Abstract

Fourier Ptychographic Microscopy (FPM) is a newly proposed computational imaging method aimed at reconstructing a high-resolution wide-field image from a sequence of low-resolution images. These low-resolution images are captured under varied illumination angles and the FPM recovery routine then stitches them together in the Fourier domain iteratively. Although FPM has achieved success with static sample reconstructions, the long acquisition time inhibits real-time application. To address this problem, we propose here a self-learning based FPM which accelerates the acquisition and reconstruction procedure. We first capture a single image under normally incident illumination, and then use it to simulate the corresponding low-resolution images under other illumination angles. The simulation is based on the relationship between the illumination angles and the shift of the sample’s spectrum. We analyze the importance of the simulated low-resolution images in order to devise a selection scheme which only collects the ones with higher importance. The measurements are then captured with the selection scheme and employed to perform the FPM reconstruction. Since only measurements of high importance are captured, the time requirements of data collection as well as image reconstruction can be greatly reduced. We validate the effectiveness of the proposed method with simulation and experimental results showing that the reduction ratio of data size requirements can reach over 70%, without sacrificing image reconstruction quality.

© 2015 Optical Society of America

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References

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2015 (1)

2014 (10)

G. Zheng, X. Ou, and C. Yang, “0.5 gigapixel microscopy using a flatbed scanner,” Biomed. Opt. Express 5(1), 1–8 (2014).
[Crossref] [PubMed]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22(5), 4960–4972 (2014).
[Crossref] [PubMed]

S. Dong, Z. Bian, R. Shiradkar, and G. Zheng, “Sparsely sampled Fourier ptychography,” Opt. Express 22(5), 5455–5464 (2014).
[Crossref] [PubMed]

S. Dong, R. Shiradkar, P. Nanda, and G. Zheng, “Spectral multiplexing and coherent-state decomposition in Fourier ptychographic imaging,” Biomed. Opt. Express 5(6), 1757–1767 (2014).
[Crossref] [PubMed]

L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for Fourier ptychography with an LED array microscope,” Biomed. Opt. Express 5(7), 2376–2389 (2014).
[Crossref] [PubMed]

S. Dong, P. Nanda, R. Shiradkar, K. Guo, and G. Zheng, “High-resolution fluorescence imaging via pattern-illuminated Fourier ptychography,” Opt. Express 22(17), 20856–20870 (2014).
[Crossref] [PubMed]

S. Dong, K. Guo, P. Nanda, R. Shiradkar, and G. Zheng, “FPscope: a field-portable high-resolution microscope using a cellphone lens,” Biomed. Opt. Express 5(10), 3305–3310 (2014).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Situ, G. Zheng, F. Chen, and Q. Dai, “Content adaptive sparse illumination for Fourier ptychography,” Opt. Lett. 39(23), 6648–6651 (2014).
[Crossref] [PubMed]

R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with Fourier ptychography,” Comput. Med. Imag. Graph. 42, 38–43 (2014).
[Crossref]

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

2013 (5)

2011 (1)

2009 (2)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microsc. 235(2), 144–162 (2009).
[Crossref] [PubMed]

2008 (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

2007 (3)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

V. Mico, Z. Zalevsky, and J. Garca, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).
[Crossref]

2004 (3)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

2003 (1)

1996 (1)

1995 (1)

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

1992 (1)

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. Royal Soc. London Ser. A 339(1655), 521–553 (1992).
[Crossref]

1982 (1)

1978 (1)

Andalman, A.

Bates, R. H. T.

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. Royal Soc. London Ser. A 339(1655), 521–553 (1992).
[Crossref]

Bian, L.

Bian, Z.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Broxton, M.

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Carney, P. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Chen, F.

Choi, W.

Choi, Y.

Cohen, N.

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Dai, Q.

L. Bian, J. Suo, G. Situ, G. Zheng, F. Chen, and Q. Dai, “Content adaptive sparse illumination for Fourier ptychography,” Opt. Lett. 39(23), 6648–6651 (2014).
[Crossref] [PubMed]

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

Dasari, R. R.

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Deisseroth, K.

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Dong, S.

Dorsch, R. G.

Fang-Yen, C.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

Feld, M. S.

Ferreira, C.

Fienup, J. R.

Garca, J.

V. Mico, Z. Zalevsky, and J. Garca, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).
[Crossref]

Gesell, L. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Goodman, J.

