Abstract

This paper explores the use of single-shot digital holography data and a novel algorithm, referred to as multiplane iterative reconstruction (MIR), for imaging through distributed-volume aberrations. Such aberrations result in a linear, shift-varying or “anisoplanatic” physical process, where multiple-look angles give rise to different point spread functions within the field of view of the imaging system. The MIR algorithm jointly computes the maximum a posteriori estimates of the anisoplanatic phase errors and the speckle-free object reflectance from the single-shot digital holography data. Using both simulations and experiments, we show that the MIR algorithm outperforms the leading multiplane image-sharpening algorithm over a wide range of anisoplanatic conditions.

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References

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  1. T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: With MATLAB (Cambridge University, 2014).
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    [Crossref]
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    [Crossref]
  4. M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
    [Crossref]
  5. C. Pellizzari, M. F. Spencer, and C. A. Bouman, “Phase-error estimation and image reconstruction from digital-holography data using a Bayesian framework,” J. Opt. Soc. Am. A 34, 1659–1669 (2017).
    [Crossref]
  6. C. Pellizzari, M. T. Banet, M. F. Spencer, and C. A. Bouman, “Demonstration of single-shot digital holography using a Bayesian framework,” J. Opt. Soc. Am. A 35, 103–107 (2018).
    [Crossref]
  7. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).
  8. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  9. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  10. S. T. Thurman and J. R. Fienup, “Correction of anisoplanatic phase errors in digital holography,” J. Opt. Soc. Am. A 25, 995–999 (2008).
    [Crossref]
  11. A. E. Tippie and J. R. Fienup, “Phase-error correction for multiple planes using a sharpness metric,” Opt. Lett. 34, 701–703 (2009).
    [Crossref]
  12. A. E. Tippie and J. R. Fienup, “Multiple-plane anisoplanatic phase correction in a laboratory digital holography experiment,” Opt. Lett. 35, 3291–3293 (2010).
    [Crossref]
  13. C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
    [Crossref]
  14. V. V. Protopopov, Laser Heterodyning, Vol. 149 of Springer Series in Optical Sciences (Springer, 2009).
  15. C. A. Bouman, Model Based Image Processing, 2013, https://engineering.purdue.edu/~bouman/publications/pdf/MBIP-book.pdf .
  16. J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
    [Crossref]
  17. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).
  18. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).
  19. C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).
  20. A. E. Tippie, “Aberration correction in digital holography,” Ph.D. dissertation (University of Rochester, 2012).

2018 (1)

2017 (3)

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
[Crossref]

C. Pellizzari, M. F. Spencer, and C. A. Bouman, “Phase-error estimation and image reconstruction from digital-holography data using a Bayesian framework,” J. Opt. Soc. Am. A 34, 1659–1669 (2017).
[Crossref]

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

2010 (1)

2009 (2)

2008 (2)

2007 (1)

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Banet, M. T.

C. Pellizzari, M. T. Banet, M. F. Spencer, and C. A. Bouman, “Demonstration of single-shot digital holography using a Bayesian framework,” J. Opt. Soc. Am. A 35, 103–107 (2018).
[Crossref]

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
[Crossref]

Bouman, C.

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

Bouman, C. A.

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Evans, M.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Fienup, J. R.

Forbes, C.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

Grow, T. D.

Hastings, N.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Höft, T. A.

Hsieh, J.

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

Kendrick, R. L.

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Liu, J.-P.

T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: With MATLAB (Cambridge University, 2014).

Marker, D. K.

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
[Crossref]

Marron, J. C.

Nemati, B.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Peacock, B.

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

Pellizzari, C.

Pellizzari, C. J.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Poon, T.-C.

T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: With MATLAB (Cambridge University, 2014).

Protopopov, V. V.

V. V. Protopopov, Laser Heterodyning, Vol. 149 of Springer Series in Optical Sciences (Springer, 2009).

Raynor, R. A.

