparse and low-rank optimization problem for reconstruction that exploits the structure of the spectral X-ray data cube.

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

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References

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2017 (2)

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

B. D. Arhatari and T. E. Gureyev, “Elemental contrast X-ray tomography using Ross filter pairs with a polychromatic laboratory source,” Sci. Reports 7, 218 (2017).

2015 (4)

K. Kim, J. C. Ye, W. Worstell, J. Ouyang, Y. Rakvongthai, G. E. Fakhri, and Q. Li, “Sparse-view spectral CT reconstruction using spectral patch-based low-rank penalty,” IEEE Trans. on Medical Imaging 34, 748–760 (2015).
[Crossref]

L. Li, Z. Chen, W. Cong, and G. Wang, “Spectral CT modeling and reconstruction with hybrid detectors in dynamic-threshold-based counting and integrating modes,” IEEE Transactions on Med. Imaging 34, 716–728 (2015).
[Crossref]

Y. Rakvongthai, W. Worstell, G. E. Fakhri, J. Bian, A. Lorsakul, and J. Ouyang, “Spectral CT using multiple balanced K-edge filters,” IEEE Transactions on Med. Imaging 34, 740–747 (2015).
[Crossref]

L. Galvis, H. Arguello, and G. R. Arce, “Coded aperture design in mismatched compressive spectral imaging,” Appl. Opt. 54, 9875–9882 (2015).
[Crossref]

2014 (3)

K. Hamalainen, L. Harhanen, A. Hauptmann, A. Kallonen, E. Niemi, and S. Siltanen, “Total variation regularization for large-scale X-ray tomography,” Int. J. Tomogr. Simul. 25, 1 – 25 (2014).

L. Li, Z. Chen, G. Wang, J. Chu, and H. Gao, “A tensor PRISM algorithm for multi-energy CT reconstruction and comparative studies,” J. X-Ray Sci. Technol. 22, 147–163 (2014).

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, 1369–1394 (2014).
[Crossref]

2013 (4)

J. Chu, W. Cong, L. Li, and G. Wang, “Combination of current-integrating and photon-counting detector modules for spectral CT,” Phys. Medicine Biol. 58, 7009 (2013).
[Crossref]

K. Taguchi and J. S. Iwanczyk, “Vision 20/20: Single photon counting x-ray detectors in medical imaging,” Med. Phys. 40, 100901 (2013).
[Crossref] [PubMed]

A. Maier, H. G. Hofmann, M. Berger, P. Fischer, C. Schwemmer, H. Wu, K. Muller, J. Hornegger, J. H. Choi, C. Riess, A. Keil, and R. Fahrig, “Conrad-a software framework for cone-beam imaging in radiology,” Med. Phys. 40, 111914 (2013).
[Crossref]

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

2011 (3)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

K. Taguchi, M. Zhang, E. C. Frey, X. Wang, J. S. Iwanczyk, E. Nygard, N. E. Hartsough, B. M. W. Tsui, and W. C. Barber, “Modeling the performance of a photon counting X-ray detector for CT: Energy response and pulse pileup effects,” Med. Phys. 38, 1089–1102 (2011).
[Crossref] [PubMed]

H. Gao, H. Yu, S. Osher, and G. Wang, “Multi-energy CT based on a prior rank, intensity and sparsity model (PRISM),” Inverse Probl. 27, 115012 (2011).
[Crossref]

2010 (1)

W. Xu, F. Xu, M. Jones, B. Keszthelyi, J. Sedat, D. Agard, and K. Mueller, “High-performance iterative electron tomography reconstruction with long-object compensation using graphics processing units (gpus),” J. Struct. Biol. 171, 142 – 153 (2010).
[Crossref] [PubMed]

2008 (2)

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Baumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Medicine Biol. 53, 4031 (2008).
[Crossref]

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

2007 (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Process. 16, 2992–3004 (2007).
[Crossref]

2004 (2)

M. Saito, “Quasimonochromatic X-ray computed tomography by the balanced filter method using a conventional X-ray source,” Med. Phys. 31, 3436–3443 (2004).
[Crossref]

J. H. Siewerdsen, A. M. Waese, D. J. Moseley, S. Richard, and D. A. Jaffray, “Spektr: A computational tool for X-ray spectral analysis and imaging system optimization,” Med. Phys. 31, 3057–3067 (2004).
[Crossref] [PubMed]

2000 (1)

L. D. Lathauwer, B. D. Moor, and J. Vandewalle, “A multilinear singular value decomposition,” SIAM J. Matrix Analysis Appl. 21, 1253–1278 (2000).
[Crossref]

1967 (1)

W. Bol, “The use of balanced filters in X-ray diffraction,” J. Sci. Instruments 44, 736 (1967).
[Crossref]

1939 (1)

P. Kirkpatrick, “On the theory and use of Ross filters,” Rev. Sci. Instruments 10, 186–191 (1939).
[Crossref]

1928 (1)

1926 (1)

P. Ross, “Polarization of X-rays,” Phys. Rev 28, 425 (1926).

Agard, D.

