Abstract

We propose an alternative method for solving the Transport of Intensity equation (TIE) from a stack of through–focus intensity images taken by a microscope or lensless imager. Our method enables quantitative phase and amplitude imaging with improved accuracy and reduced data capture, while also being computationally efficient and robust to noise. We use prior knowledge of how intensity varies with propagation in the spatial frequency domain in order to constrain a fitting algorithm [Gaussian process (GP) regression] for estimating the axial intensity derivative. Solving the problem in the frequency domain inspires an efficient measurement scheme which captures images at exponentially spaced focal steps, significantly reducing the number of images required. Low–frequency artifacts that plague traditional TIE methods can be suppressed without an excessive number of captured images. We validate our technique experimentally by recovering the phase of human cheek cells in a brightfield microscope.

© 2014 Optical Society of America

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References

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  1. G. Popescu, Quantitative phase imaging of cells and tissues (McGraw-HillNew York, 2011).
  2. K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
    [CrossRef]
  3. M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  4. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  5. L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
    [CrossRef]
  6. M. Soto, E. Acosta, “Improved phase imaging from intensity measurements in multiple planes,” Appl. Opt. 46, 7978–7981 (2007).
    [PubMed]
  7. L. Waller, L. Tian, G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18, 12552–12561 (2010).
    [PubMed]
  8. L. Waller, M. Tsang, S. Ponda, S. Yang, G. Barbastathis, “Phase and amplitude imaging from noisy images by Kalman filtering,” Opt. Express 19, 2805–2814 (2011).
    [PubMed]
  9. Z. Jingshan, J. Dauwels, M. A. Vázquez, L. Waller, “Sparse ACEKF for phase reconstruction,” Opt. Express 21, 18125–18137 (2013).
    [PubMed]
  10. L. Waller, S. S. Kou, C. J. R. Sheppard, G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 18, 22817–22825 (2010).
    [PubMed]
  11. L. Waller, Y. Luo, S. Y. Yang, G. Barbastathis, “Transport of intensity phase imaging in a volume holographic microscope,” Opt. Lett. 35, 2961–2963 (2010).
    [CrossRef] [PubMed]
  12. P. M. Blanchard, A. H. Greenaway, “Simultaneous multiplane imaging with a distorted diffraction grating,” Appl. Opt. 38, 6692–6699 (1999).
    [CrossRef]
  13. S. S. Gorthi, E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett. 37, 707–709 (2012).
    [PubMed]
  14. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [PubMed]
  15. J. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [PubMed]
  16. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
  17. T. Gureyev, K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).
  18. T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).
  19. M. Soto, E. Acosta, S. Ríos, “Performance analysis of curvature sensors: optimum positioning of the measurement planes,” Opt. Express 11, 2577–2588 (2003).
    [PubMed]
  20. C. Zuo, Q. Chen, Y. Yu, A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
    [PubMed]
  21. R. W. Schafer, “What is a Savitzky-Golay filter?[lecture notes],” Signal Processing Magazine, IEEE 28, 111–117 (2011).
    [CrossRef]
  22. S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
    [CrossRef]
  23. J. Guigay, M. Langer, R. Boistel, 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]
  24. D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
    [CrossRef]
  25. C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning (the MIT Press, 2006)..
  26. B. Xue, S. Zheng, L. Cui, X. Bai, F. Zhou, “Transport of intensity phase imaging from multiple intensities measured in unequally-spaced planes,” Opt. Express 19, 20244–20250 (2011).
    [CrossRef] [PubMed]
  27. K. Falaggis, T. Kozacki, M. Kujawinska, “Optimum plane selection criteria for single beam phase retrieval techniques based on the contrast transfer function,” Opt. Lett. 39, 30–33 (2014).
    [CrossRef]
  28. N. Loomis, L. Waller, G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in “Biomedical Optics and 3-D Imaging,” (Optical Society of America, 2010), p. JMA7.
  29. D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16, 1005–1015 (1999).
    [CrossRef]
  30. L. Tian, J. C. Petruccelli, G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. 37, 4131–4133 (2012).
    [CrossRef] [PubMed]
  31. A. Kostenko, K. J. Batenburg, A. King, S. E. Offerman, L. J. van Vliet, “Total variation minimization approach in in-line X-ray phase-contrast tomography,” Opt. Express 21, 12185–12196 (2013).
    [CrossRef] [PubMed]
  32. L. Tian, J. C. Petruccelli, Q. Miao, H. Kudrolli, V. Nagarkar, G. Barbastathis, “Compressive X-ray phase tomography based on the transport of intensity equation,” Opt. Lett. 38, 3418–3421 (2013).
    [CrossRef] [PubMed]
  33. P. Sollich, C. K. I. Williams, “Using the equivalent kernel to understand Gaussian process regression,” in Advances in Neural Information Processing Systems 17,” (the MIT Press, 2005), pp. 1313–1320.

