Abstract

The penetration power of x rays allows one to image large objects, while their short wavelength allows for high spatial resolution. As a result, with synchrotron sources, one has the potential to obtain tomographic images of centimeter-sized specimens with sub-micrometer pixel sizes. However, limitations on beam and detector size make it difficult to acquire such data of this sort in a single take, necessitating strategies for combining data from multiple regions. One strategy is to acquire a tiled set of local tomograms by rotating the specimen around each of the local tomogram center positions. Another strategy, sinogram oriented acquisition, involves the collection of projections at multiple offset positions from the rotation axis followed by data merging and reconstruction. We have carried out a simulation study to compare these two approaches in terms of radiation dose applied to the specimen, and reconstructed image quality. Local tomography acquisition involves an easier data alignment problem, and immediate viewing of subregions before the entire dataset has been acquired. Sinogram oriented acquisition involves a more difficult data assembly and alignment procedure, and it is more sensitive to accumulative registration error. However, sinogram oriented acquisition is more dose efficient, involves fewer translation motions of the object, and avoids certain artifacts of local tomography.

© 2018 Optical Society of America

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References

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    [Crossref]

2018 (3)

D. M. Pelt and D. Y. Parkinson, “Ring artifact reduction in synchrotron x-ray tomography through helical acquisition,” Meas. Sci. Technol. 29, 034002 (2018).
[Crossref]

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

J. C. da Silva, M. Guizar-Sicairos, M. Holler, A. Diaz, J. A. van Bokhoven, O. Bunk, and A. Menzel, “Quantitative region-of-interest tomography using variable field of view,” Opt. Express 26, 16752–16768 (2018).
[Crossref]

2017 (4)

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
[Crossref]

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

I. V. Oikonomidis and G. Lovric, “Imaging samples larger than the field of view: the SLS experience,” J. Phys. Conf. Ser. 849, 012004 (2017).
[Crossref]

R. F. C. Vescovi, M. B. Cardoso, and E. X. Miqueles, “Radiography registration for mosaic tomography,” J. Synchrotron Radiat. 24, 686–694 (2017).
[Crossref]

2016 (1)

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

2014 (1)

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

2012 (3)

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

2011 (1)

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

2009 (1)

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

2006 (1)

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

2003 (1)

P. Perez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. 22, 313–318 (2003).
[Crossref]

1995 (1)

P. Kuchment, K. Lancaster, and L. Mogilevskaya, “On local tomography,” Inverse Prob. 11, 571–589 (1995).
[Crossref]

1994 (1)

W. A. Kalender, “Technical foundations of spiral CT,” Semin. Ultrasound CT MRI 15, 81–89 (1994).
[Crossref]

1989 (1)

J. Dengler, “A multi-resolution approach to the 3D reconstruction from an electron microscope tilt series solving the alignment problem without gold particles,” Ultramicroscopy 30, 337–348 (1989).
[Crossref]

1985 (1)

L. Reimer and A. Schmidt, “The shrinkage of bulk polymers by radiation damage in an SEM,” Scanning 7, 47–53 (1985).
[Crossref]

1978 (1)

R. M. Lewitt and R. H. T. Bates, “Image reconstruction from projections: I: general theoretical considerations,” Optik 50, 19–33 (1978).

1970 (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
[Crossref]

Abela, R.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Andrews, J. C.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Bamberg, F.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Batenburg, K. J.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Bates, R. H. T.

R. M. Lewitt and R. H. T. Bates, “Image reconstruction from projections: I: general theoretical considerations,” Optik 50, 19–33 (1978).

Blake, A.

P. Perez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. 22, 313–318 (2003).
[Crossref]

Blanke, A.

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Bravin, A.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Brun, E.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Bunk, O.

Cardoso, M. B.

R. F. C. Vescovi, M. B. Cardoso, and E. X. Miqueles, “Radiography registration for mosaic tomography,” J. Synchrotron Radiat. 24, 686–694 (2017).
[Crossref]

Chen, S.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Ching, D. J.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
[Crossref]

Chu, Y. S.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Coan, P.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Connolley, T.