W. Xu, F. Xu, M. Jones, B. Keszthelyi, J. Sedat, D. Agard, and K. Mueller, “High-performance iterative electron tomography reconstruction with long-object compensation using graphics processing units (gpus),” J. Struct. Biol. 171, 142 – 153 (2010).
[Crossref] [PubMed]

Altman, A.

R. Carmi, G. Naveh, and A. Altman, “Material separation with dual-layer CT,” in IEEE Nuclear Science Symp. Conf. Record, 2005, vol. 4 (2005), pp. 3 pp.–1878.

Arce, G. R.

Arguello, H.

Arhatari, B. D.

B. D. Arhatari and T. E. Gureyev, “Elemental contrast X-ray tomography using Ross filter pairs with a polychromatic laboratory source,” Sci. Reports 7, 218 (2017).

Badea, C. T.

C. T. Badea, D. P. Clark, M. Holbrook, M. Srivastava, Y. Mowery, and K. B. Ghaghada, “Functional imaging of tumor vasculature using iodine and gadolinium-based nanoparticle contrast agents: a comparison of spectral micro-ct using energy integrating and photon counting detectors,” Phys. Medicine Biol. (2019). (posted February 1st 2019, in press).
[Crossref]

Barber, W. C.

K. Taguchi, M. Zhang, E. C. Frey, X. Wang, J. S. Iwanczyk, E. Nygard, N. E. Hartsough, B. M. W. Tsui, and W. C. Barber, “Modeling the performance of a photon counting X-ray detector for CT: Energy response and pulse pileup effects,” Med. Phys. 38, 1089–1102 (2011).
[Crossref] [PubMed]

Baumer, C.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Baumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Medicine Biol. 53, 4031 (2008).
[Crossref]

Berger, M.

A. Maier, H. G. Hofmann, M. Berger, P. Fischer, C. Schwemmer, H. Wu, K. Muller, J. Hornegger, J. H. Choi, C. Riess, A. Keil, and R. Fahrig, “Conrad-a software framework for cone-beam imaging in radiology,” Med. Phys. 40, 111914 (2013).
[Crossref]

Bian, J.

Y. Rakvongthai, W. Worstell, G. E. Fakhri, J. Bian, A. Lorsakul, and J. Ouyang, “Spectral CT using multiple balanced K-edge filters,” IEEE Transactions on Med. Imaging 34, 740–747 (2015).
[Crossref]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Process. 16, 2992–3004 (2007).
[Crossref]

Bol, W.

W. Bol, “The use of balanced filters in X-ray diffraction,” J. Sci. Instruments 44, 736 (1967).
[Crossref]

Bones, P. J.

I. Glass, A. P. H. Butler, P. H. Butler, P. J. Bones, and S. J. Weddell, “Physiological gating of the MARS spectral micro CT scanner,” in 28th Int. Conf. on Image and Vision Computing New Zealand (IVCNZ), (2013), pp. 317–322.

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Brady, D. J.

Butler, A. P. H.

I. Glass, A. P. H. Butler, P. H. Butler, P. J. Bones, and S. J. Weddell, “Physiological gating of the MARS spectral micro CT scanner,” in 28th Int. Conf. on Image and Vision Computing New Zealand (IVCNZ), (2013), pp. 317–322.

Butler, P. H.

I. Glass, A. P. H. Butler, P. H. Butler, P. J. Bones, and S. J. Weddell, “Physiological gating of the MARS spectral micro CT scanner,” in 28th Int. Conf. on Image and Vision Computing New Zealand (IVCNZ), (2013), pp. 317–322.

Candes, E. J.

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

Carin, L.

Carmi, R.

R. Carmi, G. Naveh, and A. Altman, “Material separation with dual-layer CT,” in IEEE Nuclear Science Symp. Conf. Record, 2005, vol. 4 (2005), pp. 3 pp.–1878.

Chen, Z.

