Abstract

Two computationally efficient methods for superresolution reconstruction and restoration of microscanning imaging systems are presented. Microscanning creates multiple low-resolution images with slightly different sample–scene phase shifts. The digital processing methods developed here combine the low-resolution images to produce an image with higher pixel resolution (i.e., superresolution) and higher fidelity. The methods implement reconstruction to increase resolution and restoration to improve fidelity in one-pass convolution with a small kernel. One method uses a small-kernel Wiener filter and the other method uses a parametric cubic convolution filter. Both methods are based on an end-to-end, continuous–discrete–continuous microscanning imaging system model. Because the filters are constrained to small spatial kernels they can be efficiently applied by convolution and are amenable to adaptive processing and to parallel processing. Experimental results with simulated imaging and with real microscanned images indicate that the small-kernel methods efficiently and effectively increase resolution and fidelity.

© 2006 Optical Society of America

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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]

2005

J. Ruiz-Alzola, C. Alberola-López, and C. F. Westin, "Adaptive kriging filters for multidimensional signal processing," Signal Process. 85, 413-439 (2005).
[CrossRef]

2004

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Advances and challenges in superresolution," Int. J. Imag. Syst. Technol. 14, 47-57 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multiframe super resolution," IEEE Trans. Image Process. 13, 13274-134 (2004).
[CrossRef]

E. Choi, J. Choi, and M. G. Kang, "Super-resolution approach to overcome physical limitations of imaging sensors: an overview," Int. J. Imag. Syst. Technol. 14, 36-46 (2004).
[CrossRef]

2003

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: a technical overview," IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[CrossRef]

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

M. G. Kang and S. Chaudhuri, "Super-resolution image reconstruction," IEEE Signal Process. Mag. 20(3), 19-20 (2003).
[CrossRef]

2002

H. Foroosh, J. B. Zerubia, and B. Marc, "Extension of phase correlation to subpixel registration," IEEE Trans. Image Process. 11, 188-200 (2002).
[CrossRef]

2001

M. Elad and Y. Hel-Or, "A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur," IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

N. Nguyen and P. Milanfar, "A computationally efficient super-resolution image reconstruction algorithm," IEEE Trans. Image Process. 10, 5733-58 (2001).
[CrossRef]

T. Blu, P. Thévenaz, and M. Unser, "MOMS: maximal-order interpolation of minimal support," IEEE Trans. Image Process. 10, 1069-1080 (2001).
[CrossRef]

2000

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, "Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames," IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[CrossRef]

1999

T. M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imag. 18, 1049-1075 (1999).
[CrossRef]

1998

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]

1996

R. R. Schultz and R. L. Stevenson, "Extraction of high-resolution frames from video sequences," IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

1993

S. P. Kim and W.-Y. Su, "Recursive high-resolution reconstruction of blurred multiframe images," IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

1991

S. E. Reichenbach and S. K. Park, "Small convolution kernels for high-fidelity image restoration," IEEE Trans. Signal Process. 39, 2263-2274 (1991).
[CrossRef]

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

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, "Characterizing digital image acquisition devices," Opt. Eng. 30, 170-177 (1991).
[CrossRef]

1990

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. Speech Signal Process. 38, 1013-1027 (1990).
[CrossRef]

1988

1983

S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vision Graph. Image Process. 23, 258-272 (1983).
[CrossRef]

1981

R. G. Keys, "Cubic convolution interpolation for digital image processing," IEEE Trans. Acoust. Speech Signal Process. 29, 1153-1160 (1981).
[CrossRef]

1956

Alam, M. S.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, "Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames," IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[CrossRef]

Alberola-López, C.

J. Ruiz-Alzola, C. Alberola-López, and C. F. Westin, "Adaptive kriging filters for multidimensional signal processing," Signal Process. 85, 413-439 (2005).
[CrossRef]

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]

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]

Biemond, J.

R. L. Lagendijk and J. Biemond, "Basic methods for image restoration and identification," in Handbook of Image and Video Processing, A.Bovik, ed. (Academic, 2000).

Blouke, M. M.