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
[Crossref]

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Sauer, K.

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Seldomridge, N.

Shao, M.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Spencer, M. F.

Thibault, J. B.

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

Thurman, S. T.

Tippie, A. E.

Trahan, R.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Williams, S.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Williams, S. E.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

Zhou, H.

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

IEEE Trans. Comput. Imaging (1)

C. J. Pellizzari, R. Trahan, H. Zhou, S. Williams, S. E. Williams, B. Nemati, M. Shao, and C. A. Bouman, “Synthetic aperature ladar: a model-based approach,” IEEE Trans. Comput. Imaging 3, 901–916 (2017).
[Crossref]

J. Opt. Soc. Am. A (4)

J. R. Stat. Soc. Ser. B (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Med. Phys. (1)

J. B. Thibault, K. Sauer, C. Bouman, and J. Hsieh, “A three-dimensional statistical approach to improved image quality for multi-slice helical CT,” Med. Phys. 34, 4526–4544 (2007).
[Crossref]

Opt. Eng. (1)

M. F. Spencer, R. A. Raynor, M. T. Banet, and D. K. Marker, “Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,” Opt. Eng. 56, 031213 (2017).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Other (9)

V. V. Protopopov, Laser Heterodyning, Vol. 149 of Springer Series in Optical Sciences (Springer, 2009).

C. A. Bouman, Model Based Image Processing, 2013, https://engineering.purdue.edu/~bouman/publications/pdf/MBIP-book.pdf .

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company, 2005).

C. Forbes, M. Evans, N. Hastings, and B. Peacock, Statistical Distributions (Wiley, 2011).

A. E. Tippie, “Aberration correction in digital holography,” Ph.D. dissertation (University of Rochester, 2012).

T.-C. Poon and J.-P. Liu, Introduction to Modern Digital Holography: With MATLAB (Cambridge University, 2014).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company, 2006).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

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Figures (15)

