Abstract

In this paper, we address the geometrical resolution limitation of an imaging sensor caused by the size of its pixels yielding insufficient spatial sampling of the image. The spatial blurring that is caused due to inadequate sampling can be resolved by placing a two-dimensional binary random mask in an intermediate image plane and shifting it along one direction while keeping the sensor as well as all other optical components fixed. Out of the set of images that are captured, a high resolution image can be decoded. In addition, this approach allows improved robustness to spatial noise.

© 2011 Optical Society of America

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  1. P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.
  2. R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
    [CrossRef] [PubMed]
  3. B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image from multiple-degraded misregistered low-resolution images,” Proc. SPIE 2308, 971–981 (1994).
    [CrossRef]
  4. H. Stark and P. Oskoui, “High-resolution image recovery from image-plane arrays using convex projections,” J. Opt. Soc. Am. A 6, 1715–1726 (1989).
    [CrossRef] [PubMed]
  5. F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
    [CrossRef]
  6. M. Irani and S. Peleg, “Motion analysis for image enhancement: resolution, occlusion, and transparency,” J. Visual Commun. Image Represent 4, 324–335 (1993).
    [CrossRef]
  7. Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
    [CrossRef]
  8. Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
    [CrossRef]
  9. A. Borkowski, Z. Zalevsky, and B. Javidi, “Geometrical super resolved imaging using non periodic spatial masking,” J. Opt. Soc. Am. A 26, 589–601 (2009).
    [CrossRef]
  10. I. ul-Haq and A. A. Mudassar, “Geometrical superresolution of CCD-pixel,” Opt. Lett. 35, 2705–2707 (2010).
    [CrossRef] [PubMed]
  11. M. Sohail and A. A. Mudassar, “Geometric superresolution by using an optical mask,” Appl. Opt. 49, 3000–3005 (2010).
    [CrossRef] [PubMed]
  12. K. A. Nugent and B. Luther-Davies, “Penumbral imaging of high energy x-rays from laser-produced plasmas,” Opt. Commun. 49, 393–396 (1984).
    [CrossRef]
  13. K. A. Nugent, “Coded aperture imaging: a Fourier space analysis,” Appl. Opt. 26, 563–569 (1987).
    [CrossRef] [PubMed]
  14. Z. Zalevsky, J. Solomon, and D. Mendlovic, “Geometrical super resolution using code division multiplexing,” Appl. Opt. 42, 32–40 (2005).
  15. C. F. Van Loan, “The ubiquitous Kronecker product,” J. Comp. Appl. Math. 123, 85–100 (2000).
    [CrossRef]
  16. R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
    [CrossRef]
  17. H. Lev-Ari, “Efficient solution of linear matrix equations with application to multistatic antenna array processing,” Commun. Inf. Syst. 5, 123–130 (2005).
  18. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996), pp. 257–258.
  19. G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  20. G. H. Golub and U. von Matt, “Generalized cross-validation for large scale problems,” in Proceedings of the Second International Workshop on Recent Advances in Total Least Squares Techniques and Errors-in-Variables Modeling, S.Van Huffel, ed. (Society for Industrial and Applied Mathematics, 1997), pp. 139–148.
  21. A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods in image restoration,” J. Comput. Appl. Math. 206, 86–98 (2007).
    [CrossRef]

2010 (2)

2009 (1)

2007 (1)

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods in image restoration,” J. Comput. Appl. Math. 206, 86–98 (2007).
[CrossRef]

2005 (4)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

H. Lev-Ari, “Efficient solution of linear matrix equations with application to multistatic antenna array processing,” Commun. Inf. Syst. 5, 123–130 (2005).

F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
[CrossRef]

Z. Zalevsky, J. Solomon, and D. Mendlovic, “Geometrical super resolution using code division multiplexing,” Appl. Opt. 42, 32–40 (2005).

2004 (1)

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

2000 (2)

C. F. Van Loan, “The ubiquitous Kronecker product,” J. Comp. Appl. Math. 123, 85–100 (2000).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

1997 (1)

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
[CrossRef] [PubMed]

1994 (1)

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image from multiple-degraded misregistered low-resolution images,” Proc. SPIE 2308, 971–981 (1994).
[CrossRef]

1993 (1)

M. Irani and S. Peleg, “Motion analysis for image enhancement: resolution, occlusion, and transparency,” J. Visual Commun. Image Represent 4, 324–335 (1993).
[CrossRef]

1989 (1)

1987 (1)

1984 (1)

K. A. Nugent and B. Luther-Davies, “Penumbral imaging of high energy x-rays from laser-produced plasmas,” Opt. Commun. 49, 393–396 (1984).
[CrossRef]

1979 (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Armstrong, E. E.