C. L. L. Hendriks and L. J. V. Vliet, "Improving resolution to reduce aliasing in an undersampled image sequence," in Sensors and Camera Systems for Scientific, Industrial, and Digital Photography Applications, M. M. Blouke, N. Sampat, G. M. Williams, and T. Yeh, eds., Proc. SPIE 3965, 1-9 (2000).

Blu, T.

T. Blu, P. Thévenaz, and M. Unser, "MOMS: maximal-order interpolation of minimal support," IEEE Trans. Image Process. 10, 1069-1080 (2001).
[CrossRef]

Bognar, J. G.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, "Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames," IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[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]

Bose, N. K.

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. Speech Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, 1979).

Chaudhuri, S.

M. G. Kang and S. Chaudhuri, "Super-resolution image reconstruction," IEEE Signal Process. Mag. 20(3), 19-20 (2003).
[CrossRef]

Choi, E.

E. Choi, J. Choi, and M. G. Kang, "Super-resolution approach to overcome physical limitations of imaging sensors: an overview," Int. J. Imag. Syst. Technol. 14, 36-46 (2004).
[CrossRef]

Choi, J.

E. Choi, J. Choi, and M. G. Kang, "Super-resolution approach to overcome physical limitations of imaging sensors: an overview," Int. J. Imag. Syst. Technol. 14, 36-46 (2004).
[CrossRef]

Elad, M.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multiframe super resolution," IEEE Trans. Image Process. 13, 13274-134 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Advances and challenges in superresolution," Int. J. Imag. Syst. Technol. 14, 47-57 (2004).
[CrossRef]

M. Elad and Y. Hel-Or, "A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur," IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

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]

Fales, C. L.

Farsiu, S.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Advances and challenges in superresolution," Int. J. Imag. Syst. Technol. 14, 47-57 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multiframe super resolution," IEEE Trans. Image Process. 13, 13274-134 (2004).
[CrossRef]

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]

Foroosh, H.

H. Foroosh, J. B. Zerubia, and B. Marc, "Extension of phase correlation to subpixel registration," IEEE Trans. Image Process. 11, 188-200 (2002).
[CrossRef]

Gonner, C.

T. M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imag. 18, 1049-1075 (1999).
[CrossRef]

Hardie, R. C.

M. S. Alam, J. G. Bognar, R. C. Hardie, and B. J. Yasuda, "Infrared image registration and high-resolution reconstruction using multiple translationally shifted aliased video frames," IEEE Trans. Instrum. Meas. 49, 915-923 (2000).
[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]

Hel-Or, Y.

M. Elad and Y. Hel-Or, "A fast super-resolution reconstruction algorithm for pure translational motion and common space-invariant blur," IEEE Trans. Image Process. 10, 1187-1193 (2001).
[CrossRef]

Hendriks, C. L. L.

C. L. L. Hendriks and L. J. V. Vliet, "Improving resolution to reduce aliasing in an undersampled image sequence," in Sensors and Camera Systems for Scientific, Industrial, and Digital Photography Applications, M. M. Blouke, N. Sampat, G. M. Williams, and T. Yeh, eds., Proc. SPIE 3965, 1-9 (2000).

Huang, T. S.

R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI Press1984), pp. 317-339.

Huck, F. O.

Irani, M.

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

Kang, M. G.

E. Choi, J. Choi, and M. G. Kang, "Super-resolution approach to overcome physical limitations of imaging sensors: an overview," Int. J. Imag. Syst. Technol. 14, 36-46 (2004).
[CrossRef]

M. G. Kang and S. Chaudhuri, "Super-resolution image reconstruction," IEEE Signal Process. Mag. 20(3), 19-20 (2003).
[CrossRef]

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: a technical overview," IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[CrossRef]

Katsaggelos, A. K.

A. K. Katsaggelos, Digital Image Restoration (Springer-Verlag, 1991).
[CrossRef]

Keys, R. G.

R. G. Keys, "Cubic convolution interpolation for digital image processing," IEEE Trans. Acoust. Speech Signal Process. 29, 1153-1160 (1981).
[CrossRef]

Kim, S. P.

