Abstract

Adaptive Optics corrected flood imaging of the retina has been in use for more than a decade and is now a well-developed technique. Nevertheless, raw AO flood images are usually of poor contrast because of the three-dimensional nature of the imaging, meaning that the image contains information coming from both the in-focus plane and the out-of-focus planes of the object, which also leads to a loss in resolution. Interpretation of such images is therefore difficult without an appropriate post-processing, which typically includes image deconvolution. The deconvolution of retina images is difficult because the point spread function (PSF) is not well known, a problem known as blind deconvolution. We present an image model for dealing with the problem of imaging a 3D object with a 2D conventional imager in which the recorded 2D image is a convolution of an invariant 2D object with a linear combination of 2D PSFs. The blind deconvolution problem boils down to estimating the coefficients of the PSF linear combination. We show that the conventional method of joint estimation fails even for a small number of coefficients. We derive a marginal estimation of the unknown parameters (PSF coefficients, object Power Spectral Density and noise level) followed by a MAP estimation of the object. We show that the marginal estimation has good statistical convergence properties and we present results on simulated and experimental data.

© 2011 OSA

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References

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  1. J. Liang, D. R. Williams, and D. T. Miller, “Supernormal vision and high-resolution retinal imaging through adaptive optics,” J. Opt. Soc. Am. A 14, 2884–2892 (1997).
    [CrossRef]
  2. M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
    [CrossRef]
  3. J. Rha, R. S. Jonnal, K. E. Thorn, J. Qu, Y. Zhang, and D. T. Miller, “Adaptive optics flood-illumination camera for high speed retinal imaging,” Opt. Express 14, 4552–4569 (2006).
    [CrossRef] [PubMed]
  4. A. Roorda, F. Romero-Borja, I. William Donnelly, H. Queener, T. Hebert, and M. Campbell, “Adaptive optics scanning laser ophthalmoscopy,” Opt. Express 10, 405–412 (2002).
    [PubMed]
  5. L. Blanc-Féraud, L. Mugnier, and A. Jalobeanu, “Blind image deconvolution,” in “Inverse Problems in Vision and 3D Tomography,”, A. Mohammad-Djafari, ed. (ISTE / John Wiley, London, 2010), chap. 3, pp. 97–121.
  6. G. R. Ayers and J. C. Dainty, “Iterative blind deconvolution and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  7. L. M. Mugnier, T. Fusco, and J.-M. Conan, “Mistral: a myopic edge-preserving image restoration method, with application to astronomical adaptive-optics-corrected long-exposure images,” J. Opt. Soc. Am. A 21, 1841–1854 (2004).
    [CrossRef]
  8. R. J. A. Little and D. B. Rubin, “On jointly estimating parameters and missing data by maximizing the complete-data likelihood,” The American Statistician 37, 218–220 (1983).
    [CrossRef]
  9. J. C. Christou, A. Roorda, and D. R. Williams, “Deconvolution of adaptive optics retinal images,” J. Opt. Soc. Am. A 21, 1393–1401 (2004).
    [CrossRef]
  10. G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Processing8, 202–219 (1999).
    [CrossRef]
  11. J. Idier, L. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Processing14, 2107–2116 (2005).
    [CrossRef]
  12. E. Lehmann, Theory of point estimation (John Wiley, New York, NY, 1983).
  13. A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035–1045 (2003).
    [CrossRef]
  14. Y. Goussard, G. Demoment, and J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in “ICASSP,” (1990), pp. 1547–1550.
  15. J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, and G. Rousset, “Myopic deconvolution of adaptive optics images by use of object and point-spread function power spectra,” Appl. Opt. 37, 4614–4622 (1998).
    [CrossRef]
  16. É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in “Astronomical Data Analysis II,”, vol. 4847, J.-L. Starck and F. D. Murtagh, eds. (Proc. Soc. Photo-Opt. Instrum. Eng., 2002), vol. 4847, pp. 174–183.

2006 (1)

2004 (3)

2003 (1)

2002 (1)

1998 (1)

1997 (1)

1988 (1)

1983 (1)

R. J. A. Little and D. B. Rubin, “On jointly estimating parameters and missing data by maximizing the complete-data likelihood,” The American Statistician 37, 218–220 (1983).
[CrossRef]

Ayers, G. R.

Blanc, A.

A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035–1045 (2003).
[CrossRef]

J. Idier, L. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Processing14, 2107–2116 (2005).
[CrossRef]

Blanc-Féraud, L.

L. Blanc-Féraud, L. Mugnier, and A. Jalobeanu, “Blind image deconvolution,” in “Inverse Problems in Vision and 3D Tomography,”, A. Mohammad-Djafari, ed. (ISTE / John Wiley, London, 2010), chap. 3, pp. 97–121.

Bresler, Y.

