Abstract

The problem of image restoration of space variant blur is common and important. One of the most useful descriptions of this problem is in its algebraic form I=H∗O, where O is the object represented as a column vector, I is the blur image represented as a column vector and H is the PSF matrix that represents the optical system. When inverting the problem to restore the geometric object from the blurred image and the known system matrix, restoration is limited in speed and quality by the system condition. Current optical design methods focus on image quality, therefore if additional image processing is needed the matrix condition is taken “as is”. In this paper we would like to present a new optical approach which aims to improve the system condition by proper optical design. In this new method we use Singular Value Decomposition (SVD) to define the weak parts of the matrix condition. We design a second optical system based on those weak SVD parts and then we add the second system parallel to the first one. The original and second systems together work as an improved parallel optics system. Following that, we present a method for designing such a “parallel filter” for systems with a spread SVD pattern. Finally we present a study case in which by using our new method we improve a space variant image system with an initial condition number of 8.76e4, down to a condition number of 2.29e3. We use matrix inversion to simulate image restoration. Results show that the new parallel optics immunity to Additive White Gaussian Noise (AWGN) is much better then that of the original simple lens. Comparing the original and the parallel optics systems, the parallel optics system crosses the MSEIF=0 [db] limit in SNR value which is more than 50db lower then the SNR value in the case of the original simple lens. The new parallel optics system performance is also compared to another method based on the MTF approach.

©2009 Optical Society of America

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References

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  • Year
  • Author
  • Publication
  1. G.M. Robbins, “Inverse Filtering for linear shift variant imaging system”, Proc. IEEE,  60,862–872 (1972).
    [Crossref]
  2. H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE. Trans. ASSP. 26608–609 (1978).
    [Crossref]
  3. H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” Proc. IEEE. ASSP, 3,196–198 (1978).
  4. T.P. Costello and W.B. Mikhael, “Efficient restoration of space variant blurs from physical optics by sectioning with modified wiener filtering,” Digital and image processing, 13,1–22 (2003).
    [Crossref]
  5. J.M. Varah, “On the numerical solution of ill-conditioned linear system with applications to ill posed problems,” SIAM J. Numer. Anal, 10, 257–265 (1972).
    [Crossref]
  6. J.H. Wilkinson, “Rounding Errors in Algebraic Processes,” 91–93 (Her Majesty’s stationery office, 1963).
  7. G.H. Golub and C.F. Van-loan, “Matrix Computation,” 17–185 (North oxford academic,1983).
  8. C.L. Lawson and R.J. Hanson, “Solving Least Squares Problems,” SIAM J. Numer. Anal, 185–8 (1995)
  9. I.F. Gorodnitsky and D. Rao, “Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers,”  1, 25–29 (1994).
  10. C.M. Leung and W.S. Lu, “An L curve approach to optimal determination of regularization parameter in image restoration,” Proc. IEEE,1021–1024 (1993).
  11. X. Wang, “Effect of small rank modification on the condition number of matrix,” Computers and mathematics with applications 54,819–825 (2007).
    [Crossref]
  12. S. Twomay, “Information content in remote sensing,” Appl. Opt,  13, 942–945 (1974).
  13. M. Bertero and P. Boccacci, “Introduction to inverse problems in imaging,”  86,252 (IOP,1998).
  14. A. A. Sawchuk and M. J. Peyrovian, “Restoration of astigmatism and curvature of field,” J. Opt. Soc. Am. 65, 712–715 (1975).
    [Crossref]
  15. H.C. Andrews and C.L. Paterson, “Singular value Decomposition and digital image processing”, IEEE. Trans. ASSP. 24, 26–53 (1976).
    [Crossref]
  16. J.W. Goodman, “Introduction to Fourier Optics,” (Mcgraw-Hill, 1996).
  17. W.T. Welford, “Aberrations of Optical Systems,” (Adam-Hilger, 1991).
  18. V. Shaoulov et al, “Model of wide angle optical field propagation using scalar diffraction theory,” Opt. Eng,  43, 1561–1567 (2004).
    [Crossref]
  19. A.W. Lohmann and W.T. Rhodes, “Two pupil synthesis of optical transfer function,” Appl. Opt. 17, 1141–1146 (1978).
    [Crossref] [PubMed]
  20. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [Crossref] [PubMed]
  21. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877, (2001).
    [Crossref]
  22. S. Mezouari and A.R. Harvey, “Phase pupil function for reduction of defocus and spherical aberration,” Opt. Lett. 28, 771–773, (2003).
    [Crossref] [PubMed]
  23. S. Mezouari et al, “Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism,” J, Opt,.Soc.Am.A. 23,1058–1062 (2006).
    [Crossref]
  24. N.S. Kopeika, “A system Engineering approach to imaging,” 517–520 (SPIE, 1998).