S. P. Kim and W.-Y. Su, "Recursive high-resolution reconstruction of blurred multiframe images," IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

S. P. Kim, N. K. Bose, and H. M. Valenzuela, "Recursive reconstruction of high resolution image from noisy undersampled multiframes," IEEE Trans. Acoust. Speech Signal Process. 38, 1013-1027 (1990).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk and J. Biemond, "Basic methods for image restoration and identification," in Handbook of Image and Video Processing, A.Bovik, ed. (Academic, 2000).

Lehmann, T. M.

T. M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imag. 18, 1049-1075 (1999).
[CrossRef]

Linfoot, E. H.

Marc, B.

H. Foroosh, J. B. Zerubia, and B. Marc, "Extension of phase correlation to subpixel registration," IEEE Trans. Image Process. 11, 188-200 (2002).
[CrossRef]

McCormick, J. A.

Milanfar, P.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Advances and challenges in superresolution," Int. J. Imag. Syst. Technol. 14, 47-57 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multiframe super resolution," IEEE Trans. Image Process. 13, 13274-134 (2004).
[CrossRef]

N. Nguyen and P. Milanfar, "A computationally efficient super-resolution image reconstruction algorithm," IEEE Trans. Image Process. 10, 5733-58 (2001).
[CrossRef]

P. Milanfar, Resolution enhancement software, www.soe.ucsc.edu/∼milanfar/SR-software.htm (2004).

Narayanswamy, R.

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, "Characterizing digital image acquisition devices," Opt. Eng. 30, 170-177 (1991).
[CrossRef]

Ng, M. K.

M. K. Ng and N. K. Bose, "Mathematical analysis of super-resolution methodology," IEEE Signal Process. Mag. 20, 62-74 (2003).
[CrossRef]

Nguyen, N.

N. Nguyen and P. Milanfar, "A computationally efficient super-resolution image reconstruction algorithm," IEEE Trans. Image Process. 10, 5733-58 (2001).
[CrossRef]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: a technical overview," IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, "Super-resolution image reconstruction: a technical overview," IEEE Signal Process. Mag. 20(3), 21-36 (2003).
[CrossRef]

Park, S. K.

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, "Characterizing digital image acquisition devices," Opt. Eng. 30, 170-177 (1991).
[CrossRef]

S. E. Reichenbach and S. K. Park, "Small convolution kernels for high-fidelity image restoration," IEEE Trans. Signal Process. 39, 2263-2274 (1991).
[CrossRef]

C. L. Fales, F. O. Huck, J. A. McCormick, and S. K. Park, "Wiener restoration of sampled image data: end-to-end analysis," J. Opt. Soc. Am. A 5, 300-314 (1988).
[CrossRef]

S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vision Graph. Image Process. 23, 258-272 (1983).
[CrossRef]

Peleg, S.

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

Reichenbach, S. E.

S. E. Reichenbach and S. K. Park, "Small convolution kernels for high-fidelity image restoration," IEEE Trans. Signal Process. 39, 2263-2274 (1991).
[CrossRef]

S. E. Reichenbach, S. K. Park, and R. Narayanswamy, "Characterizing digital image acquisition devices," Opt. Eng. 30, 170-177 (1991).
[CrossRef]

S. E. Reichenbach and J. Shi, "Two-dimensional cubic convolution for one-pass image restoration and reconstruction," in International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, 2004), pp. 2074-2076.

J. Shi and S. E. Reichenbach, "Image image interpolation by two-dimensional parametric cubic convolution," IEEE Trans. Image Process. (to be published).

Robinson, D.

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Fast and robust multiframe super resolution," IEEE Trans. Image Process. 13, 13274-134 (2004).
[CrossRef]

S. Farsiu, D. Robinson, M. Elad, and P. Milanfar, "Advances and challenges in superresolution," Int. J. Imag. Syst. Technol. 14, 47-57 (2004).
[CrossRef]

Ruiz-Alzola, J.