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
[CrossRef] [PubMed]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
[CrossRef] [PubMed]

Barrett, E. B.

F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
[CrossRef]

Borkowski, A.

Bouhamidi, A.

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods in image restoration,” J. Comput. Appl. Math. 206, 86–98 (2007).
[CrossRef]

Cheeseman, P.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996), pp. 257–258.

G. H. Golub and U. von Matt, “Generalized cross-validation for large scale problems,” in Proceedings of the Second International Workshop on Recent Advances in Total Least Squares Techniques and Errors-in-Variables Modeling, S.Van Huffel, ed. (Society for Industrial and Applied Mathematics, 1997), pp. 139–148.

Gray, R. M.

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

Hanson, R.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

Hardie, R. C.

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
[CrossRef] [PubMed]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Hoctor, R. T.

F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
[CrossRef]

Irani, M.

M. Irani and S. Peleg, “Motion analysis for image enhancement: resolution, occlusion, and transparency,” J. Visual Commun. Image Represent 4, 324–335 (1993).
[CrossRef]

Javidi, B.

Jbilou, K.

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods in image restoration,” J. Comput. Appl. Math. 206, 86–98 (2007).
[CrossRef]

Kanefsky, B.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

Katsaggelos, A. K.

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image from multiple-degraded misregistered low-resolution images,” Proc. SPIE 2308, 971–981 (1994).
[CrossRef]

Kraft, R.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

Lev-Ari, H.

H. Lev-Ari, “Efficient solution of linear matrix equations with application to multistatic antenna array processing,” Commun. Inf. Syst. 5, 123–130 (2005).

Luther-Davies, B.

K. A. Nugent and B. Luther-Davies, “Penumbral imaging of high energy x-rays from laser-produced plasmas,” Opt. Commun. 49, 393–396 (1984).
[CrossRef]

Marom, E.

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Mendlovic, D.

Z. Zalevsky, J. Solomon, and D. Mendlovic, “Geometrical super resolution using code division multiplexing,” Appl. Opt. 42, 32–40 (2005).

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Mudassar, A. A.

Nugent, K. A.

K. A. Nugent, “Coded aperture imaging: a Fourier space analysis,” Appl. Opt. 26, 563–569 (1987).
[CrossRef] [PubMed]

K. A. Nugent and B. Luther-Davies, “Penumbral imaging of high energy x-rays from laser-produced plasmas,” Opt. Commun. 49, 393–396 (1984).
[CrossRef]

Oskoui, P.

Peleg, S.

M. Irani and S. Peleg, “Motion analysis for image enhancement: resolution, occlusion, and transparency,” J. Visual Commun. Image Represent 4, 324–335 (1993).
[CrossRef]

Shamir, N.

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Sohail, M.

Solomon, J.

Z. Zalevsky, J. Solomon, and D. Mendlovic, “Geometrical super resolution using code division multiplexing,” Appl. Opt. 42, 32–40 (2005).

Stark, H.

Stutz, J.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

Tom, B. C.

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image from multiple-degraded misregistered low-resolution images,” Proc. SPIE 2308, 971–981 (1994).
[CrossRef]

ul-Haq, I.

Van Loan, C. F.

C. F. Van Loan, “The ubiquitous Kronecker product,” J. Comp. Appl. Math. 123, 85–100 (2000).
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996), pp. 257–258.

von Matt, U.

G. H. Golub and U. von Matt, “Generalized cross-validation for large scale problems,” in Proceedings of the Second International Workshop on Recent Advances in Total Least Squares Techniques and Errors-in-Variables Modeling, S.Van Huffel, ed. (Society for Industrial and Applied Mathematics, 1997), pp. 139–148.

Wahba, G.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Wheeler, F. W.

F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
[CrossRef]

Zalevsky, Z.

A. Borkowski, Z. Zalevsky, and B. Javidi, “Geometrical super resolved imaging using non periodic spatial masking,” J. Opt. Soc. Am. A 26, 589–601 (2009).
[CrossRef]

Z. Zalevsky, J. Solomon, and D. Mendlovic, “Geometrical super resolution using code division multiplexing,” Appl. Opt. 42, 32–40 (2005).

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Appl. Opt. (3)

Commun. Inf. Syst. (1)

H. Lev-Ari, “Efficient solution of linear matrix equations with application to multistatic antenna array processing,” Commun. Inf. Syst. 5, 123–130 (2005).

