Abstract

Coherent imaging has a wide range of applications in, for example, microscopy, astronomy, and radar imaging. Particularly interesting is the field of microscopy, where the optical quality of the lens is the main limiting factor. In this article, novel algorithms for the restoration of blurred images in a system with known optical aberrations are presented. Physically motivated by the scalar diffraction theory, the new algorithms are based on Haugazeau POCS and FISTA, and are faster and more robust than methods presented earlier. With the new approach the level of restoration quality on real images is very high, thereby blurring and ringing caused by defocus can be effectively removed. In classical microscopy, lenses with very low aberration must be used, which puts a practical limit on their size and numerical aperture. A coherent microscope using the novel restoration method overcomes this limitation. In contrast to incoherent microscopy, severe optical aberrations including defocus can be removed, hence the requirements on the quality of the optics are lower. This can be exploited for an essential price reduction of the optical system. It can be also used to achieve higher resolution than in classical microscopy, using lenses with high numerical aperture and high aberration. All this makes the coherent microscopy superior to the traditional incoherent in suited applications.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2015 (2)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Processing Magazine 32, 87–109 (2015).
[Crossref]

S. Bubeck, “Convex Optimization: Algorithms and Complexity,” Foundations and Trends in Machine Learning 8, 231–357 (2015).
[Crossref]

2014 (1)

M. Molaei and J. Sheng, “Imaging bacterial 3d motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Optics Express 22, 32119 (2014).
[Crossref]

2013 (2)

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

A. Kumar, W. Drexler, and R. A. Leitgeb, “Subaperture correlation based digital adaptive optics for full field optical coherence tomography,” Optics Express 21, 10850 (2013).
[Crossref] [PubMed]

2012 (1)

P. Getreuer, “Total Variation Deconvolution using Split Bregman,” Image Processing On Line 2, 158–174 (2012).
[Crossref]

2011 (1)

2009 (4)

W. Qu, C. O. Choo, V. R. Singh, Y. Yingjie, and A. Asundi, “Quasi-physical phase compensation in digital holographic microscopy,” J. Opt. Soc. Am. A 26, 2005–2011 (2009).
[Crossref]

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM Journal on Imaging Sciences 2, 183–202 (2009).
[Crossref]

J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Optics express 17, 11638–11651 (2009).
[Crossref] [PubMed]

A. Beck and M. Teboulle, “Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems,” IEEE Transactions on Image Processing 18, 2419–2434 (2009).
[Crossref] [PubMed]

2007 (2)

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Review of Scientific Instruments 78, 011301 (2007).
[Crossref]

S. Marchesini, “Invited Article: A unified evaluation of iterative projection algorithms for phase retrieval,” Review of Scientific Instruments 78, 011301 (2007).
[Crossref]

2006 (3)

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space,” Journal of Approximation Theory 141, 63–69 (2006).
[Crossref]

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Transactions on Graphics (TOG) 25, 787–794 (2006).
[Crossref]

2005 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Problems 21, 37–50 (2005).
[Crossref]

2002 (1)

1998 (1)

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” Image Processing, IEEE Transactions on 7, 370–375 (1998).
[Crossref]

1992 (1)

1982 (1)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics 21, 2758–2769 (1982).
[Crossref] [PubMed]

1976 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

Aspert, N.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

Asundi, A.

Balbekin, N. S.

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” in “Proc. SPIE 9271, Holography, Diffractive Optics, and Applications VI,”, 9271, 927128-927128-9 (2014).

Bauschke, H. H.

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space,” Journal of Approximation Theory 141, 63–69 (2006).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

Beck, A.

A. Beck and M. Teboulle, “Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems,” IEEE Transactions on Image Processing 18, 2419–2434 (2009).
[Crossref] [PubMed]

A. Beck and M. Teboulle, “A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems,” SIAM Journal on Imaging Sciences 2, 183–202 (2009).
[Crossref]

Bubeck, S.

S. Bubeck, “Convex Optimization: Algorithms and Complexity,” Foundations and Trends in Machine Learning 8, 231–357 (2015).
[Crossref]

Chan, T.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision17 (2005).

Chan, T. F.

T. F. Chan and C.-K. Wong, “Total variation blind deconvolution,” Image Processing, IEEE Transactions on 7, 370–375 (1998).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Processing Magazine 32, 87–109 (2015).
[Crossref]

Charriere, F.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

Choo, C. O.

