Abstract

In our previous work we showed the ability to improve the optical system’s matrix condition by optical design, thereby improving its robustness to noise. It was shown that by using singular value decomposition, a target point-spread function (PSF) matrix can be defined for an auxiliary optical system, which works parallel to the original system to achieve such an improvement. In this paper, after briefly introducing the all optics implementation of the auxiliary system, we show a method to decompose the target PSF matrix. This is done through a series of shifted responses of auxiliary optics (named trajectories), where a complicated hardware filter is replaced by postprocessing. This process manipulates the pixel confined PSF response of simple auxiliary optics, which in turn creates an auxiliary system with the required PSF matrix. This method is simulated on two space variant systems and reduces their system condition number from 18,598 to 197 and from 87,640 to 5.75, respectively. We perform a study of the latter result and show significant improvement in image restoration performance, in comparison to a system without auxiliary optics and to other previously suggested hybrid solutions. Image restoration results show that in a range of low signal-to-noise ratio values, the trajectories method gives a significant advantage over alternative approaches. A third space invariant study case is explored only briefly, and we present a significant improvement in the matrix condition number from 1.9160e+013 to 34,526.

© 2011 Optical Society of America

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References

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  1. 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]
  2. J. H. Wilkinson, Rounding Errors in Algebraic Processes (Her Majesty’s Stationery Office, 1963), Chap. 3, p. 91.
  3. G. H. Golub and C. F. Van-loan, Matrix Computation (North Oxford Academic, 1983).
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    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
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    [CrossRef] [PubMed]
  8. P. C. Hansen and J. G. Nagy, Deblurring Images Matrices, Spectra, and Filtering (SIAM, 2006).
  9. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
  10. R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. M. G. Lofdahl, “Multiframe deconvolution with space variant point spread function by use of inverse filtering and fast Fourier transform,” Appl. Opt. 46, 4686–4693 (2007).
    [CrossRef] [PubMed]
  13. J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).
  14. R. G. Paxman and J. H. Seldin, “Phase-diversity data set and processing strategies,” in High Resolution Solar Physics: Theory, Observations, and Techniques, T.R.Rimmele, ed., ASP Conference Series (Astronomical Society of the Pacific, 1999), Vol.  183.
  15. M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi aperture systems utilizing individual sub aperture control,” Proc. SPIE 5896, 58960G (2005).
    [CrossRef]
  16. T. C. Zaugg and R. G. Paxman, “Complementary compensation in the presence of fixed aberrations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper SMB5.
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  17. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  18. S. Mezouari and A. R. Harvey, “Phase pupil function for reduction of defocus and spherical aberration,” Opt. Lett. 28, 771–773(2003).
    [CrossRef] [PubMed]
  19. S. Mezouari, G. Moyo, and A. R. Harvey, “Circularly symmetric phase filter for control of primary third order aberration: coma and astigmatism,” J. Opt. Soc. Am. A. 23, 1058–1062(2006).
    [CrossRef]
  20. 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.
  21. I. Klapp and D. Mendlovic, “Optical design for improving matrix condition,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA7.
  22. I. Klapp and D. Mendlovic, “Imaging system and method for imaging object with reduced image blur,” U.S. patent WO/2010/103527 A2 (March 13, 2009).
  23. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1991).
  24. M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
    [CrossRef]
  25. I. Klapp and D. Mendlovic, “Trajectories by a blurred auxiliary system,” J. Opt. Soc. Am. A 28, 1796–1802 (2011).
    [CrossRef]

2011

2009

2007

2006

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

2005

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi aperture systems utilizing individual sub aperture control,” Proc. SPIE 5896, 58960G (2005).
[CrossRef]

2003

1995

1994

1992

1986

1982

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

1978

1976

H. C. Andrews and C. L. Paterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 24, 26–53 (1976).
[CrossRef]

Andrews, H. C.

H. C. Andrews and C. L. Paterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 24, 26–53 (1976).
[CrossRef]

Bardsley, J.

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

Bolcar, M. R.

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi aperture systems utilizing individual sub aperture control,” Proc. SPIE 5896, 58960G (2005).
[CrossRef]

Cathey, W. T.

Dowski, E. R.

Fienup, J. R.

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi aperture systems utilizing individual sub aperture control,” Proc. SPIE 5896, 58960G (2005).
[CrossRef]

R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van-loan, Matrix Computation (North Oxford Academic, 1983).

