Abstract

The multichannel exact blind image deconvolution theory tells us that exact recovery of unknown blur kernels is possible from multiple measurements of an identical scene through distinct blur channels. However, in many biological applications, there often exist technical difficulties in obtaining multiple distinct blur measurements, since the image content may vary for various reasons, including specimen drift between snapshots, specimen damage due to prolonged exposure, or physiological changes in live cell imaging. The main contribution of this paper is a new non-iterative single channel blind deconvolution method that eliminates the need of multiple blur measurements, but still guarantees an accurate estimation of the blurring kernel. The basic idea behind this novel method is to exploit the radial symmetry of a certain class of PSFs. This approach simplifies the PSF estimation to a 1-D channel identification problem with multiple excitations, which can be solved using a standard subspace method. Since radially symmetric PSFs are quite often encountered in many practical applications, such as optical imaging systems and electron microscopy, our theory may have great influence on many practical imaging applications. Simulation results as well as real experimental results using optical and electron microscopy confirm our theory.

© 2007 Optical Society of America

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  1. P. Sarder and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Processing Mag.23, 32-45 May (2006).
    [CrossRef]
  2. D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Processing Mag.13, 43-64 May (1996).
    [CrossRef]
  3. J. Frank, Three Dimensional Electron Microscopy of Macromolecular Assemblies, (Academic Press, San Diego, CA, USA, 1996).
  4. D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, "Blind deconvolution by means of the Richardson- Lucy algorithm," J. Opt. Soc. Am. A 12, 58-65 (1995).
    [CrossRef]
  5. J. A. Conchello, "Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images," J. Opt. Soc. Am. A 15, 2609-2620 (1998).
    [CrossRef]
  6. G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999).
    [CrossRef]
  7. G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999).
    [CrossRef]
  8. E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
    [CrossRef]
  9. C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996).
    [CrossRef]
  10. M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing  43, 134-149 (1995).
    [CrossRef]
  11. X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
    [CrossRef] [PubMed]

2000

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

1999

G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999).
[CrossRef]

G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999).
[CrossRef]

1998

1996

C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996).
[CrossRef]

1995

M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing  43, 134-149 (1995).
[CrossRef]

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

D. A. Fish, A. M. Brinicombe, E. R. Pike, and J. G. Walker, "Blind deconvolution by means of the Richardson- Lucy algorithm," J. Opt. Soc. Am. A 12, 58-65 (1995).
[CrossRef]

Baker, T.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Bergoin, M.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Bouman, C.

C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996).
[CrossRef]

Bresler, Y.

G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999).
[CrossRef]

G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999).
[CrossRef]

Brinicombe, A. M.

Cardoso, J.-F.

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

Conchello, J. A.

Duhamel, P.

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

Fish, D. A.

Gurelli, M.

M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing  43, 134-149 (1995).
[CrossRef]

Harikumar, G.

G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999).
[CrossRef]

G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999).
[CrossRef]

Hatzinakos, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Processing Mag.13, 43-64 May (1996).
[CrossRef]

Kundur, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Processing Mag.13, 43-64 May (1996).
[CrossRef]

Mayrargue, S.

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

Moulines, E.

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

Nehorai, A.

P. Sarder and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Processing Mag.23, 32-45 May (2006).
[CrossRef]

Nikias, C.

M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing  43, 134-149 (1995).
[CrossRef]

Olson, N.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Pike, E. R.

Rossmann, M.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Sarder, P.

P. Sarder and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Processing Mag.23, 32-45 May (2006).
[CrossRef]

Sauer, K.

C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996).
[CrossRef]

Van Etten, J.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Walker, J. G.

Yan, X.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

IEEE Trans. on Image Processing

G. Harikumar and Y. Bresler, "Exact image deconvolution from multiple FIR blurs," IEEE Trans. on Image Processing 8, 846-862 (1999).
[CrossRef]

G. Harikumar and Y. Bresler, "Perfect blind restoration of images blurred by multiple filters: Theory and efficient algorithms," IEEE Trans. on Image Processing 8, 202-219 (1999).
[CrossRef]

C. Bouman, and K. Sauer, "A unified approach to statistical tomography using coordinate descent optimization," IEEE Trans. on Image Processing 5, 480-492 (1996).
[CrossRef]

IEEE Trans. on Signal Processing

M. Gurelli and C. Nikias, "EVAM: An eigenvector-based algorithm for multichannel blind deconvolution of input colored signals," IEEE Trans. on Signal Processing  43, 134-149 (1995).
[CrossRef]

E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, "Subspace methods for the blind identification of multichannel FIR filters," IEEE Trans. on Signal Processing 43, 516-525 (1995).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Struct. Biol.