2014 (1)

2013 (4)

2012 (2)

2011 (3)

2010 (3)

2007 (2)

2005 (1)

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

2004 (2)

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

2003 (1)

2001 (2)

K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

1999 (2)

1998 (1)

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

1997 (1)

T. Gureyev, K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (1)

1982 (1)

1978 (1)

Acosta, E.

Allen, L. J.

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Asundi, A.

Bai, X.

Barbastathis, G.

Baruchel, J.

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Batenburg, K. J.

Blanchard, P. M.

Boistel, R.

Chen, Q.

Cloetens, P.

J. Guigay, M. Langer, R. Boistel, 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]

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Cui, L.

Dauwels, J.

Falaggis, K.

Fienup, J.

Fienup, J. R.

Gorthi, S. S.

Greenaway, A. H.

Guigay, J.

J. Guigay, M. Langer, R. Boistel, 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]

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Gureyev, T.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
[CrossRef]

T. Gureyev, K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

Gureyev, T. E.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

Jingshan, Z.

King, A.

Kitchen, M.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

Kostenko, A.

Kou, S. S.

Kozacki, T.

Kudrolli, H.

Kujawinska, M.

Langer, M.

Lee, D. J.

Lewis, R. A.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

Loomis, N.

N. Loomis, L. Waller, G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in “Biomedical Optics and 3-D Imaging,” (Optical Society of America, 2010), p. JMA7.

Luo, Y.

Miao, Q.

Nagarkar, V.

Nugent, K.

K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
[CrossRef]

T. Gureyev, K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

Nugent, K. A.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Offerman, S. E.

Oxley, M. P.

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

Paganin, D.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
[CrossRef]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Pavlov, K. M.

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

Petruccelli, J. C.

Pogany, A.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

Ponda, S.

Popescu, G.

G. Popescu, Quantitative phase imaging of cells and tissues (McGraw-HillNew York, 2011).

Rasmussen, C. E.

C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning (the MIT Press, 2006)..

Ríos, S.

Roggemann, M. C.

Schafer, R. W.

R. W. Schafer, “What is a Savitzky-Golay filter?[lecture notes],” Signal Processing Magazine, IEEE 28, 111–117 (2011).
[CrossRef]

Schlenker, M.

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Schonbrun, E.

Sheppard, C. J. R.

Sollich, P.

P. Sollich, C. K. I. Williams, “Using the equivalent kernel to understand Gaussian process regression,” in Advances in Neural Information Processing Systems 17,” (the MIT Press, 2005), pp. 1313–1320.

Soto, M.

Streibl, N.

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M. R.

Tian, L.

Tsang, M.

van Vliet, L. J.

Vázquez, M. A.

Waller, L.

Welsh, B. M.

Wilkins, S.

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

Williams, C. K. I.

C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning (the MIT Press, 2006)..

P. Sollich, C. K. I. Williams, “Using the equivalent kernel to understand Gaussian process regression,” in Advances in Neural Information Processing Systems 17,” (the MIT Press, 2005), pp. 1313–1320.

Xue, B.

Yang, S.

Yang, S. Y.

Yu, Y.

Zabler, S.

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Zheng, S.

Zhou, F.