Gonsalves, R. A.

R. A. Gonsalves, “Nonisoplanatic imaging by phase diversity,” Opt. Lett. 19, 493–495 (1994).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Hansen, P. C.

P. C. Hansen and J. G. Nagy, Deblurring Images Matrices, Spectra, and Filtering (SIAM, 2006).

Harvey, A. R.

S. Mezouari, G. Moyo, and A. R. Harvey, “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]

Jefferies, S.

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).

Klapp, I.

I. Klapp and D. Mendlovic, “Trajectories by a blurred auxiliary system,” J. Opt. Soc. Am. A 28, 1796–1802 (2011).
[CrossRef]

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 and D. Mendlovic, “Optical design for improving matrix condition,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA7.

I. Klapp and D. Mendlovic, “Imaging system and method for imaging object with reduced image blur,” U.S. patent WO/2010/103527 A2 (March 13, 2009).

Lofdahl, M. G.

Lohmann, A. W.

Mait, J. N.

Mendlovic, D.

I. Klapp and D. Mendlovic, “Trajectories by a blurred auxiliary system,” J. Opt. Soc. Am. A 28, 1796–1802 (2011).
[CrossRef]

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 and D. Mendlovic, “Imaging system and method for imaging object with reduced image blur,” U.S. patent WO/2010/103527 A2 (March 13, 2009).

I. Klapp and D. Mendlovic, “Optical design for improving matrix condition,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA7.

Mezouari, S.

S. Mezouari, G. Moyo, and A. R. Harvey, “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]

Moyo, G.

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

Nagy, J.

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).

Nagy, J. G.

P. C. Hansen and J. G. Nagy, Deblurring Images Matrices, Spectra, and Filtering (SIAM, 2006).

Paterson, C. L.

H. C. Andrews and C. L. Paterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 24, 26–53 (1976).
[CrossRef]

Paxman, R. G.

R. G. Paxman, T. J. Schultz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman and J. H. Seldin, “Phase-diversity data set and processing strategies,” in High Resolution Solar Physics: Theory, Observations, and Techniques, T.R.Rimmele, ed., ASP Conference Series (Astronomical Society of the Pacific, 1999), Vol.  183.

T. C. Zaugg and R. G. Paxman, “Complementary compensation in the presence of fixed aberrations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper SMB5.
[PubMed]

Plemmons, R.

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).

Rhodes, W. T.

Schultz, T. J.

Seldin, J. H.

R. G. Paxman and J. H. Seldin, “Phase-diversity data set and processing strategies,” in High Resolution Solar Physics: Theory, Observations, and Techniques, T.R.Rimmele, ed., ASP Conference Series (Astronomical Society of the Pacific, 1999), Vol.  183.

Van-loan, C. F.

G. H. Golub and C. F. Van-loan, Matrix Computation (North Oxford Academic, 1983).

Welford, W. T.

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

Wilkinson, J. H.

J. H. Wilkinson, Rounding Errors in Algebraic Processes (Her Majesty’s Stationery Office, 1963), Chap. 3, p. 91.

Zaugg, T. C.

T. C. Zaugg and R. G. Paxman, “Complementary compensation in the presence of fixed aberrations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper SMB5.
[PubMed]

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

H. C. Andrews and C. L. Paterson, “Singular value decomposition and digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 24, 26–53 (1976).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

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

Opt. Eng.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).

Opt. Express

Opt. Lett.

Proc. SPIE

M. R. Bolcar and J. R. Fienup, “Method of phase diversity in multi aperture systems utilizing individual sub aperture control,” Proc. SPIE 5896, 58960G (2005).
[CrossRef]

Other

T. C. Zaugg and R. G. Paxman, “Complementary compensation in the presence of fixed aberrations,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings, Technical Digest (CD) (Optical Society of America, 2005), paper SMB5.
[PubMed]

J. Bardsley, S. Jefferies, J. Nagy, and R. Plemmons, “Blind iterative restoration of image with spatially-varying blur,” presented at the Advanced Maui Optical and Space Surveillance Technologies Conference, Maui, Hawaii, (September 5–9, 2005).

R. G. Paxman and J. H. Seldin, “Phase-diversity data set and processing strategies,” in High Resolution Solar Physics: Theory, Observations, and Techniques, T.R.Rimmele, ed., ASP Conference Series (Astronomical Society of the Pacific, 1999), Vol.  183.