J. Ruiz-Alzola, C. Alberola-López, and C. F. Westin, "Adaptive kriging filters for multidimensional signal processing," Signal Process. 85, 413-439 (2005).
[CrossRef]

Sampat, N.

C. L. L. Hendriks and L. J. V. Vliet, "Improving resolution to reduce aliasing in an undersampled image sequence," in Sensors and Camera Systems for Scientific, Industrial, and Digital Photography Applications, M. M. Blouke, N. Sampat, G. M. Williams, and T. Yeh, eds., Proc. SPIE 3965, 1-9 (2000).

Schowengerdt, R. A.

S. K. Park and R. A. Schowengerdt, "Image reconstruction by parametric cubic convolution," Comput. Vision Graph. Image Process. 23, 258-272 (1983).
[CrossRef]

R. A. Schowengerdt, Remote Sensing: Models and Methods for Image Processing, 2nd ed. (Academic, 1997).

Schultz, R. R.

R. R. Schultz and R. L. Stevenson, "Extraction of high-resolution frames from video sequences," IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

Shi, J.

S. E. Reichenbach and J. Shi, "Two-dimensional cubic convolution for one-pass image restoration and reconstruction," in International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, 2004), pp. 2074-2076.

J. Shi and S. E. Reichenbach, "Image image interpolation by two-dimensional parametric cubic convolution," IEEE Trans. Image Process. (to be published).

Spitzer, K.

T. M. Lehmann, C. Gonner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imag. 18, 1049-1075 (1999).
[CrossRef]

Stein, M. L.

M. L. Stein, Interpolation of Spatial Data: Some Theory for Kriging (Springer-Verlag, 1999).
[CrossRef]

Stevenson, R. L.

R. R. Schultz and R. L. Stevenson, "Extraction of high-resolution frames from video sequences," IEEE Trans. Image Process. 5, 996-1011 (1996).
[CrossRef] [PubMed]

Su, W.-Y.

S. P. Kim and W.-Y. Su, "Recursive high-resolution reconstruction of blurred multiframe images," IEEE Trans. Image Process. 2, 534-539 (1993).
[CrossRef] [PubMed]

Thévenaz, P.

T. Blu, P. Thévenaz, and M. Unser, "MOMS: maximal-order interpolation of minimal support," IEEE Trans. Image Process. 10, 1069-1080 (2001).
[CrossRef]

Tsai, R. Y.

R. Y. Tsai and T. S. Huang, "Multiframe image restoration and registration," in Advances in Computer Vision and Image Processing, T. S. Huang, ed. (JAI Press1984), pp. 317-339.

Unser, M.

T. Blu, P. Thévenaz, and M. Unser, "MOMS: maximal-order interpolation of minimal support," IEEE Trans. Image Process. 10, 1069-1080 (2001).
[CrossRef]

Valenzuela, H. M.

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Figures (9)

Fig. 1
Fig. 1

End-to-end model of the digital imaging process.

Fig. 2
Fig. 2

Microscanning produces multiple images.

Fig. 3
Fig. 3

Simulation results.

Fig. 4
Fig. 4

Small reconstruction and restoration kernels for the simulation experiment.

Fig. 5
Fig. 5

Low-resolution infrared image of a four-bar target used for estimating the acquisition transfer function.

Fig. 6
Fig. 6

Superresolution average scan of the bar target.

Fig. 7
Fig. 7

Estimated acquisition transfer function h ^ x ( u ) .

Fig. 8
Fig. 8

Superresolution results for a microscanned infrared system.

Fig. 9
Fig. 9

Small reconstruction and restoration kernels for the real image experiment.