X. Yan, N. Olson, J. Van Etten, M. Bergoin, M. Rossmann, and T. Baker, "Structure and assembly of large lipid-containing dsDNA viruses," Nat. Struct. Biol.  7, 101-103 (2000).
[CrossRef] [PubMed]

Other

P. Sarder and A. Nehorai, "Deconvolution methods for 3-D fluorescence microscopy images," IEEE Signal Processing Mag.23, 32-45 May (2006).
[CrossRef]

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Processing Mag.13, 43-64 May (1996).
[CrossRef]

J. Frank, Three Dimensional Electron Microscopy of Macromolecular Assemblies, (Academic Press, San Diego, CA, USA, 1996).

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Figures (7)

Fig. 1.
Fig. 1.

Original Lena image.

Fig. 2.
Fig. 2.

OTFs for (a) a Gaussian blur, and (b) an electron microscope.

Fig. 3.
Fig. 3.

Gaussian blurred image (left column), and proposed blind deconvolution results (right column). Reconstruction at SNR=50dB (1st row), 30dB (2nd row), and 20dB(3rd row), respectively.

Fig. 4.
Fig. 4.

CTF blurred image (left column), and proposed blind deconvolution results (right column). Reconstruction at SNR=30dB (1st row), 10dB (2nd row), and 0dB(3rd row), respectively.

Fig. 5.
Fig. 5.

Root mean square error (RMSE) of the reconstruction using the classic blind de-convolution (Lucy-Richardson) and the proposed method: (left) Gaussian blur, and (right) CTF blur experiments.

Fig. 6.
Fig. 6.

Optical microscope images from Olympus BX51 microscope. (a) In-focus image, (b) the defocused image, and (c) blind deconvolution result by the proposed method.

Fig. 7.
Fig. 7.

Micrograph of PBCV-1 virus [11]: (a) Original de-focused measurement, and (b) the refined image. White arrows are added to indicate some examples of refined areas.

Equations (24)

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

y ( r ) = h 2 D ( r r ´ ) x ( r ´ ) + n ( r ) = ( h 2 D x ) ( r ) + n ( r )
y ( i ) = h ( i ) 2 D x + n ( i ) ,
Y ( k ) = H 2 D ( k ) X ( k ) + N ( k ) ,
Y ( k , Θ ) = H 2 D ( k , Θ ) X ( k , Θ ) + N ( k , Θ )
Y ( k , Θ ) = H ( k ) X ( k , Θ ) + N ( k , Θ ) ,
y Θ = h x Θ + n Θ ,
h = [ h M h 0 h M ] T ,
y ( i ) = Hx ( i ) + n ( i ) ,
y ( i ) = [ y M ( i ) y 0 ( i ) y N + M 1 ( i ) ] T , x ( i ) = [ x 0 ( i ) x N 1 ( i ) ] T
H = [ h M 0 0 h M h M 0 h M ] ( N + 2 M ) × N .
R y = 1 P i = 1 P y ( i ) ( y ( i ) ) H = HR x H H + σ n 2 I
S = [ s 0 s 1 s N 1 ]
G = [ g 0 g 1 g 2 M 1 ]
g i H H = 0 , i = 0 , , 2 M 1 .
c ( H ) = i = 0 2 M 1 g i H H 2 = i = 0 2 M 1 g i H HH H g i .
g i H H = h H i
i = [ g 0 i g 1 i g N 1 i g 1 i g 2 i g N i g 2 M i g 2 M + 1 i g N + 2 M 1 i ]
g i H H = h H i e
c ( h ) = h T Q h
c ( h ) = N h 2 h H Q ˜ h
h ( x , y ) = 1 2 π σ 2 exp ( x 2 + y 2 2 σ 2 ) .
CTF ( k ) = sin ( γ ( k ) + ψ )
γ ( k ) = π 2 ( C s λ 3 k 4 2 Δ z λ k 2 )
ψ = tan 1 ( Q / 1 Q 2 )

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