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 and D. Mendlovic, “Optical design for improving matrix condition,” in Signal Recovery and Synthesis, OSA Technical Digest (CD) (Optical Society of America, 2009), paper STuA7.

I. Klapp and D. Mendlovic, “Imaging system and method for imaging object with reduced image blur,” U.S. patent WO/2010/103527 A2 (March 13, 2009).

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

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (Institute of Physics, 1998).
[CrossRef]

P. C. Hansen and J. G. Nagy, Deblurring Images Matrices, Spectra, and Filtering (SIAM, 2006).

J. H. Wilkinson, Rounding Errors in Algebraic Processes (Her Majesty’s Stationery Office, 1963), Chap. 3, p. 91.

G. H. Golub and C. F. Van-loan, Matrix Computation (North Oxford Academic, 1983).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (16)

Fig. 1
Fig. 1

Hybrid optical and image processing PO imaging system.

Fig. 2
Fig. 2

Realizing PO by the rim-ring approach. The PO lens divided into the lens zone and the rim-ring zone [1].

Fig. 3
Fig. 3

Trajectories block scheme for PO system implemen tation. The auxiliary system is composed of auxiliary optics imaging and a series of shifted and weighted operations which are done by postprocessing.

Fig. 4
Fig. 4

Relation between the shift of the image to the associated trajectory (PSF matrix). Example of a 3 × 2 FOV ( 6 × 6 matrix).

Fig. 5
Fig. 5

Trajectories PSF matrices for a 3 × 2 FOV . In the example, the unshifted PSF matrix is a unit matrix (O8). For each shift relative to the initial FOV it gets a different trajectory matrix representation. The 3 × 2 FOV allows 15 different trajectories matrices. In the 6 × 6 matrix there are 36 matrix entries, where the black and white color stands for 1 and 0 values, respectively.

Fig. 6
Fig. 6

(a) Postprocessing step: in the 3 × 2 field there are 15 available translations. The 15 weighted and shifted images of the 3 × 2 chessboard object are related. (b) Composite image of the postprocessing gain by summation of 15 weighted and shifted images of the 3 × 2 chessboard object. Only the central 3 × 2 in the FOV image is accounted for image restoration.

Fig. 7
Fig. 7

PO image. (a) is the object, (b) is the image of the main system H, (c) is the composite image of the postprocessing after shifting and weighting, and (d) is the PO system image (before restoration).

Fig. 8
Fig. 8

Restoration results under noise. The object gray level is in red asterisks designated obj, PO by trajectories method (parallel sys) restoration is in blue rings, and simple main system restoration (simple sys) is in black diamonds.

Fig. 9
Fig. 9

Demonstration of the trajectories schema implementation. (a) is the object, (b) is the image of the auxiliary lens ( O aux ), (c) is the image of the main system (H), (d) is the main system image restoration result by matrix inversion, (e) is the postprocessing composite image after shifting and weighting, and (f) is the restoration of the PO system, using the trajectories method for realizing the auxiliary system.

Fig. 10
Fig. 10

Object ensemble.

Fig. 11
Fig. 11

Main system singular values graph and the chosen BMSD matrix.

Fig. 12
Fig. 12

Comparison between the restoration results of two PO with different trajectories implementations. The diamonds represent the perfect auxiliary lens image restoration by matrix inversion. The rectangles represent the simple auxiliary lens image restoration by matrix inversion. The triangles represent the perfect auxiliary lens image restoration by regularization. The crosses represent the simple auxiliary lens image restoration by regularization. The main system SNR is the x axis.

Fig. 13
Fig. 13

Comparison between study case restoration (as is) and trajectories. The crosses represent the trajectories with simple auxiliary lens image restoration by regularization. The triangles represent the study case lens restoration by matrix inversion. The circles represent the study case lens restoration by regularization. The main system SNR is the x axis.

Fig. 14
Fig. 14

Comparison between the restoration quality of the trajectories and other hybrid methods. The triangles represent the trajectories with simple auxiliary lens (SSAL). The circles represent the QF filtering. The crosses represent the rim-ring. The blue dashes represent the perfect trajectories. The red line represents the study case. Image restoration is done by regularization. The main system SNR is the x axis.