Tables (1)

Tables Icon

Table 1 Fidelity and Computational Costs of Various Methods

Equations (108)

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p [ m , n ] = s ( x , y ) h ( m x , n y ) d x d y + e [ m , n ] ,
[ m , n ]
p   and   ( x , y )
r ( x , y ) = [ m n p [ m , n ] f ( x m , y n ) ] d ( x x , y y ) d x d y .
p k , k = 0 K - 1 ,
p k
p k [ m , n ] = s ( x x k , y y k ) h k ( m x , n y ) d x d y + e k [ m , n ] ,
( x k , y k )
h k
e k
p k
p ^ k ( u , v ) = μ ν s ^ ( u μ , v ν ) h ^ k ( u μ , v ν ) × exp { i 2 π [ ( u μ ) x k + ( v ν ) y k ] } + e ^ k ( u , v ) ,
p ̂ ( u , v ) = 1 K k = 0 K 1 p ^ k ( u , v ) exp { i 2 π [ u ( x k + α k ) + v ( y k + β k ) ] } = 1 K μ ν s ^ ( u μ , v ν ) k = 0 K 1 h ^ k ( u μ , v ν ) exp [ i 2 π ( u α k + v β k ) ] exp [ i 2 π ( μ x k + ν y k ) ] + 1 K k = 0 K 1 e k ( u , v ) exp { i 2 π [ u ( x k + α k ) + v ( y k + β k ) ] } ,
( α k , β k )
p k
α k = β k = 0
f ^
r ^ ( u , v ) = p ^ ( u , v ) f ^ ( u , v ) d ^ ( u , v ) .
ϵ 2 = ε { | r ( x , y ) s ( x , y ) | 2  d x d y } = ε { | r ^ ( u , v ) s ^ ( u , v ) | 2  d u d v } .
ε [ s ̂ ( u , v ) s ̂ * ( u μ , v ν ) ] = { Φ ̂ s ( u , v ) ( μ , ν ) = ( 0 , 0 ) 0  otherwise ,
ε [ e ̂ j ( u , v ) e ̂ k * ( u , v ) ] = { Φ ̂ e k ( u , v ) j = k 0  otherwise ,
ε [ s ̂ ( u μ , v ν ) e ̂ k * ( u , v ) ] = 0 ,
Φ ̂ s
Φ ̂ e k
MSE   ϵ 2
ϵ 2 = [ Φ ̂ s ( u , v ) f ̂ ( u , v ) d ̂ ( u , v ) Φ ̂ s , p * ( u , v ) f ̂ * ( u , v ) d ^ * ( u , v ) Φ ̂ s , p ( u , v ) + | f ̂ ( u , v ) | 2 | d ^ ( u , v ) | 2 Φ ̂ p ( u , v ) ] d u d v ,
Φ ̂ p
Φ ̂ s , p
Φ ̂ p ( u , v ) = ε [ | p ^ ( u , v ) | 2 ] = 1 K 2 μ ν Φ ̂ s ( u μ , v ν ) | k = 0 K 1 h ^ k ( u μ , v ν ) ε { exp [ i 2 π ( u α k + v β k ) ] exp [ i 2 π ( μ x k + ν y k ) ] } | 2 + 1 K 2 k = 0 K 1 Φ ̂ e k ( u , v ) ,
Φ ̂ s , p ( u , v ) = ε [ s ^ ( u , v ) p ^ * ( u , v ) ] = Φ ̂ s ( u , v ) 1 K k = 0 K 1 h ^ k * ( u , v ) ε { exp [ i 2 π ( u α k + v β k ) ] } .
α k   and   β k
[ 1 2 W x , 1 2 W x ] ,       [ 1 2 W y , 1 2 W y ] ,
1 2 W x     1 2 W x 1 2 W y     1 2 W y   exp [ i 2 π ( u α k + v β k ) ] W x W y d α k d β k = s i n c ( u / W x ) sinc ( v / W y ) ,