L. Li, Z. Chen, W. Cong, and G. Wang, “Spectral CT modeling and reconstruction with hybrid detectors in dynamic-threshold-based counting and integrating modes,” IEEE Transactions on Med. Imaging 34, 716–728 (2015).
[Crossref]

L. Li, Z. Chen, G. Wang, J. Chu, and H. Gao, “A tensor PRISM algorithm for multi-energy CT reconstruction and comparative studies,” J. X-Ray Sci. Technol. 22, 147–163 (2014).

W. Miao, Y. Ding, K. Sun, D. Lai, Z. Chen, H. Wang, J. Cheng, Z. Zheng, and H. Peng, “Design of the Ross pairs for soft X-ray spectrometer,” in IEEE Int. Conf. on Plasma Science (Cat. No.02CH37340), (2002), p. 161.

Cheng, J.

W. Miao, Y. Ding, K. Sun, D. Lai, Z. Chen, H. Wang, J. Cheng, Z. Zheng, and H. Peng, “Design of the Ross pairs for soft X-ray spectrometer,” in IEEE Int. Conf. on Plasma Science (Cat. No.02CH37340), (2002), p. 161.

Choi, J. H.

A. Maier, H. G. Hofmann, M. Berger, P. Fischer, C. Schwemmer, H. Wu, K. Muller, J. Hornegger, J. H. Choi, C. Riess, A. Keil, and R. Fahrig, “Conrad-a software framework for cone-beam imaging in radiology,” Med. Phys. 40, 111914 (2013).
[Crossref]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Chu, J.

L. Li, Z. Chen, G. Wang, J. Chu, and H. Gao, “A tensor PRISM algorithm for multi-energy CT reconstruction and comparative studies,” J. X-Ray Sci. Technol. 22, 147–163 (2014).

J. Chu, W. Cong, L. Li, and G. Wang, “Combination of current-integrating and photon-counting detector modules for spectral CT,” Phys. Medicine Biol. 58, 7009 (2013).
[Crossref]

Clark, D. P.

C. T. Badea, D. P. Clark, M. Holbrook, M. Srivastava, Y. Mowery, and K. B. Ghaghada, “Functional imaging of tumor vasculature using iodine and gadolinium-based nanoparticle contrast agents: a comparison of spectral micro-ct using energy integrating and photon counting detectors,” Phys. Medicine Biol. (2019). (posted February 1st 2019, in press).
[Crossref]

Cong, W.

L. Li, Z. Chen, W. Cong, and G. Wang, “Spectral CT modeling and reconstruction with hybrid detectors in dynamic-threshold-based counting and integrating modes,” IEEE Transactions on Med. Imaging 34, 716–728 (2015).
[Crossref]

J. Chu, W. Cong, L. Li, and G. Wang, “Combination of current-integrating and photon-counting detector modules for spectral CT,” Phys. Medicine Biol. 58, 7009 (2013).
[Crossref]

Cuadros, A. P.

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

A. P. Cuadros and G. R. Arce, “Coded aperture compressive X-ray spectral CT,” in 2017 Int. Conf. on Sampling Theory and Applications (SampTA), (2017), pp. 548–551.

Dill, S.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Baumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Medicine Biol. 53, 4031 (2008).
[Crossref]

Ding, Y.

W. Miao, Y. Ding, K. Sun, D. Lai, Z. Chen, H. Wang, J. Cheng, Z. Zheng, and H. Peng, “Design of the Ross pairs for soft X-ray spectrometer,” in IEEE Int. Conf. on Plasma Science (Cat. No.02CH37340), (2002), p. 161.

Dorscheid, R.

J. P. Schlomka, E. Roessl, R. Dorscheid, S. Dill, G. Martens, T. Istel, C. Baumer, C. Herrmann, R. Steadman, G. Zeitler, A. Livne, and R. Proksa, “Experimental feasibility of multi-energy photon-counting K-edge imaging in pre-clinical computed tomography,” Phys. Medicine Biol. 53, 4031 (2008).
[Crossref]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Fahrig, R.

A. Maier, H. G. Hofmann, M. Berger, P. Fischer, C. Schwemmer, H. Wu, K. Muller, J. Hornegger, J. H. Choi, C. Riess, A. Keil, and R. Fahrig, “Conrad-a software framework for cone-beam imaging in radiology,” Med. Phys. 40, 111914 (2013).
[Crossref]

Fakhri, G. E.