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

Cossairt, O.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Crowther, R. A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
[Crossref]

da Silva, J. C.

De Andrade, V.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

De Carlo, F.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

Dengler, J.

J. Dengler, “A multi-resolution approach to the 3D reconstruction from an electron microscope tilt series solving the alignment problem without gold particles,” Ultramicroscopy 30, 337–348 (1989).
[Crossref]

DeRosier, D. J.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
[Crossref]

Diaz, A.

Diemoz, P. C.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Gangnet, M.

P. Perez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. 22, 313–318 (2003).
[Crossref]

Ge, M.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Guizar-Sicairos, M.

Gursoy, D.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Gürsoy, D.

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
[Crossref]

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

He, K.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Heinzer, S.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Holler, M.

Hong, Y. P.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Hujsak, K.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Ibison, M.

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

Irvine, S.

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Jacobsen, C.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2012).

Kalender, W. A.

W. A. Kalender, “Technical foundations of spiral CT,” Semin. Ultrasound CT MRI 15, 81–89 (1994).
[Crossref]

Katsaggelos, A. K.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Kingston, A. M.

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Klug, A.

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
[Crossref]

Krucker, T.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Kuchment, P.

P. Kuchment, K. Lancaster, and L. Mogilevskaya, “On local tomography,” Inverse Prob. 11, 571–589 (1995).
[Crossref]

Kyrieleis, A.

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

Lancaster, K.

P. Kuchment, K. Lancaster, and L. Mogilevskaya, “On local tomography,” Inverse Prob. 11, 571–589 (1995).
[Crossref]

Latham, S. J.

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Lewitt, R. M.

R. M. Lewitt and R. H. T. Bates, “Image reconstruction from projections: I: general theoretical considerations,” Optik 50, 19–33 (1978).

Li, Y.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Liu, Y.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Lovric, G.

I. V. Oikonomidis and G. Lovric, “Imaging samples larger than the field of view: the SLS experience,” J. Phys. Conf. Ser. 849, 012004 (2017).
[Crossref]

Ludwig, W.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Mancini, L.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Marone, F.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Mayr, D.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Meirer, F.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Menzel, A.

Meyer, E. P.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Miller, L. M.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Miqueles, E. X.

R. F. C. Vescovi, M. B. Cardoso, and E. X. Miqueles, “Radiography registration for mosaic tomography,” J. Synchrotron Radiat. 24, 686–694 (2017).
[Crossref]

Mogilevskaya, L.

P. Kuchment, K. Lancaster, and L. Mogilevskaya, “On local tomography,” Inverse Prob. 11, 571–589 (1995).
[Crossref]

Mokso, R.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Müller, R.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Myers, G. R.

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Oikonomidis, I. V.

I. V. Oikonomidis and G. Lovric, “Imaging samples larger than the field of view: the SLS experience,” J. Phys. Conf. Ser. 849, 012004 (2017).
[Crossref]

Parkinson, D. Y.

D. M. Pelt and D. Y. Parkinson, “Ring artifact reduction in synchrotron x-ray tomography through helical acquisition,” Meas. Sci. Technol. 29, 034002 (2018).
[Crossref]

Pelt, D. M.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

D. M. Pelt and D. Y. Parkinson, “Ring artifact reduction in synchrotron x-ray tomography through helical acquisition,” Meas. Sci. Technol. 29, 034002 (2018).
[Crossref]

Perez, P.

P. Perez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. 22, 313–318 (2003).
[Crossref]

Pianetta, P.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Quaroni, L.

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Recur, B.

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Reimer, L.

L. Reimer and A. Schmidt, “The shrinkage of bulk polymers by radiation damage in an SEM,” Scanning 7, 47–53 (1985).
[Crossref]

Reiser, M. F.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Rivers, M.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Schlossbauer, T.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Schmidt, A.

L. Reimer and A. Schmidt, “The shrinkage of bulk polymers by radiation damage in an SEM,” Scanning 7, 47–53 (1985).
[Crossref]

Schneider, P.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Schuler, A.