Φ ̂ p ( u , v ) = 1 K 2   sinc 2 ( u / W x ) sinc 2 ( v / W y ) μ ν Φ ̂ s ( u μ , v ν ) | k = 0 K 1 h ^ k ( u μ , v ν ) exp [ i 2 π ( μ x k + ν y k ) ] | 2 + 1 K 2 k = 0 K 1 Φ ̂ e k ( u , v ) ,
Φ ̂ s , p ( u , v ) = sinc ( u / W x ) sinc ( v / W y ) Φ ̂ s ( u , v ) 1 K     k = 0 K 1 h ^ k * ( u , v ) .
Φ ̂ p
( x k , y k )
K   is   2, Φ ̂ p
{ ( x 0 , y 0 ) = ( 0 , 0 ) ,
( x 1 , y 1 ) = ( 0 , 0.5 ) }
{ ( x 0 , y 0 ) = ( 0.5 , 0 ) , ( x 1 , y 1 ) = ( 0.5 , 0.5 ) }
{ ( x 0 , y 0 ) = ( 0 , 0 ) , ( x 1 , y 1 ) = ( 0.5 , 0 ) }
{ ( x 0 , y 0 ) = ( 0 , 0.5 ) , ( x 1 , y 1 ) = ( 0.5 , 0.5 ) }
Φ ̂ p
= 1 ϵ 2 Φ ̂ s ( u , v ) d u d v .
MSE,   ϵ 2
f ^ ( u , v )
ϵ 2 ( f ^ ) = L ( u , v , f ^ ) d u d v ,
L ( u , v , f ^ ) = Φ ̂ s ( u , v ) f ^ ( u , v ) d ^ ( u , v ) Φ ̂ s , p ( u , v ) f ^ * ( u , v ) d ^ * ( u , v ) Φ ̂ s , p * ( u , v ) + | f ^ ( u , v ) | 2 | d ^ ( u , v ) | 2 Φ ̂ p ( u , v ) .
L f ^ = f ^ * ( u , v ) | d ^ ( u , v ) | 2 Φ ̂ p ( u , v ) d ^ ( u , v ) Φ ̂ s , p * ( u , v ) = 0 ,
f ^ w ( u , v ) = Φ ̂ s , p ( u , v ) Φ ̂ p ( u , v ) d ^ * ( u , v ) | d ^ ( u , v ) | 2 = Φ ̂ s , p ˜ ( u , v ) Φ ̂ p ˜ ( u , v ) ,
Φ ̂ p ˜   and   Φ ̂ s , p ˜
f c
f c ( x , y ) = 0 , ( x , y ) C .
MSE   ϵ 2
ϵ 2 f c ( x , y ) = 0 ,      ( x , y ) C .
( x , y ) C Φ p ˜ ( x x , y y ) f c ( x , y ) = Φ s , p ˜ ( x , y ) , ( x , y ) C ,
Φ ̂ p ˜
Φ ̂ s , p ˜
| C |
| C |
[ 2 , 2 ] × [ 2 , 2 ]
{ a 1 , a 2 , a 3 , a 4 , a 5 } :
f p ( x , y ) = f 0 ( x , y ) + a 1 f 1 ( x , y ) + a 2 f 2 ( x , y ) + a 3 f 3 ( x , y ) + a 4 f 4 ( x , y ) + a 5 f 5 ( x , y ) ,
f 0 f 5
f p
MSE   ϵ 2
ϵ 2
ϵ 2 a 1 = ϵ 2 a 2 = ϵ 2 a 3 = ϵ 2 a 4 = ϵ 2 a 5 = 0 ,
f ^ i ( u , v ) { Re [ Φ ̂ s , p ¯ ( u , v ) ] f ^ 0 ( u , v ) Φ ̂ p ̃ ( u , v ) } d u d v = f ^ i ( u , v ) [ f ^ p ( u , v ) f ^ 0 ( u , v ) ] Φ ̂ p ̃ ( u , v ) d u d v ,
i = 1 5.
L 1
256 × 256
h ^ ( u , v ) = exp [ ( u 2 + v 2 ) ] ,
h ^ ( 0.0 , 0.5 ) = h ^ ( 0.5 , 0.0 ) = 0.779
64 × 64
30   dB
BSNR = 10   log10 ( σ p       2 / σ e       2 ) ,
σ p     2
σ e     2