Zuo, C.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (5)

D. Paganin, T. E. Gureyev, K. M. Pavlov, R. A. Lewis, M. Kitchen, “Phase retrieval using coherent imaging systems with linear transfer functions,” Opt. Commun. 234, 87–105 (2004).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

L. J. Allen, M. P. Oxley, “Phase retrieval from series of images obtained by defocus variation,” Opt. Commun. 199, 65–75 (2001).
[CrossRef]

T. Gureyev, K. Nugent, “Rapid quantitative phase imaging using the transport of intensity equation,” Opt. Commun. 133, 339–346 (1997).

T. Gureyev, A. Pogany, D. Paganin, S. Wilkins, “Linear algorithms for phase retrieval in the Fresnel region,” Opt. Commun. 231, 53–70 (2004).

Opt. Express (8)

Opt. Lett. (7)

Phys. Rev. Lett. (1)

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Physics Today (1)

K. Nugent, D. Paganin, T. Gureyev, “A phase odyssey,” Physics Today 54, 27–32 (2001).
[CrossRef]

Review of Scientific Instruments (1)

S. Zabler, P. Cloetens, J. Guigay, J. Baruchel, M. Schlenker, “Optimization of phase contrast imaging using hard x rays,” Review of Scientific Instruments 76, 073705 (2005).
[CrossRef]

Signal Processing Magazine, IEEE (1)

R. W. Schafer, “What is a Savitzky-Golay filter?[lecture notes],” Signal Processing Magazine, IEEE 28, 111–117 (2011).
[CrossRef]

Other (4)

G. Popescu, Quantitative phase imaging of cells and tissues (McGraw-HillNew York, 2011).

C. E. Rasmussen, C. K. I. Williams, Gaussian Processes for Machine Learning (the MIT Press, 2006)..

N. Loomis, L. Waller, G. Barbastathis, “High-speed phase recovery using chromatic transport of intensity computation in graphics processing units,” in “Biomedical Optics and 3-D Imaging,” (Optical Society of America, 2010), p. JMA7.

P. Sollich, C. K. I. Williams, “Using the equivalent kernel to understand Gaussian process regression,” in Advances in Neural Information Processing Systems 17,” (the MIT Press, 2005), pp. 1313–1320.

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Figures (9)

Fig. 1
Fig. 1

Generalized experimental setup for an imaging system (e.g. a microscope) which captures intensity images at a range of axial defocus distances in order to recover phase.

Fig. 2
Fig. 2

Intensity evolution in frequency space is more predictable than in real space, demonstrating approximately sinusoidal behavior. (Top) Intensity images through–focus and their corresponding Fourier spectra’s real part, from simulated data with dz = 1μm, size 200μm × 200μm using the object in Fig. 6. (Bottom left) Intensity variations over z for 4 sample pixels (x, y) denoted as dots above. (Bottom right) Real part of intensity spectrum over z for 4 spatial frequency (u, v) values denoted as dots above.

Fig. 3
Fig. 3

Rationale for exponential spacing measurement scheme. Plot shows phase transfer functions for exponentially spaced z steps, which ensure a minimum sensitivity of α across a range of frequencies. g(u, v, z1), g(u, v, z2), and g(u, v, z3) are the phase transfer functions at z1, z2, and z3, respectively. The minimum sensitivity plot shows the frequencies which are transferred at the sensitivity higher than α by choosing z1, z2, and z3. Larger z brings more low–frequency sensitivity.

Fig. 4
Fig. 4

Simulated data sets for equal and exponential spacing of z steps. (a) Equally Spaced Stack: images equally spaced by 5μm. (b) Exponentially Spaced Stack: images exponentially spaced, with z of ±5μm, ±20μm, ±80μm, and ±320μm. The exponential spacing data contains more low–frequency phase information due to the larger range of z steps.

Fig. 5
Fig. 5

Comparison of mean square error (MSE) in phase results for various methods as noise level increases. GP TIE with exponential spacing yields the best error performance.

Fig. 6
Fig. 6

Phase images recovered from simulated data with SNR of 11.1dB. (Top) Results of the Equally Spaced Stack, using higher order TIE (m=5, MSE 0.1194 in radian), SGDF TIE (0.0295), and the proposed GP TIE (0.0279). (Bottom) Results for Exponentially Spaced Stack, using Higher order TIE (m=5, MSE 0.0237) and GP TIE (0.0065). GP TIE using exponential spacing provides the best phase result.

