Abstract

The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images by the use of a maximum-likelihood approach, the in-focus image and the defocus image, which contaminated by noise, will greatly reduce the solution accuracy of the phase diversity (PD) algorithm. In the study, we introduce the deep denoising convolutional neural networks (DnCNNs) into the image preprocessing of PD to denoise the in-focus image and defocus the image containing gaussian white noise to improve the robustness of PD to noise. The simulation results show that the composite PD algorithm with DnCNNs is better than the traditional PD algorithm in both RMSE of phase estimation and SSIM, and the mean of the RMSE of the phase estimation of the improved PD algorithm is reduced by 78.48%, 82.35%, 71.09% and 73.67% compared with the mean of the RMSE of the phase estimation of the traditional PD algorithm. The well-trained DnCNNs runs fast, which does not increase the running time of traditional PD algorithms, and the compound approach may be widely used in various domains, such as the measurements of intrinsic aberrations in optical systems and compensations for atmospheric turbulence.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (3)

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

L. Nasser and T. Boudier, “A novel generic dictionary-based denoising method for improving noisy and densely packed nuclei segmentation in 3D time-lapse fluorescence microscopy images,” Sci. Rep. 9(1), 5654 (2019).
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[Crossref] [PubMed]

2018 (2)

D. Li, S. Xu, X. Qi, D. Wang, and X. Cao, “Variable step size adaptive cuckoo search optimization algorithm for phase diversity,” Appl. Opt. 57(28), 8212–8219 (2018).
[Crossref] [PubMed]

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2017 (1)

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

2016 (2)

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

S. Huang, W. Huang, and T. Zhang, “A New SAR Image Segmentation Algorithm for the Detection of Target and Shadow Regions,” Sci. Rep. 6(1), 38596 (2016).
[Crossref] [PubMed]

2015 (1)

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

2014 (1)

M. J. Booth, “Adaptive optical microscopy: the ongoing quest for a perfect image,” Light: Sci. Appl. 3(4), e165 (2014).
[Crossref]

2007 (3)

M. Elad and M. Aharon, “Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2007).

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

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[Crossref] [PubMed]

2000 (1)

1997 (1)

1994 (1)

1982 (1)

R. A. Gonsalves, “Phase Retrieval And Diversity In Adaptive Optics,” Opt. Eng. 21, 215829 (1982).

Aharon, M.

M. Elad and M. Aharon, “Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2007).

Bliss, E. S.

Booth, M. J.

M. J. Booth, “Adaptive optical microscopy: the ongoing quest for a perfect image,” Light: Sci. Appl. 3(4), e165 (2014).
[Crossref]

Boudier, T.

L. Nasser and T. Boudier, “A novel generic dictionary-based denoising method for improving noisy and densely packed nuclei segmentation in 3D time-lapse fluorescence microscopy images,” Sci. Rep. 9(1), 5654 (2019).
[Crossref] [PubMed]

Cao, X.

Cao, Z.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

Chen, Y.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

Cobra, D.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Comunello, E.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Dailey, M. J.

Demirci, U.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Elad, M.

M. Elad and M. Aharon, “Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2007).

Ellerbroek, B. L.

Eraslan, G.

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Feldman, M.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Freeman, W. T.

Y. Weiss and W. T. Freeman, “What makes a good model of natural images?” in 2007 IEEE Conference on Computer Vision and Pattern Recognition (2007), pp. 1–8.
[Crossref]

Gamiz, V. L.

Goda, M. E.

Gonsalves, R. A.

R. A. Gonsalves, “Phase Retrieval And Diversity In Adaptive Optics,” Opt. Eng. 21, 215829 (1982).

Grey, A. A.

He, K.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification,” in 2015 IEEE International Conference on Computer Vision (ICCV) (IEEE, 2015), pp. 1026–1034.
[Crossref]

Holdener, F. R.

Huang, S.

S. Huang, W. Huang, and T. Zhang, “A New SAR Image Segmentation Algorithm for the Detection of Target and Shadow Regions,” Sci. Rep. 6(1), 38596 (2016).
[Crossref] [PubMed]

Huang, W.

S. Huang, W. Huang, and T. Zhang, “A New SAR Image Segmentation Algorithm for the Detection of Target and Shadow Regions,” Sci. Rep. 6(1), 38596 (2016).
[Crossref] [PubMed]

Inci, F.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Johnson, P. M.