Found. Trends Commun. Inf. Theory (1)

R. M. Gray, “Toeplitz and circulant matrices: a review,” Found. Trends Commun. Inf. Theory 2, 155–239 (2005).
[CrossRef]

IEEE Trans. Image Process. (1)

R. C. Hardie, K. J. Barnard, and E. E. Armstrong, “Joint MAP registration and high-resolution image estimation using a sequence of undersampled images,” IEEE Trans. Image Process. 6, 1621–1633 (1997).
[CrossRef] [PubMed]

J. Comp. Appl. Math. (1)

C. F. Van Loan, “The ubiquitous Kronecker product,” J. Comp. Appl. Math. 123, 85–100 (2000).
[CrossRef]

J. Comput. Appl. Math. (1)

A. Bouhamidi and K. Jbilou, “Sylvester Tikhonov-regularization methods in image restoration,” J. Comput. Appl. Math. 206, 86–98 (2007).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Visual Commun. Image Represent (1)

M. Irani and S. Peleg, “Motion analysis for image enhancement: resolution, occlusion, and transparency,” J. Visual Commun. Image Represent 4, 324–335 (1993).
[CrossRef]

Opt. Commun. (1)

K. A. Nugent and B. Luther-Davies, “Penumbral imaging of high energy x-rays from laser-produced plasmas,” Opt. Commun. 49, 393–396 (1984).
[CrossRef]

Opt. Eng. (2)

Z. Zalevsky, D. Mendlovic, and E. Marom, “Special sensor masking for exceeding system geometrical resolving power,” Opt. Eng. 39, 1936–1942 (2000).
[CrossRef]

Z. Zalevsky, N. Shamir, and D. Mendlovic, “Geometrical super-resolution in infra-red sensor: experimental verification,” Opt. Eng. 43, 1401–1406 (2004).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

B. C. Tom and A. K. Katsaggelos, “Reconstruction of a high-resolution image from multiple-degraded misregistered low-resolution images,” Proc. SPIE 2308, 971–981 (1994).
[CrossRef]

F. W. Wheeler, R. T. Hoctor, and E. B. Barrett, “Super-resolution image synthesis using projections onto convex sets in the frequency domain,” Proc. SPIE , 5674, 479–490 (2005).
[CrossRef]

Technometrics (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Other (3)

G. H. Golub and U. von Matt, “Generalized cross-validation for large scale problems,” in Proceedings of the Second International Workshop on Recent Advances in Total Least Squares Techniques and Errors-in-Variables Modeling, S.Van Huffel, ed. (Society for Industrial and Applied Mathematics, 1997), pp. 139–148.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins, 1996), pp. 257–258.

P. Cheeseman, B. Kanefsky, R. Kraft, J. Stutz, and R. Hanson, “Super-resolved surface reconstruction from multiple images,” in Maximum Entropy and Bayesian Methods, G.R.Heidbreder, ed. (Kluwer, 1996), pp. 293–308.

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Figures (13)

Fig. 1
Fig. 1

(a) Random mask with a size of ( K M + M N 1 ) × ( L N ) . (b) Optical configuration including the binary mask. The mask has to be in an intermediate imaging position. The mask can be moved only in one direction to get different images each time.

Fig. 2
Fig. 2

Procedure of matrix generation. (a) Low-resolution matrix, K = 2 , L = 2 . (b) Latitudinal resolution of M = 2 . (c) Latitudinal and longitudinal resolution of M = 2 , N = 2 . (d) Duplicating rows by an SR factor of M N = 4 . (e) A mask size of ( K M + M N 1 ) × ( L N ) = 7 × 4 . The pixels of the mask that were on the first image shall be designated as s, and parts that are intended to be inserted will be marked by t. (f) Moving the mask down (single axis) after the first image for M N 1 times. (g) Representing each column of the mask as Toeplitz matrix. (h) The matrix after performing a Khatri–Rao product. (i) Fast computing: instead of computing the inverse matrix with a size of K L M N × K L M N , one needs to invert only K L matrices having the size of M N × M N .

Fig. 3
Fig. 3

Dependence of the algorithm’s performance with respect to the blocked percentage area for a random spatially distributed mask.

Fig. 4
Fig. 4

Simulation results depicting the dependence of the algorithm on the percentage of the blocked area for a random spatially distributed mask. Image size: 240 × 320 , SR factor = 8 × 8 , number of images taken during the process = 128 , noise variance = 0.0001 . (a) Reference image. (b) Low-resolution image. (c)–(f) Superresolved reconstruction when having mask with blocking area portion of 10%, 30%, 50%, and 80%, respectively.