Fig. 1.
Fig. 1. Example DH system using an off-axis IPRG. A laser source is used to flood-illuminate an object that has a reflectance function, r , and corresponding reflection coefficient, g . In this example, the return signal is corrupted by atmospheric turbulence, which causes APEs, ϕ . We model ϕ as a layered atmosphere consisting of multiple phase screens, ϕ = [ ϕ 1 , ϕ 2 , , ϕ K ] , where K = 3 in this example. In our discrete model, each propagation plane, k , has an associated grid sample spacing, δ k , and the distance between grids is given by Δ z i . The blue and green dotted cones represent ray traces from two different points on the object. The ray traces show that light from different points on the object will encounter different phase errors, resulting in anisoplanatic conditions. The returning signal passes through the pupil-aperture function, a , is collimated, and is imaged onto a focal-plane detector array, where it is mixed with a reference field to form the digital hologram.
Fig. 2.
Fig. 2. Conventional image-processing steps for a DH system using an off-axis IPRG. Here, we show the magnitude squared of any complex-valued quantities. To extract the signal, we start with (a) a real-valued digital hologram, then take a 2D FFT to obtain the (b) complex-valued holographic spectrum. Note that the digital hologram, shown in (a), naturally has low contrast. Next, a section of the spectrum is isolated (c), as indicated by the dashed white line. This subset of the spectrum, y , represents a complex image of the signal field in the pupil plane. Finally, for basic image formation, we take an inverse FFT (IFFT) of y to form the image shown in (d).
Fig. 3.
Fig. 3. MIR algorithm for the MAP estimates of r and ϕ . Here, S is the set of all samples in r , K is the number of partial-propagation planes, S ¯ is the set of all samples in the low-resolution phase screens, and i is the iteration index.
Fig. 4.
Fig. 4. Example single-shot reconstructions (simulated) for D / r 0 , s w = 5 . Images are displayed on a log-based decibel (dB) scale obtained according to r db = 10 log { r / max ( r ) } , where max ( · ) indicates the maximum value in the 2D array.
Fig. 5.
Fig. 5. Example of single-shot reconstructions (simulated) for D / r 0 , s w = 10 .
Fig. 6.
Fig. 6. Example of single-shot reconstructions (simulated) for D / r 0 , s w = 15 .
Fig. 7.
Fig. 7. Average peak Strehl ratio, S p , as a function of θ ˜ 0 for D / r 0 , s w = 5 (top), D / r 0 , s w = 10 (middle), and D / r 0 , s w = 15 (bottom). The error bars show the standard deviation at each point.
Fig. 8.
Fig. 8. Average NRMSE as a function of θ ˜ 0 for D / r 0 , s w = 5 (top), D / r 0 , s w = 10 (middle), and D / r 0 , s w = 15 (bottom). The error bars show the standard deviation at each point.
Fig. 9.
Fig. 9. Peak Strehl ratio as a function of run time for both weak (blue) and strong (red) APEs.
Fig. 10.
Fig. 10. Example of the intermediate samples of r ^ taken at various times during the algorithm run time corresponding to Case 1 in Fig. 9. The top images are the MIS estimates at t = 0 , 341 , 706 , 2086 , 3134 , and 4501 s. The bottom images are the MIR estimates at t = 0 , 9.9 , 21.4 , 33.2 , 163 , and 1324 s (ordered from left to right first, then top to bottom).
Fig. 11.
Fig. 11. Experimental results for three, four, five, and six plastic phase screens. The original reconstruction with no phase-error compensation is shown in the left column, the MIS reconstructions are in the second column, and the MIR reconstructions are in the third column. Additionally, we show reconstructions from the isoplanatic MBIR algorithm.
Fig. 12.
Fig. 12. MIR algorithm for the MAP estimates of r and ϕ . Here, | · | ° 2 indicates the element-wise magnitude square of a vector, and s 2 ( · ) computes the sample variance of a vector’s elements [19]. Additionally, z = [ Δ z 1 , Δ z 2 , , Δ z K ] is a vector of the partial-propagation distances, δ = [ δ 0 , δ K ] is a vector of the object and pupil-plane sample spacing, γ is a unitless parameter introduced to tune the amount of regularization in r , S is the set of all samples in r , K is the number of partial-propagation planes, S ¯ is the set of all samples in the low-resolution phase screens, and i is the iteration index.
Fig. 13.
Fig. 13. Iterative process used to initialize ϕ .
Fig. 14.
Fig. 14. Diagram of our experimental setup. The output from a 1064 nm laser was split into two paths. One path was sent through a beam expander to illuminate the object. The other path created an ideal reference that interfered with the scattered signal using digital-holographic detection in the off-axis IPRG. Phase screens were placed in front of the imaging lens to simulate the effects of anisoplanatic aberrations.
Fig. 15.
Fig. 15. Images of the object (left) and the transmitter and receiver optics (right) used in our experiment.

Equations (51)

