Abstract

Simple optical imaging systems tend to suffer from a space variant (SV) blur and ill matrix condition and, thus, tend to amplify additive noise in the essential stage of image restoration. Previously, we showed that the matrix condition of systems with SV blur can be improved by adding a system-tailored parallel auxiliary system. We introduced such a solution with the “trajectories” method, where we used auxiliary optics with a pixel confined point spread function to decompose the required auxiliary system. In this paper, by removing the pixel confined requirement, we extend the trajectories to a “blurred trajectories” method, which relies on the more common case of auxiliary optics with blurred response. The method is simulated and shown to be effective in two cases. First, we show that the matrix condition is significantly improved. In one case the condition number of a space variant system is reduced from 87640 down to 1212. In a second case of a highly defocused system, the matrix condition number is reduced from 6412.5 to 238.7. We then investigate the influence of the improvement in the matrix condition on image restoration by regularization with and without the auxiliary system. Blurred trajectories with regularization yields better restoration than regularization only. The new (to our knowledge) system is compared to other previously suggested optical designs. The method’s flexibility is demonstrated when applied as postprocessing on a system that includes the original ill-conditioned system and a quartic phase filter and yields an improvement in the overall matrix condition.

© 2011 Optical Society of America

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References

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  1. A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
    [CrossRef] [PubMed]
  2. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  3. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
    [CrossRef]
  4. S. Mezouari and A. R. Harvey, “Phase pupil function for reduction of defocus and spherical aberration,” Opt. Lett. 28, 771–773(2003).
    [CrossRef] [PubMed]
  5. S. Mezouari, G. Moyo, and A. R. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A 23, 1058–1062 (2006).
    [CrossRef]
  6. G. M. Robbins, “Inverse filtering for linear shift variant imaging system,” Proc. IEEE 60, 862–872 (1972).
    [CrossRef]
  7. H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust. Speech Signal Process. 26, 608–609 (1978).
    [CrossRef]
  8. H. Trussell and B. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) ’78, Vol. 3 (IEEE, 1978), pp. 196–198.
  9. T. P. Costello and W. B. Mikhael, “Efficient restoration of space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1–22 (2003).
    [CrossRef]
  10. P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006), Chap. 1.4.
  11. I. Klapp and D. Mendlovic, “Improvement of matrix condition of hybrid, space variant optics by the means of parallel optics design,” Opt. Express 17, 11673–11689 (2009).
    [CrossRef] [PubMed]
  12. I. Klapp, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel, and D. Mendlovic, “Trajectories in parallel optics” (in preparation).
  13. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 86, 252.
  14. I. Klapp and D. Mendlovic, “Parallel optics for improving system matrix condition,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CWB3.
  15. S. G. Lipson and H. Lipson, Optical Physics, 2nd ed. (Cambridge University, 1981), p. 170.

2009 (1)

2006 (1)

2003 (2)

T. P. Costello and W. B. Mikhael, “Efficient restoration of space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1–22 (2003).
[CrossRef]

S. Mezouari and A. R. Harvey, “Phase pupil function for reduction of defocus and spherical aberration,” Opt. Lett. 28, 771–773(2003).
[CrossRef] [PubMed]

2001 (1)

1995 (1)

1978 (2)

A. W. Lohmann and W. T. Rhodes, “Two-pupil synthesis of optical transfer functions,” Appl. Opt. 17, 1141–1151 (1978).
[CrossRef] [PubMed]

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust. Speech Signal Process. 26, 608–609 (1978).
[CrossRef]

1972 (1)

G. M. Robbins, “Inverse filtering for linear shift variant imaging system,” Proc. IEEE 60, 862–872 (1972).
[CrossRef]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 86, 252.

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 86, 252.

Cathey, W. T.

Chi, W.

Costello, T. P.

T. P. Costello and W. B. Mikhael, “Efficient restoration of space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1–22 (2003).
[CrossRef]

Dowski, E. R.

George, N.

Hansen, P. C.

P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006), Chap. 1.4.

Harvey, A. R.

Hunt, B.

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust. Speech Signal Process. 26, 608–609 (1978).
[CrossRef]

H. Trussell and B. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) ’78, Vol. 3 (IEEE, 1978), pp. 196–198.

Klapp, I.