G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Processing8, 202–219 (1999).
[CrossRef]

Campbell, M.

Christou, J. C.

Conan, J.-M.

Dainty, J. C.

Demoment, G.

Y. Goussard, G. Demoment, and J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in “ICASSP,” (1990), pp. 1547–1550.

Fusco, T.

Gendron, E.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Glanc, M.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Goussard, Y.

Y. Goussard, G. Demoment, and J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in “ICASSP,” (1990), pp. 1547–1550.

Harikumar, G.

G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Processing8, 202–219 (1999).
[CrossRef]

Hebert, T.

Idier, J.

A. Blanc, L. M. Mugnier, and J. Idier, “Marginal estimation of aberrations and image restoration by use of phase diversity,” J. Opt. Soc. Am. A 20, 1035–1045 (2003).
[CrossRef]

J. Idier, L. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Processing14, 2107–2116 (2005).
[CrossRef]

Y. Goussard, G. Demoment, and J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in “ICASSP,” (1990), pp. 1547–1550.

Jalobeanu, A.

L. Blanc-Féraud, L. Mugnier, and A. Jalobeanu, “Blind image deconvolution,” in “Inverse Problems in Vision and 3D Tomography,”, A. Mohammad-Djafari, ed. (ISTE / John Wiley, London, 2010), chap. 3, pp. 97–121.

Jonnal, R. S.

Lacombe, F.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Lafaille, D.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Le Gargasson, J.-F.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Lehmann, E.

E. Lehmann, Theory of point estimation (John Wiley, New York, NY, 1983).

Léna, P.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Liang, J.

Little, R. J. A.

R. J. A. Little and D. B. Rubin, “On jointly estimating parameters and missing data by maximizing the complete-data likelihood,” The American Statistician 37, 218–220 (1983).
[CrossRef]

Michau, V.

Miller, D. T.

Mugnier, L.

L. Blanc-Féraud, L. Mugnier, and A. Jalobeanu, “Blind image deconvolution,” in “Inverse Problems in Vision and 3D Tomography,”, A. Mohammad-Djafari, ed. (ISTE / John Wiley, London, 2010), chap. 3, pp. 97–121.

J. Idier, L. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Processing14, 2107–2116 (2005).
[CrossRef]

Mugnier, L. M.

Qu, J.

Queener, H.

Rha, J.

Romero-Borja, F.

Roorda, A.

Rousset, G.

Rubin, D. B.

R. J. A. Little and D. B. Rubin, “On jointly estimating parameters and missing data by maximizing the complete-data likelihood,” The American Statistician 37, 218–220 (1983).
[CrossRef]

Thiébaut, É.

É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in “Astronomical Data Analysis II,”, vol. 4847, J.-L. Starck and F. D. Murtagh, eds. (Proc. Soc. Photo-Opt. Instrum. Eng., 2002), vol. 4847, pp. 174–183.

Thorn, K. E.

William Donnelly, I.

Williams, D. R.

Zhang, Y.

Appl. Opt. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J.-F. Le Gargasson, and P. Léna, “Towards wide-field retinal imaging with adaptive optics,” Opt. Commun. 230, 225–238 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

The American Statistician (1)

R. J. A. Little and D. B. Rubin, “On jointly estimating parameters and missing data by maximizing the complete-data likelihood,” The American Statistician 37, 218–220 (1983).
[CrossRef]

Other (6)

G. Harikumar and Y. Bresler, “Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms,” IEEE Trans. Image Processing8, 202–219 (1999).
[CrossRef]

J. Idier, L. Mugnier, and A. Blanc, “Statistical behavior of joint least square estimation in the phase diversity context,” IEEE Trans. Image Processing14, 2107–2116 (2005).
[CrossRef]

E. Lehmann, Theory of point estimation (John Wiley, New York, NY, 1983).

Y. Goussard, G. Demoment, and J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in “ICASSP,” (1990), pp. 1547–1550.

É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in “Astronomical Data Analysis II,”, vol. 4847, J.-L. Starck and F. D. Murtagh, eds. (Proc. Soc. Photo-Opt. Instrum. Eng., 2002), vol. 4847, pp. 174–183.

L. Blanc-Féraud, L. Mugnier, and A. Jalobeanu, “Blind image deconvolution,” in “Inverse Problems in Vision and 3D Tomography,”, A. Mohammad-Djafari, ed. (ISTE / John Wiley, London, 2010), chap. 3, pp. 97–121.

