Abstract

We demonstrate phase contrast enhancement of X-ray computed tomography derived from propagation based imaging. In this method, the absorption and phase components are assumed to be correlated, allowing for phase retrieval from a single image. Experimental results are shown for liquid samples. Signal-to-noise ratio is greatly enhanced relative to pure attenuation based imaging.

© 2014 Optical Society of America

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References

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  1. S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. 83, 3155 (2003).
    [CrossRef]
  2. K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. 5, 403–416 (2009).
    [CrossRef] [PubMed]
  3. E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
    [CrossRef] [PubMed]
  4. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
    [CrossRef]
  5. A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
    [CrossRef] [PubMed]
  6. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
    [CrossRef]
  7. J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
    [CrossRef] [PubMed]
  8. R. C. Chen, L. Rigon, and R. Longo, “Comparison of single distance phase retrieval algorithms by considering different object composition and the effect of statistical and structural noise,” Opt. Express 21, 7384–7399 (2013).
    [CrossRef] [PubMed]
  9. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430 (2013).
    [CrossRef] [PubMed]
  10. M. Reed Teague, “Deterministic phase retrieval: a green?s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  11. A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19, 472–480 (2002).
    [CrossRef]
  12. A. Burvall, U. Lundstrm, P. A. C. Takman, D. H. Larsson, and H. M. Hertz, “Phase retrieval in x-ray phase-contrast imaging suitable for tomography,” Opt. Express 19, 10359–10376 (2011).
    [CrossRef] [PubMed]
  13. T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
    [CrossRef] [PubMed]
  14. G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
    [CrossRef] [PubMed]
  15. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
    [CrossRef]
  16. T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
    [CrossRef]
  17. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, and G. Barbastathis, “Compressive x-ray phase tomography based on the transport of intensity equation,” Opt. Lett. 38, 3418–3421 (2013).
    [CrossRef] [PubMed]
  18. X. Wu, H. Liu, and A. Yan, “X-ray phase-attenuation duality and phase retrieval,” Opt. Lett. 30, 379–381 (2005).
    [CrossRef] [PubMed]
  19. T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15, 579–585 (1998).
    [CrossRef]
  20. D. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
    [CrossRef]
  21. S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
    [CrossRef] [PubMed]
  22. J. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing 16, 2992–3004 (2007).
    [CrossRef] [PubMed]
  23. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
    [CrossRef]

2013

2011

2009

K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. 5, 403–416 (2009).
[CrossRef] [PubMed]

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

2008

2007

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, “Mixed transfer function and transport of intensity approach for phase retrieval in the fresnel region,” Opt. Lett. 32, 1617–1619 (2007).
[CrossRef] [PubMed]

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

2006

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
[CrossRef]

2005

2004

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
[CrossRef] [PubMed]

2003

S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. 83, 3155 (2003).
[CrossRef]

2002

1998

1996

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

1995

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

1992

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
[CrossRef]

1991

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

1983

Bakal, C. W.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Barbastathis, G.

Bioucas-Dias, J.

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

Boistel, R.

Braha, S.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Bronnikov, A. V.

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

Burvall, A.

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
[CrossRef]

Chen, R. C.

Cloetens, P.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

Davis, T. J.

T. E. Gureyev, T. J. Davis, A. Pogany, S. C. Mayo, and S. W. Wilkins, “Optical phase retrieval by use of first born- and rytov-type approximations,” Appl. Opt. 43, 2418–2430 (2004).
[CrossRef] [PubMed]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
[CrossRef]

Fessler, J.

S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
[CrossRef] [PubMed]

Figueiredo, M. A. T.

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

Freeman, N. J.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Friedman, A. C.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Gao, D.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Goldstein, T.

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

Guigay, J. P.

Gureyev, T. E.

Haramati, N.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Hasebroock, K. M.

K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. 5, 403–416 (2009).
[CrossRef] [PubMed]

Hertz, H. M.

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

Kadish, E. G.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Kazantsev, I.

S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
[CrossRef] [PubMed]

Kudrolli, H.

Langer, M.

Larsson, D. H.

Lautin, E. M.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Lautin, J. L.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Liu, H.

Longo, R.

Lundstrm, U.

Matej, S.