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Processing Magazine 32, 87–109 (2015).
[Crossref]

Colomb, T.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

Combettes, P. L.

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space,” Journal of Approximation Theory 141, 63–69 (2006).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

Depeursinge, C.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in “Biomedical Imaging: From Nano to Macro, 2009. ISBI’09. IEEE International Symposium on,” (IEEE, 2009), 201–204.

Doblas, A.

Drexler, W.

A. Kumar, W. Drexler, and R. A. Leitgeb, “Subaperture correlation based digital adaptive optics for full field optical coherence tomography,” Optics Express 21, 10850 (2013).
[Crossref] [PubMed]

Durand, F.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Deconvolution using natural image priors,” Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory (2007).

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Processing Magazine 32, 87–109 (2015).
[Crossref]

Escalante, R.

R. Escalante and M. Raydan, Alternating projection methods, no. FA08 in Fundamentals of algorithms (Society for Industrial and Applied Mathematics, 2011).
[Crossref]

Esedoglu, S.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision17 (2005).

Fergus, R.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Transactions on Graphics (TOG) 25, 787–794 (2006).
[Crossref]

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Deconvolution using natural image priors,” Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory (2007).

Ferraro, P.

Fienup, J. R.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Applied Optics 21, 2758–2769 (1982).
[Crossref] [PubMed]

Freeman, W. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Transactions on Graphics (TOG) 25, 787–794 (2006).
[Crossref]

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Deconvolution using natural image priors,” Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory (2007).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237–246 (1972).

Getreuer, P.

P. Getreuer, “Total Variation Deconvolution using Split Bregman,” Image Processing On Line 2, 158–174 (2012).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

Gross, M.

M. Gross, F. Joud, F. Verpillat, M. Lesaffre, and N. Verrier, “Two-step distortion-free reconstruction scheme for holographic microscopy,” in “Digital Holography and Three-Dimensional Imaging,” (OSA, 2013), DW1A.7.
[Crossref]

Grow, T. D.

J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Optics express 17, 11638–11651 (2009).
[Crossref] [PubMed]

Haugazeau, Y.

Y. Haugazeau, Sur les inéquations variationnelles et la minimisation de fonctionnelles convexes, Thèse (Université de Paris, 1968).

Heide, F.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Heidrich, W.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Hertzmann, A.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Transactions on Graphics (TOG) 25, 787–794 (2006).
[Crossref]

Höft, T. A.

J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Optics express 17, 11638–11651 (2009).
[Crossref] [PubMed]

Hullin, M. B.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Joud, F.

M. Gross, F. Joud, F. Verpillat, M. Lesaffre, and N. Verrier, “Two-step distortion-free reconstruction scheme for holographic microscopy,” in “Digital Holography and Three-Dimensional Imaging,” (OSA, 2013), DW1A.7.
[Crossref]

Kendrick, R. L.

J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Optics express 17, 11638–11651 (2009).
[Crossref] [PubMed]

Kim, M. K.

M. K. Kim, “Diffraction and Fourier Optics,” in “Digital Holographic Microscopy,”, 162 (Springer2011),11–28.
[Crossref]

Koch, R.

C. Zelenka and R. Koch, “Restoration of images with wavefront aberrations,” in “Pattern Recognition (ICPR), 2016 23rd International Conference on,” (IEEE, 2016), 1388–1393.

Kolb, A.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Kotera, J.

J. Kotera, F. Šroubek, and P. Milanfar, “Blind deconvolution using alternating maximum a posteriori estimation with heavy-tailed priors,” in “Computer Analysis of Images and Patterns,” (Springer, 2013), 59–66.
[Crossref]

Kühn, J.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

Kumar, A.

A. Kumar, W. Drexler, and R. A. Leitgeb, “Subaperture correlation based digital adaptive optics for full field optical coherence tomography,” Optics Express 21, 10850 (2013).
[Crossref] [PubMed]

Labitzke, B.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Lane, R. G.

Leitgeb, R. A.

A. Kumar, W. Drexler, and R. A. Leitgeb, “Subaperture correlation based digital adaptive optics for full field optical coherence tomography,” Optics Express 21, 10850 (2013).
[Crossref] [PubMed]

Lesaffre, M.

M. Gross, F. Joud, F. Verpillat, M. Lesaffre, and N. Verrier, “Two-step distortion-free reconstruction scheme for holographic microscopy,” in “Digital Holography and Three-Dimensional Imaging,” (OSA, 2013), DW1A.7.
[Crossref]

Levin, A.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Deconvolution using natural image priors,” Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory (2007).