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Figures (12)

Fig. 1
Fig. 1

Simulated object

Fig. 2
Fig. 2

Simulated image

Fig. 3
Fig. 3

Joint criterion for 0 ≤ α ≤ 1

Fig. 4
Fig. 4

Jointly estimated object

Fig. 5
Fig. 5

Marginal criterion for 0 ≤ α ≤ 1

Fig. 6
Fig. 6

Marginaly estimated object

Fig. 7
Fig. 7

RMS error on the estimation of the PSF coefficients as a function of noise level in percent (noise standard deviation over image maximum). The black, red and blue lines correspond, respectively, to 32×32, 64×64 and 128×128 pixels images. Supervised case is in dashed lines, unsupervised case in solid line.

Fig. 8
Fig. 8

Experimental image

Fig. 9
Fig. 9

Estimated PSF

Fig. 10
Fig. 10

Restored object

Fig. 11
Fig. 11

PSD comparison between experimental image (dotted line), restored object (solid line) and object prior PSD (dashed line) with the estimated hyperparameters

Fig. 12
Fig. 12

Radial average of the estimated instrument optical transfer function, deconvolution transfer function and global (instrument+deconvolution) transfer function.

Equations (24)

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i 3 D = h 3 D * 3 D o 3 D + n ,
o 3 D ( x , y , z ) = o 2 D ( x , y ) α ( z ) ,
i ( x , y ) i 3 D ( x , y , z ) | z = 0 = α ( z ) h 3 D ( x , y , z ) o 2 D ( x x , y y ) d x d y d z + n ( x , y ) = ( h 2 D o 2 D * 2 D ) ( x , y ) + n ( x , y ) ,
h 2 D ( x , y ) = α ( z ) h 3 D ( x , y , z ) d z .
h 2 D ( x , y ) j α j h j ( x , y ) ,
( o ^ , α ^ ) = argmax o , α p ( i , o , α ; θ )
= argmax o , α p ( i | o , α ; θ ) × p ( o ; θ ) × p ( α ; θ )
p ( i , o , α ; θ ) = 1 ( 2 π ) N 2 2 σ N 2 exp ( 1 2 σ 2 ( i H o ) t ( i H o ) ) × 1 ( 2 π ) N 2 2 det ( R o ) 1 / 2 exp ( 1 2 ( o o m ) t R o 1 ( o o m ) ) ,
J jmap ( o , α ) = J i ( o , α ) + J o ( o , α ) ,
J jmap ( o , α ) = ln p ( i | o , α ; θ ) ln p ( o ; θ )
J jmap ( o , α ) = 1 2 N 2 ln σ 2 + 1 2 σ 2 ( i H o ) t ( i H o ) + 1 2 lndet ( R o ) + 1 2 ( o o m ) t R o 1 ( o o m ) + C ,
o ^ ( α , θ ) = ( H t H + σ 2 R 0 1 ) 1 ( H t i + σ 2 R 0 1 o m )
J jmap ( o , α ) = 1 2 N 2 ln S n + 1 2 ν | i ˜ ( ν ) h ˜ ( ν ) o ˜ ( ν ) | 2 S n + 1 2 ν ln S o ( ν ) + 1 2 ν | o ˜ ( ν ) o ˜ m ( ν ) | 2 S o ( ν )
and o ˜ ^ ( α ) = h ˜ * ( ν ) i ˜ ( ν ) + S n S o ( ν ) o ˜ m ( ν ) | h ˜ ( ν ) | 2 + S n S o ( ν ) ,
J jmap ( α ) = 1 2 N 2 ln S n + 1 2 ν ln S o ( ν ) + 1 2 ν 1 S o ( ν ) | i ˜ ( ν ) h ˜ ( ν ) o ˜ m ( ν ) | 2 | h ˜ ( ν ) | 2 + S n S o ( ν ) .
i = ( α * h foc + ( 1 α ) h defoc ) * o + n ,
α ^ = argmax α p ( i , o , α ; θ ) d o .
α ^ ML = argmax α p ( i , α ; θ ) = argmax α p ( i | α ; θ ) p ( α ; θ ) .
p ( i | α ; θ ) = A ( det R i ) 1 / 2 exp ( 1 2 ( i i m ) t R i 1 ( i i m ) ) ,
J ML ( α ) = 1 2 lndet ( R i ) + 1 2 ( i i m ) t ( i i m ) + B
J ML ( α ) = 1 2 ν ln S o ( ν ) + 1 2 ν ln ( | h ˜ ( ν ) | 2 + S n S o ( ν ) ) + 1 2 ν 1 S o ( ν ) | i ˜ ( ν ) h ˜ ( ν ) o ˜ m ( ν ) | 2 | h ˜ ( ν ) | 2 + S n S o ( ν ) + B .
J ML ( α ) = J jmap ( α ) + 1 2 ν ln ( | h ˜ ( ν ) | 2 + S n S o ( ν ) ) 1 2 N 2 ln S n .
( α ^ , S ^ n , S ^ o ) = argmax α , S n , S o f ( i , α ; S n , S o ) .
S o ( ν ) = k 1 + ( ν ν 0 ) p .

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