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

Koch, J. A.

Lee, D. J.

Li, D.

Meng, D.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

Mircea, M.

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Mu, Q.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

Mueller, N. S.

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Nasser, L.

L. Nasser and T. Boudier, “A novel generic dictionary-based denoising method for improving noisy and densely packed nuclei segmentation in 3D time-lapse fluorescence microscopy images,” Sci. Rep. 9(1), 5654 (2019).
[Crossref] [PubMed]

Paxman, R.

Presta, R. W.

Qi, X.

Ren, S.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification,” in 2015 IEEE International Conference on Computer Vision (ICCV) (IEEE, 2015), pp. 1026–1034.
[Crossref]

Roggemann, M. C.

Sacks, R. A.

Salmon, J. T.

Seldin, J. H.

Seppala, L. G.

Simon, L. M.

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Sobieranski, A. C.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Sun, J.

K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification,” in 2015 IEEE International Conference on Computer Vision (ICCV) (IEEE, 2015), pp. 1026–1034.
[Crossref]

Tekin, H. C.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Theis, F. J.

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Thelen, B.

Toeppen, J. S.

Van Atta, L.

Van Wonterghem, B. M.

von Wangenheim, A.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Wang, D.

Weiss, Y.

Y. Weiss and W. T. Freeman, “What makes a good model of natural images?” in 2007 IEEE Conference on Computer Vision and Pattern Recognition (2007), pp. 1–8.
[Crossref]

Welsh, B. M.

Whistler, W. T.

Winters, S. E.

Woods, B. W.

Wu, D.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

Xu, H.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

Xu, S.

Xu, Z.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

Xuan, L.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

Yan, D.

Yang, C.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

H. Yu, C. Yang, X. Zihao, P.-G. Zhang, H. Xu, Z. Cao, Q. Mu, and l. Xuan, “Analysis and reduction of errors caused by Poisson noise for phase diversity technique,” Opt. Express 24, 22034-22042 (2016).

Yu, H.

Yuksekkaya, M.

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Zacharias, R. A.

Zhang, K.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

Zhang, L.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

Zhang, P.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

Zhang, P.-G.

Zhang, T.

S. Huang, W. Huang, and T. Zhang, “A New SAR Image Segmentation Algorithm for the Detection of Target and Shadow Regions,” Sci. Rep. 6(1), 38596 (2016).
[Crossref] [PubMed]

Zhang, X.

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

K. He, X. Zhang, S. Ren, and J. Sun, “Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification,” in 2015 IEEE International Conference on Computer Vision (ICCV) (IEEE, 2015), pp. 1026–1034.
[Crossref]

Zihao, X.

Zuo, W.

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

Appl. Opt. (2)

IEEE Trans. Image Process. (3)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image Denoising by Sparse 3-D Transform-Domain Collaborative Filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

M. Elad and M. Aharon, “Image Denoising Via Sparse and Redundant Representations Over Learned Dictionaries,” IEEE Trans. Image Process. 15, 3736–3745 (2007).

K. Zhang, W. Zuo, Y. Chen, D. Meng, and L. Zhang, “Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoising,” IEEE Trans. Image Process. 26(7), 3142–3155 (2017).
[Crossref] [PubMed]

J. Astro. Tele., Instr., Syst. (1)

D. Wu, C. Yang, P. Zhang, Z. Xu, H. Xu, X. Zhang, Z. Cao, Q. Mu, and L. Xuan, “Phase diversity technique with sparse regularization in liquid crystal adaptive optics system,” J. Astro. Tele., Instr., Syst. 4, 1–8 (2018).

