Abstract

We study the reconstruction of a high-resolution image from multiple low-resolution images by using a nonlinear iterative backprojection algorithm. We exploit diversities in the imaging channels, namely, the number of imagers, magnification, position, rotation, and fill factor, to undo the degradation caused by the optical blur, pixel blur, and additive noise. We quantify the improvements gained by these diversities in the reconstruction process and discuss the trade-off among system parameters. As an example, for a system in which the pixel size is matched to the diffraction-limited optical blur size at a moderate detector noise level, we can reduce the reconstruction root-mean-square error by 570% by using 16 cameras and a large amount of diversity. The algorithm was implemented on a 56 camera array specifically constructed to demonstrate the resolution-enhancement capabilities. Practical issues associated with building and operating this device are presented and analyzed.

© 2006 Optical Society of America

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References

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  1. K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).
  2. S. Chaudhuri, ed., Super-Resolution Imaging (Norwell, 2001).
  3. R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances-Computer Vision and Image Process (JAI Press, 1984), Vol. 1, pp. 317-339.
  4. H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
    [CrossRef]
  5. M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
    [CrossRef]
  6. M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
    [CrossRef] [PubMed]
  7. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
    [CrossRef]
  8. S. Baker and T. Kanade, "Limits on super-resolution and how to break them," IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
    [CrossRef]
  9. C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
    [CrossRef]
  10. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, 1990), pp. 495-510.
  11. D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
    [CrossRef]
  12. G. B. Giannakis and C. Tepedelenlioglu, "Direct blind equalizers of multiple FIR channels: a deterministic approach," IEEE Trans. Signal Process. 47, 62-74 (1999).
    [CrossRef]
  13. J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 85-89.
  14. A. J. den Dekker and A. van den Bos, "Resolution: a survey," J. Opt. Soc. Am. A 14, 547-557 (1997).
    [CrossRef]
  15. M. A. Neifeld, "Information, resolution, and space-bandwidth product," Opt. Lett. 23, 1477-1479 (1998).
    [CrossRef]
  16. P. Milanfar and A. Shakouri, "A statistical analysis of diffraction-limited imaging," In IEEE 2000 International Conference on Image Processing (ICIP-2002) (IEEE, 2002), Vol. 1, pp. 22-25.
  17. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.
  18. M. Chiang and T. Boult, "Efficient image warping and super-resolution," in Proceedings of the Third IEEE Workshop on Applications of Computer Vision (IEEE, 1996), pp. 56-61.
    [CrossRef]
  19. M. C. Chiang and T. E. Boult, "Local blur estimation and super-resolution," in Proceedings of Computer Vision and Pattern Recognition, IEEE Computer Society Conference (IEEE 1997), pp. 821-826.

2003

C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
[CrossRef]

2002

S. Baker and T. Kanade, "Limits on super-resolution and how to break them," IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

1999

G. B. Giannakis and C. Tepedelenlioglu, "Direct blind equalizers of multiple FIR channels: a deterministic approach," IEEE Trans. Signal Process. 47, 62-74 (1999).
[CrossRef]

1998

M. A. Neifeld, "Information, resolution, and space-bandwidth product," Opt. Lett. 23, 1477-1479 (1998).
[CrossRef]

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

1997

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

A. J. den Dekker and A. van den Bos, "Resolution: a survey," J. Opt. Soc. Am. A 14, 547-557 (1997).
[CrossRef]

D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
[CrossRef]

1992

K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

1991

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Aizawa, K.

K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).

Armstrong, E. E.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Baker, S.

S. Baker and T. Kanade, "Limits on super-resolution and how to break them," IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Bognar, J. G.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Boult, T.

M. Chiang and T. Boult, "Efficient image warping and super-resolution," in Proceedings of the Third IEEE Workshop on Applications of Computer Vision (IEEE, 1996), pp. 56-61.
[CrossRef]

Boult, T. E.

M. C. Chiang and T. E. Boult, "Local blur estimation and super-resolution," in Proceedings of Computer Vision and Pattern Recognition, IEEE Computer Society Conference (IEEE 1997), pp. 821-826.

Chaudhuri, S.

S. Chaudhuri, ed., Super-Resolution Imaging (Norwell, 2001).

Chiang, M.

M. Chiang and T. Boult, "Efficient image warping and super-resolution," in Proceedings of the Third IEEE Workshop on Applications of Computer Vision (IEEE, 1996), pp. 56-61.
[CrossRef]

Chiang, M. C.

M. C. Chiang and T. E. Boult, "Local blur estimation and super-resolution," in Proceedings of Computer Vision and Pattern Recognition, IEEE Computer Society Conference (IEEE 1997), pp. 821-826.

den Dekker, A. J.