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Sheppard, A. P.

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Sijber, J.

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2012).

Stampanoni, M.

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Sztrókay, A.

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Titarenko, V.

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

van Bokhoven, J. A.

Vescovi, R. F. C.

R. F. C. Vescovi, M. B. Cardoso, and E. X. Miqueles, “Radiography registration for mosaic tomography,” J. Synchrotron Radiat. 24, 686–694 (2017).
[Crossref]

Vila-Comamala, J.

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

Wang, J.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Williams, P. A.

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

Withers, P. J.

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

Xiao, X.

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

Yoo, S.

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

ACM Trans. Graph. (1)

P. Perez, M. Gangnet, and A. Blake, “Poisson image editing,” ACM Trans. Graph. 22, 313–318 (2003).
[Crossref]

IEEE Trans. Image Process. (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Inverse Prob. (1)

P. Kuchment, K. Lancaster, and L. Mogilevskaya, “On local tomography,” Inverse Prob. 11, 571–589 (1995).
[Crossref]

J. Microsc. (1)

A. Kyrieleis, V. Titarenko, M. Ibison, T. Connolley, and P. J. Withers, “Region-of-interest tomography using filtered backprojection: assessing the practical limits,” J. Microsc. 241, 69–82 (2011).
[Crossref]

J. Phys. Conf. Ser. (1)

I. V. Oikonomidis and G. Lovric, “Imaging samples larger than the field of view: the SLS experience,” J. Phys. Conf. Ser. 849, 012004 (2017).
[Crossref]

J. Struct. Biol. (1)

R. Mokso, L. Quaroni, F. Marone, S. Irvine, J. Vila-Comamala, A. Blanke, and M. Stampanoni, “X-ray mosaic nanotomography of large microorganisms,” J. Struct. Biol. 177, 233–238 (2012).
[Crossref]

J. Synchrotron Radiat. (4)

Y. Liu, F. Meirer, P. A. Williams, J. Wang, J. C. Andrews, and P. Pianetta, “TXM-wizard: a program for advanced data collection and evaluation in full-field transmission x-ray microscopy,” J. Synchrotron Radiat. 19, 281–287 (2012).
[Crossref]

R. F. C. Vescovi, M. B. Cardoso, and E. X. Miqueles, “Radiography registration for mosaic tomography,” J. Synchrotron Radiat. 24, 686–694 (2017).
[Crossref]

D. J. Ching and D. Gürsoy, “XDesign: an open-source software package for designing x-ray imaging phantoms and experiments,” J. Synchrotron Radiat. 24, 537–544 (2017).
[Crossref]

D. Gürsoy, F. De Carlo, X. Xiao, and C. Jacobsen, “TomoPy: a framework for the analysis of synchrotron tomographic data,” J. Synchrotron Radiat. 21, 1188–1193 (2014).
[Crossref]

Meas. Sci. Technol. (2)

D. M. Pelt and D. Y. Parkinson, “Ring artifact reduction in synchrotron x-ray tomography through helical acquisition,” Meas. Sci. Technol. 29, 034002 (2018).
[Crossref]

F. De Carlo, D. Gursoy, D. J. Ching, K. J. Batenburg, W. Ludwig, L. Mancini, F. Marone, R. Mokso, D. M. Pelt, J. Sijber, and M. Rivers, “TomoBank: a tomographic data repository for computational x-ray science,” Meas. Sci. Technol. 29, 034004 (2018).
[Crossref]

NeuroImage (1)

S. Heinzer, T. Krucker, M. Stampanoni, R. Abela, E. P. Meyer, A. Schuler, P. Schneider, and R. Müller, “Hierarchical microimaging for multiscale analysis of large vascular networks,” NeuroImage 32, 626–636 (2006).
[Crossref]

Nucl. Instrum. Methods Phys. Res. A (1)

A. Kyrieleis, M. Ibison, V. Titarenko, and P. J. Withers, “Image stitching strategies for tomographic imaging of large objects at high resolution at synchrotron sources,” Nucl. Instrum. Methods Phys. Res. A 607, 677–684 (2009).
[Crossref]

Opt. Express (1)

Optik (1)

R. M. Lewitt and R. H. T. Bates, “Image reconstruction from projections: I: general theoretical considerations,” Optik 50, 19–33 (1978).

