Abstract

We address the problem of computational representation of image formation in 3D widefield fluorescence microscopy with depth varying spherical aberrations. We first represent 3D depth-dependent point spread functions (PSFs) as a weighted sum of basis functions that are obtained by principal component analysis (PCA) of experimental data. This representation is then used to derive an approximating structure that compactly expresses the depth variant response as a sum of few depth invariant convolutions pre-multiplied by a set of 1D depth functions, where the convolving functions are the PCA-derived basis functions. The model offers an efficient and convenient trade-off between complexity and accuracy. For a given number of approximating PSFs, the proposed method results in a much better accuracy than the strata based approximation scheme that is currently used in the literature. In addition to yielding better accuracy, the proposed methods automatically eliminate the noise in the measured PSFs.

© 2010 Optical Society of America

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  1. D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
    [CrossRef] [PubMed]
  2. P. Sarder and A. Nehorai,"Deconvolution methods for 3-D fluorescence microscopy images," IEEE Sig. Proc. Mag. 23, 32-45 (2006).
    [CrossRef]
  3. S. Gibson and F. Lanni, "Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy, " J. Opt. Soc. Am. A 8, 1601-1613 (1991).
    [CrossRef]
  4. B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
    [CrossRef] [PubMed]
  5. J. Shaevitz and D. Fletcher, "Enhanced three-dimensional deconvolution microscopy using a measured depthvarying point-spread function," J. Opt. Soc. Am. A 24, 2622-2627 (2007).
    [CrossRef]
  6. C. Preza and J. Conchello, "Image estimation account for point-spread function depth variation in threedimensional fluorescence microscopy," Proc. SPIE 1-8 (2003).
  7. C. Preza and J. Conchello, "Depth-variant maximum likelihood restoration for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. A 21, 1593-1601 (2004).
    [CrossRef]
  8. E. Hom, F. Marchis, T. Lee, S. Haase, D. Agard, and J. Sedat, "AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data," J. Opt. Soc. Am. A 24, 1580-1600 (2007).
    [CrossRef]
  9. C. Vonesch and M. Unser, "A Fast Thresholded Landweber Algorithm forWavelet-Regularized Multidimensional Deconvolution," IEEE Tran. Imag. Proc. 17, 539-549 (2008).
    [CrossRef]
  10. C. Vonesch and M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Tran. Imag. Proc. 18, 509-523 (2009).
    [CrossRef]
  11. I. Jolliffe, Principal component analysis (Springer, 2002).
  12. S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
    [CrossRef]

2009 (1)

C. Vonesch and M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Tran. Imag. Proc. 18, 509-523 (2009).
[CrossRef]

2008 (1)

C. Vonesch and M. Unser, "A Fast Thresholded Landweber Algorithm forWavelet-Regularized Multidimensional Deconvolution," IEEE Tran. Imag. Proc. 17, 539-549 (2008).
[CrossRef]

2007 (2)

2006 (1)

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

2004 (2)

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

C. Preza and J. Conchello, "Depth-variant maximum likelihood restoration for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. A 21, 1593-1601 (2004).
[CrossRef]

1997 (1)

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

1991 (1)

1989 (1)

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Agard, D.

E. Hom, F. Marchis, T. Lee, S. Haase, D. Agard, and J. Sedat, "AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data," J. Opt. Soc. Am. A 24, 1580-1600 (2007).
[CrossRef]

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Conchello, J.

Fletcher, D.

Gibson, S.

Gustafsson, M.

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

Haase, S.

Hanser, B.

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

Hiraoka, Y.

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Hom, E.

Lanni, F.

Lee, T.

Marchis, F.

Nehorai, A.

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

Preza, C.

Sarder, P.

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

Sedat, J.

E. Hom, F. Marchis, T. Lee, S. Haase, D. Agard, and J. Sedat, "AIDA: an adaptive image deconvolution algorithm with application to multi-frame and three-dimensional data," J. Opt. Soc. Am. A 24, 1580-1600 (2007).
[CrossRef]

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Shaevitz, J.

Shaw, P.

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Torok, P.

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

Unser, M.

C. Vonesch and M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Tran. Imag. Proc. 18, 509-523 (2009).
[CrossRef]

C. Vonesch and M. Unser, "A Fast Thresholded Landweber Algorithm forWavelet-Regularized Multidimensional Deconvolution," IEEE Tran. Imag. Proc. 17, 539-549 (2008).
[CrossRef]

Varga, P.

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

Visser, T.