I. Klapp and D. Mendlovic, “Improvement of matrix condition of hybrid, space variant optics by the means of parallel optics design,” Opt. Express 17, 11673–11689 (2009).
[CrossRef] [PubMed]

I. Klapp and D. Mendlovic, “Parallel optics for improving system matrix condition,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CWB3.

I. Klapp, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel, and D. Mendlovic, “Trajectories in parallel optics” (in preparation).

Lipson, H.

S. G. Lipson and H. Lipson, Optical Physics, 2nd ed. (Cambridge University, 1981), p. 170.

Lipson, S. G.

S. G. Lipson and H. Lipson, Optical Physics, 2nd ed. (Cambridge University, 1981), p. 170.

Lohmann, A. W.

Mendlovic, D.

I. Klapp and D. Mendlovic, “Improvement of matrix condition of hybrid, space variant optics by the means of parallel optics design,” Opt. Express 17, 11673–11689 (2009).
[CrossRef] [PubMed]

I. Klapp and D. Mendlovic, “Parallel optics for improving system matrix condition,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CWB3.

I. Klapp, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel, and D. Mendlovic, “Trajectories in parallel optics” (in preparation).

Mezouari, S.

Mikhael, W. B.

T. P. Costello and W. B. Mikhael, “Efficient restoration of space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1–22 (2003).
[CrossRef]

Moyo, G.

Nagy, J. G.

P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006), Chap. 1.4.

O’Leary, D. P.

P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006), Chap. 1.4.

Rhodes, W. T.

Robbins, G. M.

G. M. Robbins, “Inverse filtering for linear shift variant imaging system,” Proc. IEEE 60, 862–872 (1972).
[CrossRef]

Trussell, H.

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust. Speech Signal Process. 26, 608–609 (1978).
[CrossRef]

H. Trussell and B. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) ’78, Vol. 3 (IEEE, 1978), pp. 196–198.

Appl. Opt. (2)

Digital Signal Process. (1)

T. P. Costello and W. B. Mikhael, “Efficient restoration of space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1–22 (2003).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust. Speech Signal Process. 26, 608–609 (1978).
[CrossRef]

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

Opt. Express (1)

Opt. Lett. (2)

Proc. IEEE (1)

G. M. Robbins, “Inverse filtering for linear shift variant imaging system,” Proc. IEEE 60, 862–872 (1972).
[CrossRef]

Other (6)

H. Trussell and B. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) ’78, Vol. 3 (IEEE, 1978), pp. 196–198.

I. Klapp, Tel Aviv University, Ramat-Aviv, Tel Aviv 69978, Israel, and D. Mendlovic, “Trajectories in parallel optics” (in preparation).

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (IOP, 1998), pp. 86, 252.

I. Klapp and D. Mendlovic, “Parallel optics for improving system matrix condition,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CWB3.

S. G. Lipson and H. Lipson, Optical Physics, 2nd ed. (Cambridge University, 1981), p. 170.

P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering (SIAM, 2006), Chap. 1.4.

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Figures (15)

Fig. 1
Fig. 1

A hybrid system includes these blocks.

Fig. 2
Fig. 2

Hybrid optical and image processing parallel optics imaging system. The ideal restored image designate (Res) will be close to the object quality (Obj).

Fig. 3
Fig. 3

Trajectories method scheme for PO system implementation; the auxiliary system is composed of auxiliary optics imaging and a series of shifted and weighted operations that are done by postprocessing. The ideal restored image will be close to the object quality (Obj).

Fig. 4
Fig. 4

Relation between the image shift and the trajectories matrix in a 6 × 6 matrix. In the top left we present the 3x2 “chessboard” object (Obj). The shifted images that associate with ( Δ n , Δ m ) = ( 0 , 0 ) , ( 1 , 0 ) , ( + 1 , 0 ) are designated Img1, Img2, and Img3, respectively.

Fig. 5
Fig. 5

Trajectories for the 6 × 6 PSF matrix for 3 × 2 FOV. (a) Trajectories assuming pixel confined auxiliary lens, (b) blurred trajectories with a blurred auxiliary lens. The kernel function is in the bottom right of each figure.