Duhamel, P.

D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
[CrossRef]

Elad, M.

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Feuer, A.

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.

Gesbert, D.

D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
[CrossRef]

Giannakis, G. B.

G. B. Giannakis and C. Tepedelenlioglu, "Direct blind equalizers of multiple FIR channels: a deterministic approach," IEEE Trans. Signal Process. 47, 62-74 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 85-89.

Gross, D.

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

Hardie, R. C.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Huang, T. S.

R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances-Computer Vision and Image Process (JAI Press, 1984), Vol. 1, pp. 317-339.

Irani, M.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Kanade, T.

S. Baker and T. Kanade, "Limits on super-resolution and how to break them," IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

Katsaggelos, A. K.

C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
[CrossRef]

Komatsu, T.

K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, 1990), pp. 495-510.

Mayrargue, S.

D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
[CrossRef]

Milanfar, P.

P. Milanfar and A. Shakouri, "A statistical analysis of diffraction-limited imaging," In IEEE 2000 International Conference on Image Processing (ICIP-2002) (IEEE, 2002), Vol. 1, pp. 22-25.

Molina, R.

C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
[CrossRef]

Neifeld, M. A.

Peleg, S.

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.

Saito, T.

K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).

Segall, C. A.

C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
[CrossRef]

Shakouri, A.

P. Milanfar and A. Shakouri, "A statistical analysis of diffraction-limited imaging," In IEEE 2000 International Conference on Image Processing (ICIP-2002) (IEEE, 2002), Vol. 1, pp. 22-25.

Tepedelenlioglu, C.

G. B. Giannakis and C. Tepedelenlioglu, "Direct blind equalizers of multiple FIR channels: a deterministic approach," IEEE Trans. Signal Process. 47, 62-74 (1999).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.

Tsai, R. Y.

R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances-Computer Vision and Image Process (JAI Press, 1984), Vol. 1, pp. 317-339.

Ur, H.

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

van den Bos, A.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.

Watson, E. A.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

CVGIP: Graph. Models Image Process.

H. Ur and D. Gross, "Improved resolution from subpixel shifted pictures," CVGIP: Graph. Models Image Process. 54, 181-186 (1992).
[CrossRef]

M. Irani and S. Peleg, "Improving resolution by image registration," CVGIP: Graph. Models Image Process. 53, 231-239 (1991).
[CrossRef]

IEEE Signal Process. Mag.

C. A. Segall, R. Molina, and A. K. Katsaggelos, "High-resolution images from low-resolution compressed video," IEEE Signal Process. Mag. 20, 37-48 (2003).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

K. Aizawa, T. Komatsu, and T. Saito, "A scheme for acquiring very high resolution images using multiple cameras," IEEE Trans. Acoust. Speech Signal Process. 3, 23-26 (1992).

IEEE Trans. Image Process.

M. Elad and A. Feuer, "Restoration of a single superresolution image from several blurred, noisy, and undersampled measured images," IEEE Trans. Image Process. 6, 1646-1658 (1997).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell.

S. Baker and T. Kanade, "Limits on super-resolution and how to break them," IEEE Trans. Pattern Anal. Mach. Intell. 24, 1167-1183 (2002).
[CrossRef]

IEEE Trans. Signal Process.

D. Gesbert, P. Duhamel, and S. Mayrargue, "On-line blind multichannel equalization based on mutually referenced filters," IEEE Trans. Signal Process. 45, 2307-2317 (1997).
[CrossRef]

G. B. Giannakis and C. Tepedelenlioglu, "Direct blind equalizers of multiple FIR channels: a deterministic approach," IEEE Trans. Signal Process. 47, 62-74 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Opt. Lett.

Other

J. W. Goodman, Statistical Optics (Wiley, 1985), pp. 85-89.

P. Milanfar and A. Shakouri, "A statistical analysis of diffraction-limited imaging," In IEEE 2000 International Conference on Image Processing (ICIP-2002) (IEEE, 2002), Vol. 1, pp. 22-25.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C++: the Art of Scientific Computing (Cambridge U. Press, 2002), pp. 108-132.

M. Chiang and T. Boult, "Efficient image warping and super-resolution," in Proceedings of the Third IEEE Workshop on Applications of Computer Vision (IEEE, 1996), pp. 56-61.
[CrossRef]

M. C. Chiang and T. E. Boult, "Local blur estimation and super-resolution," in Proceedings of Computer Vision and Pattern Recognition, IEEE Computer Society Conference (IEEE 1997), pp. 821-826.

S. Chaudhuri, ed., Super-Resolution Imaging (Norwell, 2001).

R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances-Computer Vision and Image Process (JAI Press, 1984), Vol. 1, pp. 317-339.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, 1990), pp. 495-510.