K. Kim, J. C. Ye, W. Worstell, J. Ouyang, Y. Rakvongthai, G. E. Fakhri, and Q. Li, “Sparse-view spectral CT reconstruction using spectral patch-based low-rank penalty,” IEEE Trans. on Medical Imaging 34, 748–760 (2015).
[Crossref]

Y. Rakvongthai, W. Worstell, G. E. Fakhri, J. Bian, A. Lorsakul, and J. Ouyang, “Spectral CT using multiple balanced K-edge filters,” IEEE Transactions on Med. Imaging 34, 740–747 (2015).
[Crossref]

Figueiredo, M. A. T.

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C. T. Badea, D. P. Clark, M. Holbrook, M. Srivastava, Y. Mowery, and K. B. Ghaghada, “Functional imaging of tumor vasculature using iodine and gadolinium-based nanoparticle contrast agents: a comparison of spectral micro-ct using energy integrating and photon counting detectors,” Phys. Medicine Biol. (2019). (posted February 1st 2019, in press).
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K. Taguchi, M. Zhang, E. C. Frey, X. Wang, J. S. Iwanczyk, E. Nygard, N. E. Hartsough, B. M. W. Tsui, and W. C. Barber, “Modeling the performance of a photon counting X-ray detector for CT: Energy response and pulse pileup effects,” Med. Phys. 38, 1089–1102 (2011).
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Y. Rakvongthai, W. Worstell, G. E. Fakhri, J. Bian, A. Lorsakul, and J. Ouyang, “Spectral CT using multiple balanced K-edge filters,” IEEE Transactions on Med. Imaging 34, 740–747 (2015).
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A. Maier, H. G. Hofmann, M. Berger, P. Fischer, C. Schwemmer, H. Wu, K. Muller, J. Hornegger, J. H. Choi, C. Riess, A. Keil, and R. Fahrig, “Conrad-a software framework for cone-beam imaging in radiology,” Med. Phys. 40, 111914 (2013).
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Figures (10)

Fig. 1
Fig. 1 (a) 80 keV X-ray source spectrum (blue), filtered with Ce (red) and Dy (cyan). Dy and Ce are a Ross filter pair; thus, a quasi-monochromatic spectrum can be obtained by subtracting the filtered spectrum corresponding to the lower K-edge filter, Ce, from the spectrum, obtained using the higher K-edge filter, Dy as shown in (b). Vertical lines in the spectra indicate the K-edges of Ce and Dy, 40.4 and 53.8 keV, respectively.
Fig. 2
Fig. 2 Top: Subsampling strategies for CSXI. (a) Uniform view (UV-CSXI): K-edge filters are assigned sequentially every F view angles (F is the number of K-edge filters used); (b) Random view (RV-CSXI): A K-edge filter is assigned randomly to every view angle; (c) Uniform detector (UD-CSXI): the same pattern of K-edge filters repeats every F coded aperture elements; (d) Random detector (RD-CSXI): random coded apertures with multiple K-edge filters are used at each view angle. Bottom: Spectrally coded sinograms, color-coded according to the material used to filter the X-ray that impinges on a particular detector element. Zoomed area is shown for each sinogram, depicting the multiplexing structure.
Fig. 3
Fig. 3 Top: X-ray downsampling in CSXI with a subsampling rate of 2. A random block/unblock coded aperture is assigned to each view angle in the (a) UVB-CSXI and (b) RVB-CSXI configurations. And, blocking elements are introduced in the K-edge coded apertures to subsample the measurements for the (c) UDB-CSXI and (d) RDB-CSXI configurations. Bottom: The spectrally coded sinograms in each case are color-coded according to the material used to filter the X-ray that impinges on a particular detector element. A zoomed area of each sinogram is shown to illustrate their multiplexing structure.
Fig. 4
Fig. 4 Filtered sinograms inpainting. (a) If, measured undersampled sinograms; (b) yf , log-normalized undersampled sinograms; (c) xf , reconstructed effective linear attenuation coefficients; (d) I ^ f, final estimated intensity sinograms, for f = 1 , , F.
Fig. 5
Fig. 5 Monochromatic sinogram estimation process. (a) I ^ f, stacked estimated intensity measurement with column vectors rj ; (b) J, energy binned intensity estimates with column vectors dj; (c) zk, log-transformed energy binned sinograms.
Fig. 6
Fig. 6 (a) 80 keV X-ray source spectrum, filtered with (b) Ce, (c) Dy, and (d) Er. Ce and Dy, and Dy and Er are Ross filter pairs. Note the filtered spectra for all filters has the same value in the bins [ 0 , E 1 ) and ( E f , E T ) highlighted in black for all the figures.
Fig. 7
Fig. 7 (a) Forbild thorax phantom [33]. (b) Energy spectra of an 80 keV X-ray source seen through the five different filters used in the CSXI simulations [Molybdenum (Mo), Cerium (Ce), Dysprosium (Dy), Erbium (Er) and Tungsten (W)]. (c) Quasi-monochromatic energy binned spectra obtained subtracting the filtered spectra of the corresponding Ross filter pairs Ce-Mo, Dy-Ce, Er-Dy, and W-Er.
Fig. 8
Fig. 8 (a) Effective linear attenuation coefficient images at energy bin k = 3. Reconstructions using a subsampling rate of 8 for (b) UV-CSXI when view angles are subsampled and (c) UVB-CSXI (block-unblock coded apertures are used to subsample the detectors). The PSNR and zoomed versions of the highlighted regions are shown for each reconstruction. (d-e) Normalized absolute errors for (b) and (c), respectively.
Fig. 9
Fig. 9 (a) 3D-IRCADb thoracic-abdominal CT scan slice, (b) Segmented CT scan according to pixel material. (c-f) Original energy binned effective linear attenuation coefficient images. Reconstructions obtained with (g-j) UV-CSXI, and (k-n) RD-CSXI using one of 10 random masks at each view. The PSNR is shown for each reconstruction. (o-v) Normalized absolute error for each configuration.
Fig. 10
Fig. 10 (a) Effective linear attenuation coefficient images at energy bin k = 1. Reconstructions obtained using (b) UV-CSXI with M = P = 512 and a single scan, and (c) the system proposed in [12] with M = 512, P = 103 and 5 different scans. The PSNR is shown for each reconstruction as well as zoomed versions of the highlighted areas in the phantom. The normalized absolute error is shown in each case.