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

Light Sci. Appl. (1)

A. C. Sobieranski, F. Inci, H. C. Tekin, M. Yuksekkaya, E. Comunello, D. Cobra, A. von Wangenheim, and U. Demirci, “Portable lensless wide-field microscopy imaging platform based on digital inline holography and multi-frame pixel super-resolution,” Light Sci. Appl. 4(10), e346 (2015).
[Crossref] [PubMed]

Light: Sci. Appl. (1)

M. J. Booth, “Adaptive optical microscopy: the ongoing quest for a perfect image,” Light: Sci. Appl. 3(4), e165 (2014).
[Crossref]

Nat. Commun. (1)

G. Eraslan, L. M. Simon, M. Mircea, N. S. Mueller, and F. J. Theis, “Single-cell RNA-seq denoising using a deep count autoencoder,” Nat. Commun. 10(1), 390 (2019).
[Crossref] [PubMed]

Opt. Eng. (1)

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Figures (3)

Fig. 1
Fig. 1 The structure of the used denoising deep CNN.
Fig. 2
Fig. 2 The training results of deep denoising convolutional neural networks.
Fig. 3
Fig. 3 The RMSEs between the true Zernike coefficients and calculated Zernike coefficients under different PSNR for four sets of aberration coefficients.

Tables (4)

Tables Icon

Table 1 The first row in the table is the ideal in-focus images of the optical system with four sets of wavefront aberration. The second row is the in-focus image contaminated by noise. The third row is the in-focus image which are the denoised image processed by deep CNN. In the second row, for each set of wavefront aberration, there are two kinds of noise images of peak signal to noise ratio, they are 20dB, 30dB.

Tables Icon

Table 2 The average SSIM values between the in-focus images contaminated by noise and the ideal in-focus images and the average SSIM values between the in-focus denoised images and the ideal in-focus images, within the PSNR range of 40-20 dB for four sets of aberration coefficients.

Tables Icon

Table 3 The first row in the table is the ideal defocus images of the optical system with four sets of wavefront aberration. The second row is the defocus images contaminated by noise. The third row is the defocus images which are the denoised image processed by deep CNN. In the second row, for each set of wavefront aberration, there are two kinds of noise images of peak signal to noise ratio, they are 20dB, 30dB.

Tables Icon

Table 4 The average SSIM values between the defocus images contaminated by noise and the ideal defocus images and the average SSIM values between the defocus denoised images and the ideal defocus images, within the PSNR range of 40-20 dB for four sets of aberration coefficients.

Equations (22)

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i(x,y)=o(x,y)PSF(x,y).
I(u,v)=O(u,v)OTF(u,v).
PSF(x,y)= | F T (P(u,v)) | 2 .
P(u,v)=A(u,v)exp(iϕ(u,v)).
ϕ(u,v)= i N a i c i (u,v).
i d (x,y)=o(x,y)PS F d (x,y).
I d (u,v)=O(u,v)OT F d (u,v).
PS F d (u,v)= | F T ( P d (u,v)) | 2 .
P d (u,v)=A(u,v)expi(ϕ(u,v)+ ϕ d (u,v)).
ϕ d (u,v)=b· c 4 (u,v).
E(o,a)= [ i(x,y)o(x,y)PSF(x,y) ] 2 + [ i d (x,y)o(x,y)PS F d (x,y) ] 2 .
E(O,a)= [ I(u,v)O(u,v)OTF(u,v) ] 2 + [ I d (u,v)O(u,v)OT F d (u,v) ] 2 .
δE(O,a) δO =0,
E(a)= uX,VY | I(u,v)OT F d (u,v) I d (u,v)OTF(u,v) | 2 | OTF(u,v) | 2 + | OT F d (u,v) | 2 .
i (x,y)=o(x,y)PSF(x,y)+n(x,y).
i d (x,y)=o(x,y)PS F d (x,y)+ n d (x,y).
E(a)= uX,VY | I (u,v)OT F d (u,v) I d (u,v)OTF(u,v) | 2 | OTF(u,v) | 2 + | OT F d (u,v) | 2 .
E(a)= uX,VY | S(u,v)OT F d (u,v) S d (u,v)OTF(u,v) | 2 | OTF(u,v) | 2 + | OT F d (u,v) | 2 .
l(Θ)= 1 2N i=1 N R( y i ;Θ)( y i x i ) F 2 .
RMSE= { i=4 n+41 ( c i t c i r ) 2 n } 1/2 .
PSNR= { MAX( i ( x,y ) 2 ) x=1 M y=1 N [ i( x,y ) i ( x,y ) ] 2 /M*N } 1/2 .
SSIM( f, f ^ )= (2 μ f μ f ^ + b 1 )(2 σ f, f ^ + b 2 ) ( μ f 2 + μ f ^ 2 + b 1 )( σ f 2 + σ f ^ 2 + b 2 ) .