Fig. 6
Fig. 6

Postprocessing step: in the 3 × 2 field there are 15 available translations. (a) The trajectories results, (b) the blurred trajectories results. The 15 weighted and shifted images of the 3 × 2 chessboard object are related.

Fig. 7
Fig. 7

Restoration results for a 3 × 2 chessboard object under noise. The object gray level is in blue asterisks designated as “obj,” restoration of parallel optics by trajectories method (PO confined) is in brown rings, restoration of parallel optics by blurred trajectories method (PO blurred) is in red triangles, and main system restoration (main) is in pink rectangles.

Fig. 8
Fig. 8

Real world blurred trajectories system illustration.

Fig. 9
Fig. 9

Objects ensemble.

Fig. 10
Fig. 10

Restoration average MSEIF ensemble results of space variant study case. In blue diamonds, the study case (SC) “as is” without additional optical design; in magenta rectangles, the rim-ring restoration; in orange triangles, the trajectories with blurred auxiliary system (Traj). Image restoration is done by regularization. The auxiliary image (AUX) is in black crosses.

Fig. 11
Fig. 11

Typical images in SNR = 45 [ db ] of space variant study case: study case image (SC-Img.), auxiliary lens image (Aux-Img.), object (Obj.), study case restoration (SC-Res.), rim-ring restoration (RR-Res.), blur trajectories restoration (Traj-Res.).

Fig. 12
Fig. 12

Main system for a deep defocused imaging system.

Fig. 13
Fig. 13

Main system singular values graph and the chosen BMSD matrix.

Fig. 14
Fig. 14

Comparison between restoration average MSEIF ensemble results of deep defocused study case. Ensemble results of trajectories with blurred auxiliary system (Traj) are in orange triangles, the study case restoration (SC) is in blue diamonds (as is without additional optics), and the rim-ring solution is in magenta rectangles. Image restoration is done by regularization. The auxiliary optics (AUX) is in black crosses.

Fig. 15
Fig. 15

Typical images in SNR = 45 [ db ] of deep defocused study case: study case image (SC-Img.), auxiliary lens image (Aux-Img.), object (Obj.), study case regularization restoration (SC-Res.), rim-ring restoration (RR-Res.), blur trajectories restoration (Traj-Res.).

Equations (21)

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I L × 1 image = H L × L · I L × 1 object ,
k 2 ( A ) = σ 1 / σ n ,
H = σ i · M i ,
O = i Ω Δ σ i · M i σ i Δ σ i .
H 1 = H + O .
BMSD = i Ω M i .
I L × 1 main = H L × L · I L × 1 object + I L × 1 noise 1 .
I L × 1 aux = O L × L aux · I L × 1 object + I L × 1 noise 2 .
( l in ) = ( l out ) = ( m 1 ) · N n + n .
( l in ) ( l out ) = ( m 1 ) · N n + n ( m + Δ m 1 ) · N n + n + Δ n .
O ^ l ( i , j ; Δ m , Δ n ) = O l ( i , j ; Δ m , Δ n ) · H aux ( i , j ) .
M l = i = 1 L j = 1 L O l ( i , j ; Δ m , Δ n ) · H aux ( i , j ) .
BMSD O = l = 1 Q W l · O ^ l .
W l = 1 M l i = 1 L j = 1 L BMSD ( i , j ) O ^ l ( i , j ; Δ m , Δ n ) .
I L × 1 aux _ sys = i = 1 Q W l · O l · I L × 1 aux = i = 1 Q W l · O ^ l · I L × 1 object + I L × 1 noise .
H 1 = H + l = 1 Q W l · O ^ l .
I L × 1 image = H L × L · I L × 1 object + I L × 1 noise 1 + l = 1 Q W l · O ^ l · I L × 1 object + I L × 1 noise .
MSEIF = 20 · log e ( I L × 1 image I L × 1 object 2 I L × 1 res I L × 1 object 2 ) .
I ^ L × 1 res = ( H 1 t · H 1 + α · I ) 1 · H 1 · I L × 1 image ,
PSF = 1 11 · [ 1 1 1 1 3 1 1 1 1 ] .
D airy = 1.22 λ · S img 0.5 · D app D app = 1.22 λ · S img 0.5 · D airy 0.03 [ mm ] .

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