2007 (1)

X. Wang, “Effect of small rank modification on the condition number of matrix,” Computers and mathematics with applications 54,819–825 (2007).
[Crossref]

2006 (1)

S. Mezouari et al, “Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism,” J, Opt,.Soc.Am.A. 23,1058–1062 (2006).
[Crossref]

2004 (1)

V. Shaoulov et al, “Model of wide angle optical field propagation using scalar diffraction theory,” Opt. Eng,  43, 1561–1567 (2004).
[Crossref]

2003 (2)

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

T.P. Costello and W.B. Mikhael, “Efficient restoration of space variant blurs from physical optics by sectioning with modified wiener filtering,” Digital and image processing, 13,1–22 (2003).
[Crossref]

2001 (1)

1998 (1)

M. Bertero and P. Boccacci, “Introduction to inverse problems in imaging,”  86,252 (IOP,1998).

1995 (1)

1994 (1)

I.F. Gorodnitsky and D. Rao, “Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers,”  1, 25–29 (1994).

1978 (3)

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE. Trans. ASSP. 26608–609 (1978).
[Crossref]

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” Proc. IEEE. ASSP, 3,196–198 (1978).

A.W. Lohmann and W.T. Rhodes, “Two pupil synthesis of optical transfer function,” Appl. Opt. 17, 1141–1146 (1978).
[Crossref] [PubMed]

1976 (1)

H.C. Andrews and C.L. Paterson, “Singular value Decomposition and digital image processing”, IEEE. Trans. ASSP. 24, 26–53 (1976).
[Crossref]

1975 (1)

1974 (1)

S. Twomay, “Information content in remote sensing,” Appl. Opt,  13, 942–945 (1974).

1972 (2)

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

J.M. Varah, “On the numerical solution of ill-conditioned linear system with applications to ill posed problems,” SIAM J. Numer. Anal, 10, 257–265 (1972).
[Crossref]

Andrews, H.C.

H.C. Andrews and C.L. Paterson, “Singular value Decomposition and digital image processing”, IEEE. Trans. ASSP. 24, 26–53 (1976).
[Crossref]

Bertero, M.

M. Bertero and P. Boccacci, “Introduction to inverse problems in imaging,”  86,252 (IOP,1998).

Boccacci, P.

M. Bertero and P. Boccacci, “Introduction to inverse problems in imaging,”  86,252 (IOP,1998).

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 and image processing, 13,1–22 (2003).
[Crossref]

Dowski, E. R.

George, N.

Golub, G.H.

G.H. Golub and C.F. Van-loan, “Matrix Computation,” 17–185 (North oxford academic,1983).

Goodman, J.W.

J.W. Goodman, “Introduction to Fourier Optics,” (Mcgraw-Hill, 1996).

Gorodnitsky, I.F.

I.F. Gorodnitsky and D. Rao, “Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers,”  1, 25–29 (1994).

Hanson, R.J.

C.L. Lawson and R.J. Hanson, “Solving Least Squares Problems,” SIAM J. Numer. Anal, 185–8 (1995)

Harvey, A.R.

Hunt, B.

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” Proc. IEEE. ASSP, 3,196–198 (1978).

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE. Trans. ASSP. 26608–609 (1978).
[Crossref]

Kopeika, N.S.

N.S. Kopeika, “A system Engineering approach to imaging,” 517–520 (SPIE, 1998).

Lawson, C.L.

C.L. Lawson and R.J. Hanson, “Solving Least Squares Problems,” SIAM J. Numer. Anal, 185–8 (1995)

Leung, C.M.