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( r ^ , ϕ ^ ) = argmin r , ϕ { log p ( r , ϕ | y ) } = argmin r , ϕ { log p ( y | r , ϕ ) log p ( r ) log p ( ϕ ) } ,
y = A ϕ g + w ,
p ( g | r ) C N ( 0 , D ( r ) , 0 ) ,
p ( w ) C N ( 0 , σ w 2 I , 0 ) ,
p ( y | r , ϕ ) C N ( 0 , A ϕ D ( r ) A ϕ H + σ w 2 I , 0 ) ,
p ( Δ ) = 1 z exp { { i , j } P b i , j ρ ( Δ ) } ,
ρ r ( Δ r σ r ) = | Δ r | p p σ r p ( | Δ r T σ r | q p 1 + | Δ r T σ r | q p ) ,
p ( ϕ ) = k = 1 K p ( ϕ k ) .
ρ ϕ ¯ k ( Δ ϕ ¯ k σ ϕ ¯ k ) = | Δ ϕ ¯ k | 2 2 σ ϕ ¯ k 2 ,
ϕ k = P ϕ ¯ k ,
c ( r , ϕ ¯ ) = log p ( y | r , ϕ ¯ ) log p ( r ) log p ( ϕ ¯ ) = log | A ϕ ¯ D ( r ) A ϕ ¯ H + σ w 2 I | + y H ( A ϕ ¯ D ( r ) A ϕ ¯ H + σ w 2 I ) 1 y + { i , j } P b i , j ρ r ( Δ r σ r ) + k = 1 K { i , j } P ¯ ρ ϕ ¯ k ( Δ ϕ ¯ k σ ϕ ¯ k ) .
Q ( r , ϕ ¯ ; r , ϕ ¯ ) = E g [ log p ( y , g | r , ϕ ¯ ) | y , r , ϕ ¯ ] log p ( r ) log p ( ϕ ¯ ) ,
( r ^ , ϕ ¯ ^ ) = argmin r , ϕ ¯ { Q ( r , ϕ ¯ ; r , ϕ ¯ ) } .
ε = r i r i 1 r i 1 .
NRMSE = α * r ^ r 2 r 2 ,
α * = argmin α { α r ^ r 2 }
S p = [ | A ϕ c H D ( a ) | 2 ] max [ | A 0 H D ( a ) | 2 ] max ,
H ˜ ( f x 0 , f y 0 ) = e j 2 π λ Δ z e j π λ Δ z ( f x 0 2 + f y 0 2 ) ,
u ˜ 1 ( x 1 , y 1 ) = CSFT 1 [ H ˜ ( f x 0 , f y 0 ) CSFT [ u ˜ 0 ( x 0 , y 0 ) ] ] ,
u 1 = F 1 H F u 0 ,
H = D ( vec [ e j π λ Δ z N 2 δ 0 2 ( p 2 + q 2 ) ] ) .
u 1 = Λ 1 F 1 H 0 , 1 F Λ 0 u 0 ,
Λ 0 = D ( vec [ e j π λ 1 β 1 Δ z δ 0 2 ( m 2 + n 2 ) ] ) , Λ 1 = D ( vec [ e j π λ β 1 1 β 1 Δ z δ 1 2 ( m 2 + n 2 ) ] ) ,
H 0 , 1 = D ( vec [ e j π λ Δ z N 2 δ 1 δ 0 ( p 2 + q 2 ) ] ) .
u K = A ϕ u 0 ,
A ϕ = D ( a ) Λ K [ k = 1 K D ( e j ϕ k ) F 1 H k 1 , k F ] Λ 0 , Λ 0 = D ( vec [ e j π λ 1 β 1 Δ z 1 δ 0 2 ( m 2 + n 2 ) ] ) , Λ K = D ( vec [ e j π λ β K 1 β K Δ z K δ K ( m 2 + n 2 ) ] ) ,
H k 1 , k = D ( vec [ e j π λ Δ z k N 2 δ k 1 δ k ( p 2 + q 2 ) ] ) .
u ˜ R ( x , y ) = R ˜ 0 e j 2 π ( f r x x + f r y y ) ,
i ˜ ( x , y ) = | u ˜ R ( x , y ) + u ˜ S ( x , y ) | 2 = | u ˜ R ( x , y ) | 2 + | u ˜ S ( x , y ) | 2 + u ˜ R ( x , y ) u ˜ S * ( x , y ) + u ˜ R * ( x , y ) u ˜ S ( x , y ) ,