Fig. 7
Fig. 7

(Left) GP regression of the intensity spectrum’s real part over z for three sample frequency points (u, v) (Data Set 1). (Right) The frequency (u, v) of the three components depicted on real part of the recovered spectrum of phase (the image size is 945 × 888 but only the central part of the spectrum is shown for clarity). According to Eq. (3), the values for πλ(u2 + v2) are 0.029 × 104m−1, 4.145 × 104m−1, and 17.299 × 104m−1, respectively.

Fig. 8
Fig. 8

Exponentially spaced defocus steps with GP TIE provide accurate phase results using less images. (a) Data Set 1: equally spaced dz = 4μm, from 256μm to 256μm. Each image has 945 × 888 pixels with effective size 0.31μm × 0.31μm. (b) Phase recovered with equally spaced z steps: (left) with all 129 images [256μm to 256μm, dz = 4μm]; (middle) subset of 15 images using minimum z step size [28μm to 28μm, dz = 4μm]; and (right) subset of 15 images using maximum z range [252μm to 252μm, dz = 36μm]. With equal spacing, there is a forced trade–off between low–frequency noise and high–frequency blurring, such that many images are required for good quality phase results. (c) Phase recovered with exponentially spaced z steps: (left) subset of 15 images (β = 2), and (right) subset of 9 images (β = 4). The minimum and maximum defocus distances are fixed at ±4μm and ±256μm, respectively.

Fig. 9
Fig. 9

(a) Data Set 2: exponentially spaced, with z of ±5.7μm, ±11.4μm, ±22.8μm, and ±45.6μm. Each image has 350 × 360 pixels of size 0.62μm × 0.62μm. (b) Phase images of Data Set 2 by Higher order TIE (m = 2, 3, and 4) and GP TIE.

Tables (1)

Tables Icon

Table 1 Algorithm of GP TIE.

Equations (18)

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I ( x , y , z ) z | z = 0 = λ 2 π [ ( I ( x , y , 0 ) ϕ ( x , y ) ] ,
Φ ( u , v ) = F ( u , v ) / [ 4 π 2 ( u 2 + v 2 ) ] ,
( u , v , z ) = δ ( u , v ) 2 U ( u , v ) cos [ π λ ( u 2 + v 2 ) z ] 2 Φ ( u , v ) sin [ π λ ( u 2 + v 2 ) z ] ,
g ( u , v , z ) = sin [ π λ ( u 2 + v 2 ) z ] .
z n + 1 = β z n .
( f 1 , f 2 , , f N | z 1 , z 2 , , z N ) ~ 𝒩 ( 0 , K ( Z , Z ) + σ n I ) ,
K i j = σ f 2 exp [ 1 2 2 ( z i z j ) 2 ] ,
[ f f ( z ) ] ~ 𝒩 ( 0 , [ K ( Z , Z ) + σ n I K ( Z , z ) K ( z , Z ) K ( z , z ) ] ) ,
( f ( z ) | f 1 , f 2 , f N , z 1 , z 2 , , z N , z ) ~ 𝒩 ( f ¯ ( z ) , K ¯ ) ,
f ¯ ( z ) = K ( z , Z ) ( K ( Z , Z ) + σ n I ) 1 f ,
K ¯ = K ( z , z ) K ( z , Z ) ( K ( Z , Z ) + σ n I ) 1 K ( Z , z ) .
h ( z ) T = K ( z , Z ) ( K ( Z , Z ) + σ n I ) 1 .
f ¯ ( z ) = h ( z ) T f .
h ˜ S E ( s ) = 1 1 + b exp ( 2 π 2 2 | s | 2 ) ,
s c 2 = log ( 1 / b ) / ( 2 π 2 2 ) .
¯ ( u m , v n , z ) = h ( u m , v n , z ) T I m n ,
( u m , v n , z ) z | z = 0 = h ( u m , v n , z ) T z | z = 0 I m n ,
h ( u m , v n , z ) T z = [ σ f 2 2 ( z z 1 ) exp [ 1 2 2 ( z z 1 ) 2 ] , , σ f 2 2 ( z z N ) exp [ 1 2 2 ( z z N ) 2 ] ] ( K ( Z , Z ) + σ n I ) 1 .

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