C.M. Leung and W.S. Lu, “An L curve approach to optimal determination of regularization parameter in image restoration,” Proc. IEEE,1021–1024 (1993).

Lohmann, A.W.

Lu, W.S.

C.M. Leung and W.S. Lu, “An L curve approach to optimal determination of regularization parameter in image restoration,” Proc. IEEE,1021–1024 (1993).

Mezouari, S.

S. Mezouari et al, “Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism,” J, Opt,.Soc.Am.A. 23,1058–1062 (2006).
[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]

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 and image processing, 13,1–22 (2003).
[Crossref]

Paterson, C.L.

H.C. Andrews and C.L. Paterson, “Singular value Decomposition and digital image processing”, IEEE. Trans. ASSP. 24, 26–53 (1976).
[Crossref]

Peyrovian, M. J.

Rao, D.

I.F. Gorodnitsky and D. Rao, “Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers,”  1, 25–29 (1994).

Rhodes, W.T.

Robbins, G.M.

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

Sawchuk, A. A.

Shaoulov, V.

V. Shaoulov et al, “Model of wide angle optical field propagation using scalar diffraction theory,” Opt. Eng,  43, 1561–1567 (2004).
[Crossref]

Trussell, H.

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE. Trans. ASSP. 26608–609 (1978).
[Crossref]

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” Proc. IEEE. ASSP, 3,196–198 (1978).

Twomay, S.

S. Twomay, “Information content in remote sensing,” Appl. Opt,  13, 942–945 (1974).

Van-loan, C.F.

G.H. Golub and C.F. Van-loan, “Matrix Computation,” 17–185 (North oxford academic,1983).

Varah, J.M.

J.M. Varah, “On the numerical solution of ill-conditioned linear system with applications to ill posed problems,” SIAM J. Numer. Anal, 10, 257–265 (1972).
[Crossref]

Wang, X.

X. Wang, “Effect of small rank modification on the condition number of matrix,” Computers and mathematics with applications 54,819–825 (2007).
[Crossref]

Welford, W.T.

W.T. Welford, “Aberrations of Optical Systems,” (Adam-Hilger, 1991).

Wilkinson, J.H.

J.H. Wilkinson, “Rounding Errors in Algebraic Processes,” 91–93 (Her Majesty’s stationery office, 1963).

Appl. Opt (1)

S. Twomay, “Information content in remote sensing,” Appl. Opt,  13, 942–945 (1974).

Appl. Opt. (2)

Computers and mathematics with applications (1)

X. Wang, “Effect of small rank modification on the condition number of matrix,” Computers and mathematics with applications 54,819–825 (2007).
[Crossref]

Digital and image processing (1)

T.P. Costello and W.B. Mikhael, “Efficient restoration of space variant blurs from physical optics by sectioning with modified wiener filtering,” Digital and image processing, 13,1–22 (2003).
[Crossref]

IEEE. Trans. ASSP. (2)

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE. Trans. ASSP. 26608–609 (1978).
[Crossref]

H.C. Andrews and C.L. Paterson, “Singular value Decomposition and digital image processing”, IEEE. Trans. ASSP. 24, 26–53 (1976).
[Crossref]

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

S. Mezouari et al, “Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism,” J, Opt,.Soc.Am.A. 23,1058–1062 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng (1)

V. Shaoulov et al, “Model of wide angle optical field propagation using scalar diffraction theory,” Opt. Eng,  43, 1561–1567 (2004).
[Crossref]

Opt. Lett. (2)

Proc. IEEE (1)

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

Proc. IEEE. ASSP (1)

H. Trussell and B. Hunt, “Image restoration of space-variant blurs by sectioned methods,” Proc. IEEE. ASSP, 3,196–198 (1978).

SIAM J. Numer. Anal (1)

J.M. Varah, “On the numerical solution of ill-conditioned linear system with applications to ill posed problems,” SIAM J. Numer. Anal, 10, 257–265 (1972).
[Crossref]

Other (9)

J.H. Wilkinson, “Rounding Errors in Algebraic Processes,” 91–93 (Her Majesty’s stationery office, 1963).

G.H. Golub and C.F. Van-loan, “Matrix Computation,” 17–185 (North oxford academic,1983).