Luke, D. R.

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space,” Journal of Approximation Theory 141, 63–69 (2006).
[Crossref]

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Problems 21, 37–50 (2005).
[Crossref]

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334–1345 (2002).
[Crossref]

Marchesini, S.

S. Marchesini, “Invited Article: A unified evaluation of iterative projection algorithms for phase retrieval,” Review of Scientific Instruments 78, 011301 (2007).
[Crossref]

S. Marchesini, “A unified evaluation of iterative projection algorithms for phase retrieval,” Review of Scientific Instruments 78, 011301 (2007).
[Crossref]

Marquet, P.

T. Colomb, J. Kühn, F. Charriere, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Optics express 14, 4300–4306 (2006).
[Crossref] [PubMed]

Marron, J. C.

J. C. Marron, R. L. Kendrick, N. Seldomridge, T. D. Grow, and T. A. Höft, “Atmospheric turbulence correction using digital holographic detection: experimental results,” Optics express 17, 11638–11651 (2009).
[Crossref] [PubMed]

Martínez-Corral, M.

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase Retrieval with Application to Optical Imaging: A contemporary overview,” IEEE Signal Processing Magazine 32, 87–109 (2015).
[Crossref]

Milanfar, P.

J. Kotera, F. Šroubek, and P. Milanfar, “Blind deconvolution using alternating maximum a posteriori estimation with heavy-tailed priors,” in “Computer Analysis of Images and Patterns,” (Springer, 2013), 59–66.
[Crossref]

Molaei, M.

M. Molaei and J. Sheng, “Imaging bacterial 3d motion using digital in-line holographic microscopy and correlation-based de-noising algorithm,” Optics Express 22, 32119 (2014).
[Crossref]

Noll, R. J.

Park, F.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision17 (2005).

Pavillon, N.

C. S. Seelamantula, N. Pavillon, C. Depeursinge, and M. Unser, “Zero-order-free image reconstruction in digital holographic microscopy,” in “Biomedical Imaging: From Nano to Macro, 2009. ISBI’09. IEEE International Symposium on,” (IEEE, 2009), 201–204.

Petrov, N. V.

I. A. Shevkunov, N. S. Balbekin, and N. V. Petrov, “Comparison of digital holography and iterative phase retrieval methods for wavefront reconstruction,” in “Proc. SPIE 9271, Holography, Diffractive Optics, and Applications VI,”, 9271, 927128-927128-9 (2014).

Qu, W.

Raydan, M.

R. Escalante and M. Raydan, Alternating projection methods, no. FA08 in Fundamentals of algorithms (Society for Industrial and Applied Mathematics, 2011).
[Crossref]

Rouf, M.

F. Heide, M. Rouf, M. B. Hullin, B. Labitzke, W. Heidrich, and A. Kolb, “High-quality computational imaging through simple lenses,” ACM Transactions on Graphics (TOG) 32, 149 (2013).
[Crossref]

Roweis, S. T.

R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake from a single photograph,” ACM Transactions on Graphics (TOG) 25, 787–794 (2006).
[Crossref]

Saavedra, G.

Sánchez-Ortiga, E.

Saxton, W. O.

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C. Zelenka and R. Koch, “Restoration of images with wavefront aberrations,” in “Pattern Recognition (ICPR), 2016 23rd International Conference on,” (IEEE, 2016), 1388–1393.

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Figures (13)

Fig. 1
Fig. 1

Optical system for phase retrieval.

Fig. 2
Fig. 2

Wavefront correction in an out of focus optical system. Compared to Fig. 1, the image plane is out of focus, which causes an additional wavefront deformation.

Fig. 3
Fig. 3

Overview of the WFC-GS algorithm.

Fig. 4
Fig. 4

Non-convexity of the image plane constraint. The circle show the set of points with identical amplitude and the arrows highlight the phase of two examples. The red connecting line is their linear combination, which is clearly not within the points of identical amplitude.

Fig. 5
Fig. 5

Comparison of restoration quality between method by [14] [13] and novel wavefront correction algorithm WFC-HAAR.

Fig. 6
Fig. 6

Restoration quality comparison for a given number of iterations by PSNR for different algorithms. Blue WFC-GS and red WFC-HIO plot are algorithms from [2], all others are novel algorithms.