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Figures (23)

Fig. 1
Fig. 1

Detector structures in two LR imagers with pixel widths of (a) D 1 and (b) D 2 (D 1 > D 2).

Fig. 2
Fig. 2

(a) Pixel-transfer functions for two pixels of different sizes, (b) reduction in SNR for two different imagers, (c) reduction in SNR for several imagers.

Fig. 3
Fig. 3

Illustration of degradations in the imaging model.

Fig. 4
Fig. 4

Comparison of the diffraction-limited blur spot and pixel size for (a) W = 1.0, D = 2 and (b) W = 1.0, D = 4. Blur spot = 2∕W and pixel size = D.

Fig. 5
Fig. 5

Illustration of IBP.

Fig. 6
Fig. 6

Original image.

Fig. 7
Fig. 7

Sample measured LR images with (a) D = 2, (b) D = 4, (c) D = 8 at SNR = 54 dB and W = 0.25.

Fig. 8
Fig. 8

(a) Residual RMSE versus iteration number and (b) reconstruction RMSE versus iteration number with different relaxation parameters for one sample realization of the multiaperture system. D = 2, M = 4, W = 1.0, SNR = 54 dB.

Fig. 9
Fig. 9

Reconstructed images at SNR = 46 dB, W = 1.0, D = 4, and M = 32 with (a) shift-only, (b) low-diversity, and (c) high-diversity configurations. These images have RMSE values of 0.13, 0.125, and 0.105, respectively.

Fig. 10
Fig. 10

Reconstruction RMSE versus number of LR images demonstrates the benefit of measurement diversity for (a) D = 2, (b) D = 4 at SNR = 54 dB and W = 0.5, (c) D = 2, (d) D = 4 at SNR = 46 dB and W = 1.0.

Fig. 11
Fig. 11

Reconstructed images obtained from the same number of measured pixels [MN 2D 2] (a) D = 2, M = 4; (b) D = 4, M = 16 at SNR = 54 dB and W = 0.25.

Fig. 12
Fig. 12

Comparison of reconstructed spectra with D = 4, W = 1.0, and SNR = 46 dB for (a) shift-only, (b) low-diversity, and (c) high-diversity cases. The reconstructed spectra in the high-diversity case are closer to the original spectrum. Also notice that the SNR at frequencies around the spectral nulls are higher in low- and high-diversity cases than in the shift-only case.

Fig. 13
Fig. 13

Comparison of reconstructed spectra for M = 32, D = 4, W = 1.0, and SNR = 60 dB for different diversity configurations. Spectra are obtained from only those pixels that are seen by all the cameras and in all the configurations.

Fig. 14
Fig. 14

Reconstruction RMSE versus number of LR images at different SNR for D = 2 with W = 1.0 at (a) low and (b) high diversity.

Fig. 15
Fig. 15

Reconstructed images at SNR = 54 dB, W = 0.5, and D = 4 with (a) M = 2, (b) M = 4, (c) M = 8, and (d) M = 32.

Fig. 16
Fig. 16

Reconstruction RMSE versus fill factor for D = 8, W = 0.25, and M = 32 for different SNR.

Fig. 17
Fig. 17

Reconstructed images with D = 4, W = 1.0, M = 32, and SNR = 54 dB with (a) 0, (b) ±1%, (c) ±3%, and (d) ±5% error in the channel parameters.

Fig. 18
Fig. 18

Reconstruction RMSE versus error in the channel parameters at SNR = 54 dB, W = 1.0, and (a) D = 2, and (b) D = 4.

Fig. 19
Fig. 19

Front view of the camera array. Each square holds an electronics board supporting four sensors. Fifty-six of the 129 sensors have lenses attached. The central area is devoted to interface electronics and a PC104 computer.

Fig. 20
Fig. 20

Demonstration of foveation showing two enhanced-resolution regions.

Fig. 21
Fig. 21

Full image from single sensor. Note the multiline chart to the left of the circular target.

Fig. 22
Fig. 22

Individual sensor with closeup of the resolution chart.

Fig. 23
Fig. 23

Estimate formed by multiaperture imaging showing a closeup of the resolution chart.