S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
[CrossRef] [PubMed]

Mayo, S. C.

Miao, Q.

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

Myers, G. R.

Nagarkar, V.

Osher, S.

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
[CrossRef]

Paganin, D.

D. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

Paganin, D. M.

Petruccelli, J. C.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

Pogany, A.

Reed Teague, M.

Rigon, L.

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
[CrossRef]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
[CrossRef]

Schoenfeld, A. H.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Serkova, N. J.

K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. 5, 403–416 (2009).
[CrossRef] [PubMed]

Singh, M.

S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. 83, 3155 (2003).
[CrossRef]

Singh, S.

S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. 83, 3155 (2003).
[CrossRef]

Sprayregen, S.

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Stevenson, A. W.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

Takman, P. A. C.

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
[CrossRef]

Tian, L.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

Wilkins, S. W.

Wu, X.

Yan, A.

American Journal of Roentgenology

E. M. Lautin, N. J. Freeman, A. H. Schoenfeld, C. W. Bakal, N. Haramati, A. C. Friedman, J. L. Lautin, S. Braha, E. G. Kadish, and S. Sprayregen, “Radiocontrast-associated renal dysfunction: incidence and risk factors,” American Journal of Roentgenology 157, 49–58 (1991).
[CrossRef] [PubMed]

Appl. Opt.

Expert Opin. Drug Metab. Toxicol.

K. M. Hasebroock and N. J. Serkova, “Toxicity of MRI and CT contrast agents,” Expert Opin. Drug Metab. Toxicol. 5, 403–416 (2009).
[CrossRef] [PubMed]

IEEE Transactions on Image Processing

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

IEEE Transactions on Information Theory

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Transactions on Information Theory 52, 489–509 (2006).
[CrossRef]

IEEE Transactions on Medical Imaging

S. Matej, J. Fessler, and I. Kazantsev, “Iterative tomographic image reconstruction using fourier-based forward and back-projectors,” IEEE Transactions on Medical Imaging 23, 401–412 (2004).
[CrossRef] [PubMed]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat Med

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phasecontrast xray computed tomography for observing biological soft tissues,” Nat Med 2, 473–475 (1996).
[CrossRef] [PubMed]

Nat Phys

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance x-ray sources,” Nat Phys 2, 258–261 (2006).
[CrossRef]

Nature

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard x-rays,” Nature 373, 595–598 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Physica D: Nonlinear Phenomena

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D: Nonlinear Phenomena 60, 259–268 (1992).
[CrossRef]

SIAM J. Imaging Sci.

T. Goldstein and S. Osher, “The split bregman method for l1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[CrossRef]

Signal Process.

S. Singh and M. Singh, “Explosives detection systems (EDS) for aviation security,” Signal Process. 83, 3155 (2003).
[CrossRef]

Other

D. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Imaging geometry for TIE tomography.

Fig. 2
Fig. 2

A projection image of the sample (a), with its profile (b) showing significant white noise at the detector level. The liquids occupy the upper portions of the tubes, and air bubbles occupy the lower portions of the tubes.

Fig. 3
Fig. 3

Reconstruction results. (a) Filtered backprojection, (b) compressive PAD TIE, (c) plot along line in (a), and (d) plot along line in (b). These reconstructions show cross sections of the Eppendorf tubes containing the liquids taken along the plane coinciding with the line in Fig. 2(a)

Equations (7)

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

k I ( x , y , z ) z = [ I ( x , y , z ) ϕ ( x , y ) ] , = ( x , y ) ,
g ( x , y ) = I ( M x , M y ) I 0 = [ 1 + d λ γ ( λ ) 4 π 2 ] exp [ 2 ϕ ( x , y ) γ ( λ ) ] ,
ϕ ( x , y ; θ ) = δ ( x , y s , z s ; λ ) D ( y y s cos θ + z s sin θ ) d y s d z s ,
ϕ ( x , y ) = γ ( λ ) 2 ln 2 D 1 { 2 D { I ( x , y ) I 0 } 1 + π z λ γ ( λ ) | u 2 + v 2 | }
δ ( x , y s , z s ; λ ) = 1 D 1 { | w | 1 D { ϕ ( x , y , θ ) } } ,
g = PRn An
n ^ = argmin n n TV such that g = An

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