Fig. 5
Fig. 5

Dependence of the algorithm and the runtime on the SR factor. Number of images taken during the process = 2 × SR factor , and the size of the ring image is 256 × 256 pixels. It has been obtained by performing 256 2 matrix inversions. The size of each matrix is 2 × SR factor rows , and SR factor columns.

Fig. 6
Fig. 6

Simulation results depicting the algorithm dependence on the SR factor. Image size: 256 × 256 , number of images taken during the process = 2 × SR factor , percentage of the image covered by the random mask = 50 % , noise variance = 0.001 . One may observe how, in the superresolved images, the high frequencies of the object are recovered while being fully undersampled in the LR images. (a) Reference image. Low-resolution images [(b), (d), (f), (h)] and their corresponding high-resolution reconstructions [(c), (e), (g), (i)] for an SR factor of 4 × 4 , 8 × 8 , 12 × 12 , and 16 × 16 , respectively.

Fig. 7
Fig. 7

Dependence of the reconstruction quality and time on the number of images taken during the reconstruction process, using the SVD with GCV regularization.

Fig. 8
Fig. 8

Demonstration of the dependence of the quality and time of the obtained reconstruction on the number of images taken in the process. Image size: 240 × 320 , SR factor = 8 × 8 . The percentage of the blocked area = 50 % , noise variance = 0.0001 . (a) High resolution reference. (b) Low-resolution image. (c)–(f) The obtained reconstruction when using 64, 128, 192, and 256 images, respectively.

Fig. 9
Fig. 9

Dependence of the quality of reconstruction on the number of bits.

Fig. 10
Fig. 10

Reconstruction results for various number of quantization bits. (a) The reference image. (b)–(d) The reference, LR, and reconstruction images with 2 quantization bits. (e)–(g) The same as (b)–(d) but with 4 quantization bits. (h)–(j) The same as (b)–(d) but with 6 quantization bits. SR factor = 88 , the number of images taken during the process = 128 , the percentage of the blocked area = 50 % , noise variance = 0.001 .

Fig. 11
Fig. 11

Effect of additive noise on the quality of the reconstructed image for different numbers of images taken during the process.

Fig. 12
Fig. 12

Simulation results showing the effect of the amount of noise on the quality of the reconstructed image for 512 images taken during the process. Image size: 240 × 320 , SR factor = 8 × 8 , the percentage of the blocked area = 50 % . (a) Reference image. (b) LR image. (c)–(f) The obtained reconstruction with noise variances of 0.0001, 0.001, 0.01, and 0.1, respectively.

Fig. 13
Fig. 13

Simulation results showing the effect of the amount of noise on the quality of the reconstructed, for Lena image when comparing the suggested algorithm with the approach from  [10]. 244 images were taken during the proposed process. Image size: 128 × 128 , SR factor = 12 × 12 . The percentage of the blocking area was 50%.

Equations (10)

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

w k , l = u ( x , y ) g ( x k Δ x , y l Δ y ) d x d y .
A 2 = I L I K [ 1 , 1 , , 1 M ] 1 M .
A 3 = I L [ 1 , 1 , , 1 N ] I K [ 1 , 1 , , 1 M ] 1 M N .
A 4 = A 3 D = I L [ 1 , 1 , , 1 N ] I K [ 1 , 1 , , 1 M ] D · 1 M N .
Mask= [ t M N 1 , 1 t M N 1 , 2 t M N 1 , n l t M N 1 , L N t M N 2 , 1 t M N 2 , 2 t M N 2 , n l t M N 2 , L N t m n , 1 t m n , 2 t m n , n l t m n , L N t 1 , 1 t 1 , 2 t 1 , L N s 1 , 1 s 1 , 2 s 1 , L N s 2 , 1 s 2 , 2 s 2 , L N s k m , 1 s k m , 2 s k m , n l s k m , L N s K M , 1 s K M , 2 s K M , n l s K M , L N ] .
s t l n = [ s 1 , l n s 2 , l n s k m , l n s K M 1 , l n s K M , l n t 1 , l n s 1 , l n s 2 , l n s k m , l n s K M 1 , l n t m n , l n s k m , l n t m n , l n t M N 2 , l n t 1 , l n s 1 , l n s 2 , l n s K M M N + 1 , l n t M N 1 , l n t M N 2 , l n t m n , l n t 1 , l n s 1 , l n s 2 , l n s K M M N + 1 , l n ] .
A 5 = A 3 S T = 1 M N { I L [ 1 , 1 , , 1 N ] I K [ 1 , 1 , , 1 M ] } S T ,
A 5 z = b .
A ^ z ^ = b ^ ,
I K = ( 1 0 0 1 ) I L = ( 1 0 0 1 ) ,

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