Phys. Med. Biol. (1)

A. Sztrókay, P. C. Diemoz, T. Schlossbauer, E. Brun, F. Bamberg, D. Mayr, M. F. Reiser, A. Bravin, and P. Coan, “High-resolution breast tomography at high energy: a feasibility study of phase contrast imaging on a whole breast,” Phys. Med. Biol. 57, 2931–2942 (2012).
[Crossref]

Proc. R. Soc. London Ser. A (1)

R. A. Crowther, D. J. DeRosier, and A. Klug, “The reconstruction of a three-dimensional structure from projections and its application to electron microscopy,” Proc. R. Soc. London Ser. A 317, 319–340 (1970).
[Crossref]

Proc. SPIE (1)

S. J. Latham, A. M. Kingston, B. Recur, G. R. Myers, and A. P. Sheppard, “Multi-resolution radiograph alignment for motion correction in x-ray micro-tomography,” Proc. SPIE 9967, 996710 (2016).
[Crossref]

Scanning (1)

L. Reimer and A. Schmidt, “The shrinkage of bulk polymers by radiation damage in an SEM,” Scanning 7, 47–53 (1985).
[Crossref]

Sci. Rep. (1)

D. Gürsoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, Y. Li, M. Ge, L. M. Miller, Y. S. Chu, V. De Andrade, K. He, O. Cossairt, A. K. Katsaggelos, and C. Jacobsen, “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7, 11818 (2017).
[Crossref]

Semin. Ultrasound CT MRI (1)

W. A. Kalender, “Technical foundations of spiral CT,” Semin. Ultrasound CT MRI 15, 81–89 (1994).
[Crossref]

Ultramicroscopy (1)

J. Dengler, “A multi-resolution approach to the 3D reconstruction from an electron microscope tilt series solving the alignment problem without gold particles,” Ultramicroscopy 30, 337–348 (1989).
[Crossref]

Other (3)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, 2012).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

https://github.com/mdw771/tomosim .

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Figures (8)