Tables (2)

Tables Icon

Table 1 Percentage RMSE Reduction with Different W, Diversity, and SNR a

Tables Icon

Table 2 Percentage Reduction in RMSE with Different W and Diversity at Noise Free, SNR = 54 dB, and SNR = 46 dB a

Equations (196)

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

570 %
D 1
D 2 ( D 1 > D 2 )
H k ( w ) = sinc ( D k w )
k = 1 , 2
D 1 = 1.0
D 2 = 0.8
σ 2
R k ( w ) = S ( w ) H k ( w ) + σ 2
R k ( w )
S ( w ) H k ( w )
σ 2
SNR k ( w ) = 10 log 10 [ S 2 ( w ) H k     2 ( w ) ] / σ 2
SNR O ( w ) = 10 log 10 S 2 ( w ) / σ 2
L k ( w ) = SNR k ( w )
SNR O ( w )
= 10
log 10 H k 2 ( w )
{ g k ( i , j ) ; k = 0 , , M 1 }
g k ( i , j ) = S { h p ( x , y ) h a ( x , y ) T k [ f ( x , y ) ] } + n k ( i , j ) ,
h a ( x , y )
h p ( x , y )
T k
n k ( i , j )
( x , y )
T k
[ x y ] = s k [ cos θ k sin θ k sin θ k cos θ k ] [ x y ] + [ a k b k ] ,
s k
θ k
a k
b k
{ s k , θ k , a k , b k : k = 1 , , M }
H ( w 1 , w 2 )
H ( w 1 , w 2 ) = Λ ( w 1 W ) Λ ( w 2 W ) ,
Λ ( )
Λ ( x ) = 1 x if   | x | < 1
= 0 otherwise,
w 1
w 2
h a ( x , y )
H ( w 1 , w 2 )
h p ( x , y )
h p ( x , y ) = 1 D 2      for   D 2 x , y D 2
= 0 otherwise .
h p ( x , y )
σ 2
W = 1.0
D = 2
D = 4
( d b )
h a ( x ) = sinc 2 ( x W ) :
d b = 2 / W
W = 1.0
d b = 2
f ( 0 )
{ g k     ( 0 ) }
{ g k } , k = 1 , , N
g k   (0) ( i , j ) = S { h p ( x , y ) h a ( x , y ) T k [ f ( 0 ) ( x , y ) ] } .
f ( 0 )
{ g k     ( 0 ) }
{ g k }
{ g k g k     ( 0 ) }
e ( n )
e ( n ) = { ( 1 / M L 2 ) k i , j [ g k ( i , j ) g k     ( n ) ( i , j ) ] 2 } 1 / 2
( S 1 )
( h b )
h b ( i , j ) = [ h a ( i , j ) ] / c
c = i , j h a     2 ( i , j )
h a ( i , j )
h b ( i , j ) = 1 / c
c = i , j h a ( i , j )
( T k     1 )
λ ( 0 < λ < 1 )
e ( n )
W = 0.25
D = 2 , 4
SNR = 54   dB
SNR = ( 1 / N 2 ) i , j f 2 ( i , j ) σ 2 .
f ( 0 )
D = 2
SNR = 54   dB
f ^
{ ( 1 / N C ) ( i , j ) C [ f ( i , j ) f ^ ( i , j ) ] 2 } 1 / 2
N c
λ = 0.5
SNR = 46   dB
W = 1.0
W = 0.5
SNR = 54   dB
D = 2
D = 4
W = 1.0
SNR = 46   dB
D = 2
D = 4
( M N 2 / D 2 )
D = 2
M = 4
D = 4
M = 16
SNR = 54   dB
W = 0.25
D = 2 , M = 4
RMSE = 0.154
D = 4
M = 16
RMSE = 0.186
[ ( RMSE M 1 RMSE M 2 )  ∗  100 % ] / RMSE M 2
M 1 = 1
M 2 = 32
D = 2
M 1 = 1
M 2 = 64
D = 4
D = 2
D = 4
SNR = 54   dB
D = 2
570 %
W = 1.0
45 %
W = 0.25
222 %
W = 0.5
SNR = 54   dB
D = 4
W = 1.0
W = 1.0
D = 4
SNR = 46   dB
M = 4
M = 32
| F ( w , w ) |
F ( w 1 , w 2 )
M = 32
M  =  4
SNR = 60   dB
M = 32
1 / M
26   dB
SNR = 54   dB
D = 4
W = 0.5
M = 2
M = 4
M = 8
M = 32
D = 8
W = 0.25
M = 32
30 %
60 %
60 %
60 %
40   dB
{ s k , θ k , a k , b k : k = 1 , , M }
D = 4
W = 1.0
M = 32
SNR = 54   dB
RMSE=0 .086
± 1%
± 2%
± 5%
D = 2
M = 4
D = 4
M = 8
D = 4
± 1 %
9.0   μm × 8 .2   μm
6 .5   mm
F / #
2.54
570 %
266 %
W = 1.0
SNR = 54   dB
D = 2
D = 4
26   dB
33%
D = 2
W = 1.0
30 %
60 %
D = 8
W = 0.25
69 %
105 %
± 1 %
± 5 %
D = 2
M = 4
SNR = 54   dB
W = 1.0

Metrics