C.L. Lawson and R.J. Hanson, “Solving Least Squares Problems,” SIAM J. Numer. Anal, 185–8 (1995)

I.F. Gorodnitsky and D. Rao, “Analysis of Error Produced by Truncated SVD and TIkhonov Regularization Methods, Conference Record of the Twenty-Eighth Asilomar Conference on Signals, Systems and Computers,”  1, 25–29 (1994).

C.M. Leung and W.S. Lu, “An L curve approach to optimal determination of regularization parameter in image restoration,” Proc. IEEE,1021–1024 (1993).

J.W. Goodman, “Introduction to Fourier Optics,” (Mcgraw-Hill, 1996).

W.T. Welford, “Aberrations of Optical Systems,” (Adam-Hilger, 1991).

M. Bertero and P. Boccacci, “Introduction to inverse problems in imaging,”  86,252 (IOP,1998).

N.S. Kopeika, “A system Engineering approach to imaging,” 517–520 (SPIE, 1998).

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Figures (8)
Fig. 1.
Fig. 1. A hybrid system includes the following blocks.

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Fig. 2.
Fig. 2. Parallel Optics Lens divided into “lens” zone and “Rim-ring” zone.

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Fig. 3.
Fig. 3. SVD and O filter information. a) the SVD, singular value of H. b) zoom in to the last 6 singular values which compose the BMSD. c) PSF shape of the BMSD after translating it back into a 2D image (for convenience, we show only every fourth column), every small rectangular is the whole FOV, We see that the PSFs across the FOV are very spread.

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Fig. 4.
Fig. 4. Optical information a) The parallel design cross-action b) Illustration of the parallel design cross-action c) OSLO simulation ray trace and Seidel sums used for the simulation.

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Fig. 5.
Fig. 5. The average value of MSEIF in systems with and without “rim-ring”, for each SNR value we average an ensemble of 10,000 measurements.

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Fig. 6.
Fig. 6. imaging simulations, in row 1 SNR=133 [db]: a) simple lens image b) simple lens image restoration in SNR=133 c) object 1: d) Parallel optics restoration in SNR=133 e) Parallel optics image. In row 2: SNR=110 [db]: f) simple lens image g) simple lens image restoration in SNR=110 h) object 1: i) Parallel optics restoration in SNR=110 j) Parallel optics image.

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Fig. 7.
Fig. 7. ensemble of MSEIF Simple lens restoration In circles, lens +rim-ring in cross. Lens+QF in diamonds. Each point is an average of 3600 realization.

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Fig. 8.
Fig. 8. Ensemble imaging simulations in SNR=130 [db], column 1 is the bare lens optical image, column 2 is the bare lens restoration, column 3 is the object, column 4 is the parallel optics restoration, column 5 is the parallel optics image, column 6 is the lens + quadratic filter restoration,column 7 is the lens + quadratic filter image. In each line we present different object.

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Tables (1)

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Table 1. - simulation log -influence of design parameters on figure of merits

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Equations (63)