Fig. 7
Fig. 7

Comparison of restoration quality between algorithms from [2] and novel algorithms at iteration 7.

Fig. 8
Fig. 8

Comparison of algorithms in restoration of noisy images. Blue WFC-GS and red WFC-HIO plot are algorithms from [2], all others are novel algorithms.

Fig. 9
Fig. 9

Microscopic experimental setting.

Fig. 10
Fig. 10

Comparison of [2] and novel algorithms on a real microscopic image.

Fig. 11
Fig. 11

Image restoration of a image of a human kidney slice. The prominent dark streaks are the blood vessels inside the kidney. The restoration makes them visible. Result 11(c) was created with method from [2], all other results with novel algorithms.

Fig. 12
Fig. 12

Restoration of a pigeon blood sample. The dark ellipses are individual blood cells.

Fig. 13
Fig. 13

Restoration of a human neuron sample. The dark structure is a neuron, dark dots are cell cores. Only slight defocus, imperfect restoration due to noisy background and 3D structure of the neuron.

Tables (1)

Tables Icon

Table 1 Wavefront Correction methods used in this article

Equations (45)

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

M = I B + N ,
A m = U B a + N a ,
I o = A m 2 .
B = B a 2 .
x = a r g m i n x ( F ( x ) ) ,
p L ( y ) = p r o x 1 / L ( g ) ( y 1 L f ( y ) )
= a r g m i n x ( L 2 x ( y 1 L f ( y ) ) 2 + g ( x ) ) ,
x k = p r o x 1 / L ( g ) ( y 1 L f ( y ) )
t k + 1 = 1 + 1 + 4 t k 2 2
y k + 1 = x k + t k 1 t k + 1 ( x k x k 1 ) .
ϕ d ( x , y ) = k 2 r ( x 2 + y 2 )
p d = exp ( j k 2 r ( x 2 + y 2 ) ) ,
ϕ p ( x , y ) = π λ z ( x 2 + y 2 ) ,
A i ( k ) = ( o i ) p s ( k )
f i ( n ) = 1 ( A i ) ( n )
f i ( n ) = f i ( n )
A i ( k ) = ( f i ) ( k )
o i ( n ) = 1 ( A i p s 1 ) ( n )
o i + 1 ( n ) = o 0 ( n ) o i ( n ) o i ( n ) .
o i + 1 ( n ) = P o P f ( o i ( n ) ) .
o i + 1 ( n ) = ( P o ( ( 1 + β ) P f 1 ) + I P f ) ( o i ( n ) ) .
o i + 1 ( n ) = 1 2 ( P o ( o i ( n ) ) + P f ( o i ( n ) ) ) ,
o i + 1 ( n ) = ( 1 2 β ( R o R f + I ) + ( 1 + β ) P f ) o i ( n ) ,
o i + 1 ( n ) = Q ( o 0 ( n ) , o i ( n ) , ( 1 μ n ) o i ( n ) + μ n T o i ( n ) ) ,
A p = F o .
W = F 1 ( F p d ) .
W f ( x ) = ( W x ) 2 .
arg min x ( W f ( x ) o 0 ) 2 2 + λ x 1 subject to : x + m × n
f ( x ) = ( W f ( x ) o 0 ) 2 2
g ( x ) = x 1 .
f ( x ) = 2 ( W T ( W x ) b ) ,
f ( x ) = 2 R e ( W T ( R e ( W ( x ) ) b ) ) .
a t + b ( 1 + t ) S , t [ 0..1 ] ,
p = a r g m i n s S s t = P S ( t ) .
y n + 1 = P A P B ( y n ) .
R = 2 P I
w X | ( x y ) ( x y ) 0 w X | ( w z ) ( y z ) 0 .
π = ( x y ) ( y z )
μ = x y 2
ν = y z 2
ρ = μ ν π 2 .
Q ( x , y , z ) = { z , p = 0 and π 0 x + ( 1 + π ν ) ( z y ) , ρ > 0 and π ν ρ y + ( ν ρ ( π ( x y ) + μ ( z y ) ) , ρ > 0 and π ν < ρ .
y n + 1 = Q ( x , Q ( x , y n , P B y n ) , P A Q ( x , y n , P B y n ) )
T = 1 2 R o R A + 1 2 ,
y n + 1 = Q ( x , y n , ( 1 μ n ) y n + μ n T y n ) ,

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