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

Grosenick, L.

Guo, K.

Horstmeyer, R.

R. Horstmeyer, X. Ou, G. Zheng, P. Willems, and C. Yang, “Digital pathology with Fourier ptychography,” Comput. Med. Imag. Graph. 42, 38–43 (2014).
[Crossref]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Hurst, A. C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Jefimovs, K.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Jiang, W.

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

Johnson, I.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Kim, M.

Lapides, J.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Levoy, M.

M. Broxton, L. Grosenick, S. Yang, N. Cohen, A. Andalman, K. Deisseroth, and M. Levoy, “Wave optics theory and 3-D deconvolution for the light field microscope,” Opt. Express 21(21), 25418–25439 (2013).
[Crossref] [PubMed]

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microsc. 235(2), 144–162 (2009).
[Crossref] [PubMed]

Li, X.

Lohmann, A. W.

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

Marcellin, M. W.

M. W. Marcellin, JPEG2000 Image Compression Fundamentals, Standards and Practice (Springer, 2002).

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

McDowall, I.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microsc. 235(2), 144–162 (2009).
[Crossref] [PubMed]

Mendlovic, D.

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Mico, V.

V. Mico, Z. Zalevsky, and J. Garca, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).
[Crossref]

Miller, J. J.

Nanda, P.

Ou, X.

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Price, C. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[Crossref] [PubMed]

Ramchandran, K.

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

J. M. Rodenburg and R. H. T. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Philos. Trans. Royal Soc. London Ser. A 339(1655), 521–553 (1992).
[Crossref]

J. M. Rodenburg, Ptychography and Related Diffractive Imaging Methods (Advances in Imaging and Electron Physics, 2008).

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Shiradkar, R.

Shum, H.

J. Sun, Z. Xu, and H. Shum, “Image super-resolution using gradient profile prior,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[Crossref] [PubMed]

Situ, G.

Sun, J.

J. Sun, Z. Xu, and H. Shum, “Image super-resolution using gradient profile prior,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Sung, Y.

Suo, J.

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Tian, L.

Turpin, T. M.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

Waller, L.

Wang, Z.

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Figures (12)

Fig. 1
Fig. 1

Our FPM prototype setup. a. The optical system includes a LED matrix illuminator on a commercial microscope. b. A raw image matrix contains 15×15 images, with each corresponding to illumination from a single LED (the red axis label denotes the lateral coordinates of the LED array). The 70 sub-images surrounded by white rectangles have been found to be the most important measurements selected by the proposed method. c. Diagram of setup. A programmable LED matrix is placed beneath the sample plane. We first capture Ic (r) with illumination from the blue LED (placed directly below the sample), and then devise a selection scheme for choosing the most informative images Im (r). LEDs in black represent less informative illumination angles which are not captured. P(k) is the pupil function and O(k − km) represents the exit wave.

Fig. 2
Fig. 2

The FPM reconstruction. a1. The FOV of the raw image. b1. The FOV of the FPM reconstruction. a2. The magnified view of the raw image. b2. The magnified view of the FPM reconstruction.

Fig. 3
Fig. 3

Simulation results of the AFP reconstruction and the FPM reconstruction. a1. and b1. are reconstructed with 119 measurements by AFP and FPM, respectively. a2. The strategy of spectrum expanding in the AFP algorithm. The algorithm updates the spectrum (enclosed by the yellow circle) circle by circle (shown as the red circles) from the center outward (along the purple arrow). b2. The strategy of spectrum expanding in the FPM algorithm. The algorithm updates the spectrum from top left to bottom right (along the yellow and purple arrows). a3. and b3. are the spectrum (zoomed in) reconstructed by AFP and FPM algorithms, respectively, and (kx,ky) denotes lateral coordinates in the Fourier domain.

Fig. 4
Fig. 4

Redundancy in Fourier Spectrum of images at different resolutions. a. Images from USC-SIPI [36]. b1. The relationship between reserved ratio of the whole Fourier spectrum and recovered image quality. Here we use images with higher resolution. b2. The relationship between reserved ratio of the whole Fourier spectrum and recovered image quality. Here we use images with lower resolution.

Fig. 5
Fig. 5

The spectrum similarity of the 1951 USAF test chart. a1 and a2 are the importance distributions of the measurements from the high-resolution and the low-resolution image, respectively. b1 and b2 are the corresponding selection schemes from the high-resolution and the low-resolution image, respectively (both using 77 measurements). Red denotes selected measurements, blue denotes discarded ones while yellow and green denote the boundary between them (red and blue).