i ˜ ( x , y ) = R ˜ 0 2 + | u ˜ S ( x , y ) | 2 + R ˜ 0 u ˜ S * ( x , y ) e j 2 π ( f r x x + f r y y ) + R ˜ 0 u ˜ S ( x , y ) e j 2 π ( f r x x + f r y y ) .
i ( m , n ) = i ˜ ( x , y ) | x = m δ x y = n δ y + w R ( m , n ) ,
i ( m , n ) = R 0 2 + | u S ( m , n ) | 2 + R 0 u S * ( m , n ) e j ( ω m m + ω n n ) + R 0 u S ( m , n ) e j ( ω m m + ω n n ) + w R ( m , n ) ,
I ( p , q ) = DFT [ i ( m , n ) ] = R 0 2 δ [ p , q ] + U S ( p , q ) U S ( p , q ) + R 0 U S * ( p + p r , q + q r ) + R 0 U S ( p p r , q q r ) + W h ( p , q ) .
y ( p , q ) = U S ( p , q ) + w ( p , q ) ,
y = u + w ,
Q ( r , ϕ ¯ ; r , ϕ ¯ ) = E g [ log p ( y , g | r , ϕ ¯ ) | y , r , ϕ ¯ ] log p ( r ) log p ( ϕ ¯ ) , = E g [ log p ( y | g , ϕ ¯ ) + log p ( g | r ) | y , r , ϕ ¯ ] log p ( r ) log p ( ϕ ¯ ) ,
Q ( r , ϕ ¯ ; r , ϕ ¯ ) = E g [ 1 σ w 2 y A ϕ ¯ g 2 | y , r , ϕ ¯ ] + log | D ( r ) | + i = 1 M 1 r i E g [ | g i | 2 | y , r , ϕ ¯ ] + { i , j } P b i , j ρ r ( Δ r σ r ) + k = 1 K [ { i , j } P b i , j ρ ϕ ¯ k ( Δ ϕ ¯ k σ ϕ ¯ k ) ] + κ ,
μ = C 1 σ w 2 A ϕ ¯ H y
C = [ 1 σ w 2 A ϕ ¯ H A ϕ ¯ + D ( r ) 1 ] 1 .
Q ( r , ϕ ¯ ; r , ϕ ¯ ) = 1 σ w 2 2 Re { y H A ϕ ¯ μ } + log | D ( r ) | + i = 1 N 1 r i ( C i , i + | μ i | 2 ) + { i , j } P b i , j ρ r ( Δ r σ r ) + k = 1 K [ { i , j } P b i , j ρ ϕ ¯ k ( Δ ϕ ¯ k σ ϕ ¯ k ) ] ,
C D ( σ w 2 1 + σ w 2 r ) ,
q s ( r s ; r , ϕ ¯ ) = log r s + C s , s + | μ s | 2 r s + j s b ˜ s , j ( r s r j ) 2 ,
b ˜ s , j = b s , j | r s r j | p 2 2 σ r p | r s r j T σ r | q p ( q p + | r s r j T σ r | q p ) ( 1 + | r s r j T σ r | q p ) 2 .
α 1 = 2 j s b ˜ s , j , α 2 = 2 j s b ˜ s , j r j , α 3 = 1 , α 4 = ( C s , s + | μ s | 2 ) .
A ϕ ¯ = Ψ k D ( e j P ϕ ¯ k ) Σ k ,
q ( ϕ ¯ k ; r , ϕ ¯ ) = 2 σ w 2 Re { ( Ψ k H y ) D ( e j P ϕ ¯ k ) ( Σ k μ ) } + 1 2 σ ϕ ¯ k 2 { i , j } P b i , j | Δ ϕ ¯ k | 2 .
q ( ϕ ¯ k , s ; r , ϕ ¯ ) = | χ | cos ( χ ϕ ¯ k , s ) + 1 2 σ ϕ ¯ k 2 j s b i , j | ϕ ¯ k , s ϕ ¯ k , j | 2 ,
χ = 2 σ w 2 i = 1 n b 2 [ Ψ k H y ] B ( i ) * [ Σ k μ ] B ( i ) .
ϕ ^ = [ Z Z Z ] c ^ ,
c ^ = argmax c { ( | A c H y | 2 ) β 1 α D ( W ) | F A c H y | 2 1 } .
SNR = s 2 ( A ϕ g ) s 2 ( w ) ,