256 × 256
Φ s ( x , y ) = exp ( x 2 + y 2 / ρ ) ,
Φ ̂ s ( u , v ) = 2 π ρ 2 [ 1 + 4 π 2 ρ 2 ( u 2 + v 2 ) ] 3 / 2 .
[ 2 , 2 ] × [ 2 , 2 ]
a 1 = 74.176 , a 2 = 95.360
a 3 = 16.804 , a 4 = 0.967 ,
a 5 = 0.238
1.8   GHz
256   MB
256 × 256
h ^ ( u , v ) = h ^ x ( u ) h ^ x ( v ) .
1024 × 1024
256 × 256
1024 × 1024
a 1 = 1.052
a 2 = 7.033 ,
a 3 = 7.584 ,
a 4 = 1.123
a 5 = 0.137
1024 × 1024
1024 × 1024
256 × 256
[ S , S ] × [ S , S ]
R × R
4 K S 2 R 2
K S 2
f 0 ( x , y ) = { x 2 y 2 x 2 y 2 + 1 0 x 1 ,     0 y 1 , ( 2 x y 2 2 x 2 y 2 + 2 ) ( x 2 ) 2 1 < x 2 ,     0 y 1 , ( 4 x y 4 y 4 x + 4 ) ( x 2 ) 2 ( y 2 ) 2 1 < x 2 ,     1 < y 2 , ( 2 x 2 y - 2 y - 2 x 2 + 2 ) ( y 2 ) 2 0 x 1 ,     1 < y 2 , f 1 ( x , y ) = { x 3 y 3 x 2 y 2 ( 5 x y 3 4 x y 2 4 y 3 + 3 y 2 ) ( x 2 ) 2 ( 9 x y 8 y 8 x + 7 ) ( x 2 ) 2 ( y 2 ) 2 ( 5 x 3 y 4 x 2 y 4 x 3 + 3 x 2 ) ( y 2 ) 2 0 x 1 ,     0 y 1 , 1 < x 2 ,     0 y 1 , 1 < x 2 ,     1 < y 2 , 0 x 1 ,     1 < y 2 , f 2 ( x , y ) = { x 3 y 2 2 x 2 y 2 + x 2 y 3 ( 4 x y 3 3 x y 2 3 y 3 + 2 y 2 ) ( x 2 ) 2 ( 8 x y 7 y 7 x + 6 ) ( x 2 ) 2 ( y 2 ) 2 ( 4 x 3 y 3 x 2 y 3 x 3 + 2 x 2 ) ( y 2 ) 2 0 x 1 ,     0 y 1 , 1 < x 2 ,     0 y 1 , 1 < x 2 ,     1 < y 2 , 0 x 1 ,     1 < y 2 , f 3 ( x , y ) = { x 3 x 2 y 2 + y 3 ( 2 x y 3 2 x y 2 + x + y 2 y 3 - 1 ) ( x 2 ) 2 ( 4 x y 3 x 3 y + 2 ) ( x 2 ) 2 ( y 2 ) 2 ( 2 x 3 y 2 x 2 y + y + x 2 x 3 1 ) ( y 2 ) 2 0 x 1 ,     0 y 1 , 1 < x 2 ,     0 y 1 , 1 < x 2 ,     1 < y 2 , 0 x 1 ,     1 < y 2 , f 4 ( x , y ) = { 5 x 2 6 x 2 y 2 + 5 y 2 4 ( 13 y 2 14 x y 2 + 12 x 11 ) ( x 2 ) 2 ( 30 x + 30 y 32 x y 28 ) ( x 2 ) 2 ( y 2 ) 2 ( 13 x 2 14 x 2 y + 12 y 11 ) ( y 2 ) 2 0 x 1 ,     0 y 1 , 1 < x 2 ,     0 y 1 , 1 < x 2 ,     1 < y 2 , 0 x 1 ,     1 < y 2 , f 5 ( x , y ) = { 4 x 2 3 x 2 y 2 + 4 y 2 4 ( 5 y 2 4 x y 2 + 8 x 8 ) ( x 2 ) 2 ( 4 x + 4 y 7 ) ( x 2 ) 2 ( y 2 ) 2 ( 5 x 2 4 x 2 y + 8 y 8 ) ( y 2 ) 2       0 x 1 ,     0 y 1 ,       1 < x 2 ,     0 y 1 ,       1 < x 2 ,     1 < y 2 ,       0 x 1 ,     1 < y 2.
h ^ x ( u ) .

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