Fig. 6
Fig. 6

Flowchart of the proposed method. Step 1: capture a single image under normally incident illumination. Step 2: generate simulated images with the measurement. Step 3: rank the simulated images by their relative importance and make the selection scheme. Step 4: capture measurements according to the selection scheme. Step 5: employ the measurements to perform the FPM reconstruction.

Fig. 7
Fig. 7

A comparison between the traditional FPM algorithm and the proposed method, with simulated data. a. Simulation of low-resolution images captured with the central LED for a 1951 USAF test chart and an image of an airport (USC-SIPI dataset [36]). b. The ground truth high-resolution image. c. The reconstruction and recovered spectrum (zoom in) of the proposed method with fewer measurements. The reconstruction of the 1951 USAF test chart utilizes 77 measurements and the reconstruction of the image of airport utilizes 89 measurements. d. The reconstruction and recovered spectrum of the FPM method with all 293 measurements.

Fig. 8
Fig. 8

FPM reconstructions with different selection schemes. a. The PSNR of the reconstructions of the 1951 USAF test chart and the image of airport with different selection schemes. b. The SSIM of the reconstructions of the 1951 USAF test chart and the image of airport with different selection schemes.

Fig. 9
Fig. 9

Experimental reconstruction of the 1951 USAF test chart. a. The captured image under the illumination of the central LED. b. The reconstruction and recovered spectrum of the FPM method with 225 measurements. c. The reconstruction and recovered spectrum of our self-learning method with 70 measurements.

Fig. 10
Fig. 10

Reconstruction of a stained dog cardiac region sample experimental dataset ([38]). a. Full-FOV raw image of a stained dog cardiac region sample. b1. Zoom-in of the region in the red rectangle. b2. The recovered spectrum and reconstructed intensity of the FPM method with 293 measurements. b3. The recovered spectrum and reconstructed intensity of our method with 65 measurements. c1. Zoom-in of the region in the blue rectangle. c2. The recovered spectrum and reconstructed intensity of the FPM method with 293 measurements. c3. The recovered spectrum and reconstructed intensity of our method with 53 measurements.

Fig. 11
Fig. 11

Comparison of an image of biological tissue with different distortions. a. The ground truth. b. A slightly blurred image with added zero-mean random noise. PSNR = 12.4975 and MSSIM = 0.49736. c. A severely blurred image. PSNR = 12.5754 and MSSIM = 0.2397.

Fig. 12
Fig. 12

The flowchart of the SSIM measurement system. The input of the SSIM assessment consists of two images, one is considered to have perfect quality and then the similarity measure can serve as a quantitative measurement of the quality of the other one. SSIM consists of three property comparisons: luminance comparison, contrast comparison and structure comparison.

Equations (9)

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i m ( r ) = | [ O ( k k m ) P ( k ) ] ( r ) | 2 ,
min O ( k ) , P ( k ) m = 1 N L E D | I m ( r ) | [ O ( k k m ) P ( k ) ] ( r ) | 2 | 2 ,
I ( u u 0 , v v 0 ) = { Θ ( x , y ) } with Θ ( x , y ) { I c ( x i , y i ) e j 2 π ( u 0 x i M + v 0 y i N ) , ( x i , y i ) ( x , y ) } ,
A x , y = { 1 , if S ( x , y ) > H 0 , otherwise .
P S N R = 10 × log 10 255 2 × M × N x , y ( I r e c o I r e f e ) 2 ,
μ l = 1 M × N ( x i , y i ) ( x , y ) I l ( x i , y i ) , l = 1 , 2 ,
σ l = 1 M × N 1 ( x i , y i ) ( x , y ) ( I l ( x i , y i ) μ l ) 2 , l = 1 , 2.
σ 1 , 2 = 1 M × N 1 ( x i , y i ) ( x , y ) ( I 1 ( x i , y i ) μ 1 ) ( I 2 ( x i , y i ) μ 2 ) .
S S I M ( I 1 , I 2 ) = ( 2 μ 1 μ 2 + C 1 ) ( 2 σ 1 , 2 + C 2 ) ( μ 1 2 + μ 2 2 + C 1 ) ( σ 1 2 + σ 2 2 + C 2 ) ,

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