Fig. 15
Fig. 15

Typical imaging results in the main system ( SNR = 35 [db] ). SC.Img is the study case image, SC.Res is the study case image restoration, Obj is the object, RR.Res is the rim-ring image restoration, RR.Img is the rim-ring image, QF.Res is the quartic phase restoration, QF.Img is the quartic phase image, and Traj.Res is the trajectories image restoration. All image restorations are done by regularization.

Fig. 16
Fig. 16

Example for an alloptics auxiliary lens.

Equations (45)

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I L × 1 image = H L × L · I L × 1 object .
( l ) = ( i 1 ) · N + j .
I ^ L × 1 res = I L × 1 object = H L × L 1 · I L × 1 image .
I ^ L × 1 res = ( I L × 1 object + Δ I L × 1 object ) = H L × L 1 · ( I L × 1 image + I L × 1 noise ) .
Δ I L × 1 object 2 I L × 1 object 2 k 2 ( H ) · I L × 1 noise 2 I L × 1 image 2 .
k 2 ( H ) = σ 1 / σ n .
H 1 = U · S 1 · V t = U · ( S + Δ S ) · V t = U · S · V t + U · Δ S · V t = H + O ,
H = U · S · V t = i = 1 n σ i · M i .
M i = u i · v i t ,
k 2 ( H ) = 1 / σ n .
O = U · Δ S · V t = i = 1 n Δ S i · M i .
Δ S i = { 0 σ i 1 κ 1 1 κ 1 σ i < 1 κ 1 .
Δ S i 1 = { 0 σ i 1 κ 1 1 σ i < 1 κ 1 .
BMSD = i = 1 n Δ S i 1 · M i .
O BMSD .
H 1 = H + O .
I L × 1 image = H 1 L × L · I L × 1 object = ( H + O ) · I L × 1 object = H L × L · I L × 1 object + O L × L · I L × 1 object + I L × 1 noise .
PSF ( x img , y img ; i ) = | h ( x img , y img ; i ) | 2 ,
h tot ( x img , y img ; i ) = T H · h H ( x img , y img ; i , A H ) + T O · h O ( x img , y img ; i , A O ) ,
PSF ( z ) = | T H · h H ( z ) | 2 + | T O · h O ( z ) | 2 + T H · T O · h H ( z ) * · h O ( z ) + T H · T O · h H ( z ) · h O ( z ) * .
PSF ( x img , y img ; i ) = | h tot ( x img , y img ; i ) | 2 | T H · h H ( x img , y img ; i ) | 2 + | T O · h O ( x img , y img ; i ) | 2 .
O BMSD .
O O ˜ = l = 1 M W l · T l { O aux } .
O l = T l { O aux } .
O l = 1 M W l · O l .
W l = 1 N l i = 1 L j = 1 L BMSD ( i , j ) O l ( i , j ; Δ m , Δ n ) .
I L × 1 auxiliary - system = O ˜ · I L × 1 object .
I L × 1 auxiliary - system = W l · O l · I L × 1 object .
I L × 1 main = H L × L · I L × 1 object .
I L × 1 image = H L × L · I L × 1 object + i = 1 M W l · O l · I L × 1 object + I L × 1 noise .
I L × 1 traj - l = O l · I L × 1 object .
I L × 1 auxiliary - system = W l · I L × 1 traj - l .
L l = W l T l .
I L × 1 auxiliary - system = L l · I L × 1 aux .
I L × 1 aux = O L × L aux · I L × 1 object + I L × 1 noise 1 = O L × L aux = [ I ] I L × 1 object + I L × 1 noise 1 .
( l ) = ( m 1 ) · N n + n .
( 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 .
H = [ 1.00 2.10 3.00 4.00 5.00 6.000 1.00 2.10 3.30 4.40 5.10 6.100 1.05 2.30 3.20 4.10 5.40 6.200 1.05 2.11 3.10 4.12 5.14 6.110 1.11 2.22 3.30 4.14 5.21 6.111 1.21 2.21 3.41 4.12 5.17 6.100 ] .
κ min = cond ( H + BMSD 1 ) = 187 . 87.
O * = l = 1 M = 15 W l · O l .
κ opt = cond ( H + O * ) = 72 . 6.
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 + λ · I ) 1 · H 1 t · I L × 1 image ,
I ^ L × 1 res = ( H + O ) L × L 1 · ( I L × 1 image - H + I L × 1 image - O ) ,

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