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A . x =b
k2 (A)=σ1⁄ σn
∥δx∥2∥x∥2
∥ δ x ∥2 =∥ xperfect −xrestored∥2
A . x ≈ b
r=A . x −b
E =min ∥ A . x −b ∥2
xLS =∑i=1ruit.bσi . v
ELS2 =∑i=r+1m(uit.b)2
∥xLS−x∥2∥x∥2 ≤ε . {2.k2(A)cos(θ)+tan(θ).(k2(A))2}+ (error)2
ε=max{∥δA∥2∥A∥2,∥δb∥2∥b∥2}
sin (θ)=ELS∥b∥2
A(k) . x(k) =b(k)
b(k) =∑i=1kβiui
∥ x˜(k) − xLS ∥2 ≤K2η1σk +[∑i=k+1n(βiσi)2]0.5
emin2 =∥ A . xλ −b∥22 +λ ∥ b . f . xλ ∥22
xλ =∑i=1mσi2σi2+λi2. uit.bσi . vi
∥ xLS −xλ ∥2 =∑i=1m(λi2σi2+λi2.uit.bσi)2
k2(A)
AREMF =∣δx∣2⁄∣x∣2∣n∣2⁄∣b∣2 =1p . (∑i=1pσi2). (∑i=1pσi−2)
ILx1image =HLxL . ILx1object
(l) =(i−1) . N +j
H=U.S.Vt
H . Ht =U . Δ . Ut
Ht . H =V . Δ . Vt
ULxL =[u1,u2,..uL]
VLxL =[v1,v2,..,vL]
SLxL =[σ100000σ2000000000000000σL]
MLxLi =ui . vit
H−1 =V . S−1 . Ut
S 1 =S+Δ S
H 1 =U.S1.Vt =U.(S+ΔS) . Vt =U . S . Vt +U . Δ S . Vt
O =U.Δ S . Vt
h˜ (ximg,yimg) =∫ ∫ P̂ (λSimgx˜,λSimgy˜).exp (−j2π(ximgx˜,yimgy˜)) d x˜ d y˜
( x˜ =xp ⁄ λ . Simg , y˜=y p ⁄ λ . Simg )
P̂ (λSimgx˜,λSimgy˜)=P (λSimgx˜,λSimgy˜) . exp (jKW(λSimgx˜,λSimgy˜))
h˜tot ximgyimg =h˜lensximgyimg+h˜rim−ringximgyimg
PSF ximgyimg =∣ h˜tot ximgyimg ∣2 =∣h˜lens∣2+∣h˜rimring∣2 +h˜lens hrimring*˜ +hrimring˜ hlens*˜
{∥PSF−(∣hlens˜∣2+∣h˜rimring∣2)∥2⁄∥PSF∥2≤0.1∣∀PSF⊂FOV}
Waberrations xpypη =18 . S 1 . (xp2+yp2)2hp2 +12 . S 2 . yp.(xp2+yp2)hp3 . ηηmax
+12 . S 3. yp2hp2 . η2ηmax2 +14 . (S3+S4) . xp2+yp2hp2 . η2ηmax2 +12 . S 5. yphp . η3ηmax3
P˜Rim−Ring =P.exp(j.K.(WO−filter+Waberrations))
BMSD =∑iMLxLi , i ⊂ mis sin g −eigenmatrix
PR=Power_Rim−ringpower_lens ≈︸Rlens>>ΔRrim 2.ΔRrim(Tlens)2·Rlens
Δ Rrim =0.5 . (Tlens)2 . Rlens . PR
h˜tot ximgyimg =Tlens. h˜lens ximgyimg;Alens +TRim−ring . h˜Rim−ring ( ximg yimg ; ARim−ring )
tan (β) =0.5(Dlens+ΔRrim)Si
R =Si⁄cos(β)
−Rn . ∂W(xprim,yprim) ∣max =Rim . cos (β)
−∂W(xprim,yprim)∂r ∣max =Rim.cos(β)R =Rim.cos2(β)Si
W (xprim,yprim) =Sshape . (xprim2+yprim2)ΔRrim2 = S shape . (Δrrim2)ΔRrim2
−∂W(xprim,yprim)∂r ∣ = 2 . Sshape.ΔrrimΔRrim2
−∂W(xprim,yprim)∂r ∣ = 2 . Sshape.ΔrrimΔRrim2
2.SshapeΔRrim =Rim.cos2(β)Si
S shape =Rim.ΔRim.cos2(β)2.Si ⇔ R im =2.Sshape.SiΔRrim.cos2(β)
ÎLx1res =HLxL−1 . ILx1image
MSEIF =20.log(∥INy×Nximage−INy×Nxobject∥2∥INy×Nxres−INy×Nxobject∥2)
Φ (r)=0.0248(r2−0.5)2+0.0161 (r2−0.5)
R P T =Pparallel−optPlens=87.5168.59 =1.276
R P R =Deffective2Dlens2 ⇒ Deffective =RPR.Dlens2 =1.276 0.04 m m =0.452mm
N Aeff =sin(atan(RPR.Deffective2.Si))=0.347
Ptransfer_to_image =PLens.TLens2 +TRim−ring . {P2−PLens}
Pcross_proudct =sum (matrix) −PLens . Tlens2 −TRim_ring . {P2−Plens}

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