Fig. 1.
Fig. 1. Comparison on the acquisition scheme of local tomography acquisition (LTA; left) versus sinogram oriented acquisition (SOA; right). The top row depicts information collection in sinogram space, where each stripe with an arrow and a distinct color represents one angular scan over 180° (which is then used to synthesize the full 360° sinogram). The bottom row shows the mapping of different scans to the full image of one object slice. For samples with roughly equal extension in both lateral dimensions, if the number of scans required in SOA is n f , then n f 2 scans are needed by LTA.
Fig. 2.
Fig. 2. Phantom object created for simulations, where the highest attenuation values are white and the lowest are black. The object has a diameter, or maximum projected thickness, of L = 2048 voxels, each set to a per-voxel linear attenuation coefficient of μ = 1 / 2048 , so that the total attenuation through the disk if solid is exp [ 1 ] . The object is designed with random spherical “pores” inside with linear attenuation coefficients ranging from μ = 0 / 2048 to μ = 1 / 2048 .
Fig. 3.
Fig. 3. Coverage on the full sinogram in an experiment using (a) sinogram oriented acquisition (SOA) and (b) local tomography acquisition (LTA) with equal field of view. Brighter values in the images correspond to the number of times that a voxel in the object is sampled by (exposed to) the illuminating beam.
Fig. 4.
Fig. 4. Schematic diagram showing the assumed pattern of data acquisition in the LTA approach of beyond-field-of-view tomography. The specimen is represented by the gray solid disk. Each local ROI that can be reconstructed using the data acquired from a scan in LTA is shown by a dashed blue circle. Each of these ROIs has a diameter of f LTA , and they are packed in a way that the distance between the centers of each pair of diagonally adjacent ROIs is exactly f LTA , so that the sample can be fully covered without gaps.
Fig. 5.
Fig. 5. Acquired data size (a) and radiation dose (b) as a function of the truncation ratio T of Eq. (6) for both local tomography acquisition (LTA) and sinogram oriented acquisition (SOA). In each subplot, the variation of n f , SOA and n f , LTA is also shown. The figure indicates that the acquired data size and radiation dose do not necessarily decrease with increasing truncation ratio; rather, both quantities are associated with the arrangement of scans in an actual experiment. These results were calculated for fixed values of γ SOA = 0.85 and γ LTA = 0.85 , as discussed in the text.
Fig. 6.
Fig. 6. Comparison of image quality between local tomography acquisition (LTA) and sinogram oriented acquisition (SOA). This comparison is made using the Structural SIMilarity index (SSIM) with regard to the ground truth image, against reconstructions for SOA (a) and LTA (b). When noise is not a factor, and registration errors are negligible, the SOA result is identical to the ground truth. Also shown (c) is the SSIM as a function of truncation ratio T .
Fig. 7.
Fig. 7. Mean registration error plotted against (a) the average photon budget per voxel for both SOA and LTA, and (b) the downsampling level in projection angles for LTA. The plot indicates that the accuracy of phase correlation degrades when projection images become noisier due to lower number of incident photons. For this particular sample, registration in reconstruction domain for LTA requires a higher incident photon flux in order to give reliable registration results. In addition, a reduction in projection angles also causes a significant deterioration in registration accuracy for LTA.
Fig. 8.
Fig. 8. Comparison of the effect of registration errors. Shown here are sinogram oriented acquisition (SOA; left column in the grid) and local tomography acquisition (LTA; right column) reconstructions at a region of interest (first row) and the center (second row) of a slice in the charcoal sample. The SOA reconstruction was done by stitching eight tiles extracted from the full sinogram. Registration errors following a Gaussian distribution with a standard deviation of 4.0 were exerted to the tile positions before stitching. The rotation center for SOA reconstruction was calibrated to optimize the quality around the object center. As a result, the central region of the charcoal reconstructed using both methods appears similar. However, at around 1000 pixels above the object center, the SOA reconstruction shows severe distortion of dot-shaped features (pointed out by colored arrows) due to the deviation of its actual position from the rotation center inputted to the reconstruction routine. The inset in the SOA figure shows the appearance of one of the distorted features when the tiles are correctly registered.

Equations (20)

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

C LTA = { ( s , θ ) | s 0 ( θ ) f / 2 s s 0 ( θ ) + f / 2 }
s 0 ( θ ) = x 2 + y 2 cos ( α θ ) + c 0 ,
C SOA = { ( s , θ ) | p f / 2 s p + f / 2 } ,
n f , SOA = ceil [ L f γ SOA f + 1 ] ,
n f , LTA = { 1 f L ceil ( 2 L γ LTA f ) f < L .
T = f L ,
d E d t = | n ¯ E 0 d I d t | ,
d E d t = n ¯ E 0 exp ( μ ¯ L / 2 ) μ r ( t ) .
E = n ¯ E 0 exp ( μ ¯ L / 2 ) μ ¯ Δ .
D s , j = N θ n ¯ E 0 exp ( μ ¯ L / 2 ) μ ¯ ρ Δ 2 ,
D = s j D s , j s ε Ω s ,
c = argmax x R 2 F 1 [ F [ I a ( x ) ] F * [ I b ( x ) ] | F [ I a ( x ) ] F * [ I b ( x ) ] | ] ( x ) .
l ( A , B ) = 2 μ A μ B + c 1 μ A 2 + μ B 2 + c 1 ,
c ( A , B ) = 2 σ A σ B + c 2 σ A 2 + σ B 2 + c 2 ,
s ( A , B ) = σ A B + c 3 σ A σ B + c 3 ,
c 1 = ( k 1 L ) 2 ,
c 2 = ( k 2 L ) 2 ,
c 3 = c 2 / 2
SSIM ( A , B ) = c ( A , B ) · s ( A , B ) .
f ( k , n ph I ) = ( n ph I ) k e n ph I k ! ,