Tables (4)

Tables Icon

Algorithm 1 Image reconstruction via the ADMM

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Table 1 Ross filter pairs and respective energy bins

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Table 2 CSXI PSNR at each k energy bin for the Thorax Forbild Phantom

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Table 3 Thorax DICOM reconstruction PSNR for UV-CSXI downsampling x8 vs K-edge filter system in [12] with P = 13

Equations (16)

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I j = E I 0 ( E ) exp  ( l μ ( l , E ) d l ) d E .
I j f = E I f ( E ) exp  ( l μ ( l , E ) d l ) d E .
I j f = k = 1 F + 1 B j ( f , k ) I 0 ( k ) exp  ( i = 1 N 2 H j , i u k ( i ) ) ,
argmin { u k ( i ) } i , k j = 1 M P [ ( k = 1 F + 1 B j ( f , k ) I 0 ( k ) exp  ( i = 1 N 2 H i j u k ( i ) ) ) I j f ] 2 .
I j f = E I f ( E ) exp  ( l μ ¯ f ( l ) d l ) d E = I 0 f exp  ( l μ ¯ f ( l ) d l ) ,
x ^ f = argmin x f 1 2 | | y f C f H x f | | 2 2 + λ 1 | | Ψ x f | | 1 , for f = 1 , , F .
J = argmin d j j = 1 M P 1 2 | | r j B j d j | | 2 2 ,
Z = H ( U ) = [ [ H u 1 ] T | | [ H K u K ] T ] T ,
θ ^ = argmin 1 2 | | Z H ( U ) | | 2 2 + η 1 | | vec ( θ ) | | 1 + η * | | U | | * ,
argmin U , D , B 1 2 | | Z H ( U ) | | 2 2 + λ * | | D | | * + λ 1 | | B | | 1 + μ * | | D U V | | 2 2 + μ 1 | | B Ψ ( U ) W | | 2 2 .
D τ + 1 = 1 ξ + 1 n = 1 ξ + 1 fold ( S ϵ { U ( n ) τ + 1 + V ( n ) τ } ) ,
B τ + 1 = softshrink { Ψ ( U τ + 1 ) + W τ , λ 1 / μ 1 } ,
softshrink ( x , δ ) = { x δ if x > δ 0 if | x | δ x + δ if x < δ
u k ^ = argmin u k | | z k A k u k | | 2 2   +   λ   T V ( u k ) ,
I j k , f = E k 1 E k I f ( E ) exp  ( l μ ( l , E ) d l ) d E ,
I j k , f ( E k 1 E k I 0 ( E ) exp   ( μ f ( E ) δ j , f ) d E ) exp   ( l μ ¯ ( l , k ) d l ) = E k 1 E k exp   ( μ f ( E ) δ j , f ) d E E k E k 1 B j ( f , k ) I 0 ( k ) exp   ( l μ ¯ ( l , k ) d l ) d j k ,

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