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

Vonesch, C.

C. Vonesch and M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Tran. Imag. Proc. 18, 509-523 (2009).
[CrossRef]

C. Vonesch and M. Unser, "A Fast Thresholded Landweber Algorithm forWavelet-Regularized Multidimensional Deconvolution," IEEE Tran. Imag. Proc. 17, 539-549 (2008).
[CrossRef]

Wiersma, S.

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

IEEE Sig. Proc. Mag. (1)

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

IEEE Tran. Imag. Proc. (2)

C. Vonesch and M. Unser, "A Fast Thresholded Landweber Algorithm forWavelet-Regularized Multidimensional Deconvolution," IEEE Tran. Imag. Proc. 17, 539-549 (2008).
[CrossRef]

C. Vonesch and M. Unser, "A Fast Multilevel Algorithm for Wavelet-Regularized Image Restoration," IEEE Tran. Imag. Proc. 18, 509-523 (2009).
[CrossRef]

J. Microsc. (1)

B. Hanser, M. Gustafsson, D. Agard, and J. Sedat, "Phase-retrieved pupil functions in wide-field fluorescent microscopy," J. Microsc. 216, 32-48 (2004).
[CrossRef] [PubMed]

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

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

S. Wiersma, P. Torok, T. Visser, and P. Varga, "Comparison of different theories for focusing through a plane interface," J. Opt. Soc. Am. B 14, 1482-1490 (1997).
[CrossRef]

Methods Cell Biol. (1)

D. Agard, Y. Hiraoka, P. Shaw, and J. Sedat, "Fluorescence microscopy in three dimensions," Methods Cell Biol. 30, 353-377 (1989).
[CrossRef] [PubMed]

Other (2)

I. Jolliffe, Principal component analysis (Springer, 2002).

C. Preza and J. Conchello, "Image estimation account for point-spread function depth variation in threedimensional fluorescence microscopy," Proc. SPIE 1-8 (2003).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic representation 3D depth dependent system

Fig. 2.
Fig. 2.

Theoretical DV-PSF sections. Horizontal axis represent x. Image size is 2925 nm× 4200 nm.

Fig. 3.
Fig. 3.

Basis functions for theoretical PSFs (xz sections). Image at the top left is the mean PSF h̄. The remaining are the first 5 principal components.

Fig. 4.
Fig. 4.

Approximation error for various values of number of basis PSFs. x-axis represents number of PSFs used to approximate 41 PSFs spanning the range z = [0,3000 nm]. PCA method yields a significant advantage over the strata based method.

Fig. 5.
Fig. 5.

Results of approximating using two basis functions only. Displayed are the approximated xz sections and error images for various values of z . For z = 0 nm, 3000 nm, strata approximation is identical to the originals because the basis function therein are originals themselves. However, for the other values of z , PCA method gives better approximation resulting in a 17 fold improvement in overall approximation error. The displayed error images are normalized to the overall maximum of the original PSFs.

Fig. 6.
Fig. 6.

xz sections of the same results in Figure 5 for various values of z

Fig. 7.
Fig. 7.

Measured DV-PSF sections. Image size is 2925nm×4200nm.

Fig. 8.
Fig. 8.

Basis functions for measured PSFs (xz sections). Image at the left is the mean PSF h̄. The right one is the first principal component.

Fig. 9.
Fig. 9.

Approximation error for various values of number of basis PSFs. x-axis represents number of PSFs used to approximate 41 PSFs spanning the range z = [0,3000 nm]. PCA method yields a better approximation.

Fig. 10.
Fig. 10.

Results of approximating using two basis functions only. Displayed are the approximated xz sections and error images for various values of z . For z = 0 nm, 3000 nm, strata approximation is identical to the originals because the basis function therein are originals themselves. However, for the other values of z , PCA method gives better approximation.

Fig. 11.
Fig. 11.

xz section of approximation result of Fig. 10 for z = 0. Note the pronounced line artifact in strata approximation.

Equations (25)

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

X ̄ = ( 1 / N ) i = 1 N X i .
R = [ v 1 v 2 v N ]
R = UD V T ,
X i = X ¯ + j=1 B c i , j P j ,
r B = d B d 1 ,
S i ( z ) ( x , y ) = S ̄ ( z ) ( x , y ) + j = 1 B 1 c i , j ( z ) P j ( z ) ( x , y ) ,
c i , j ( z ) = S i ( z ) ( x , y ) S ̄ ( z ) ( x , y ) , P j ( z ) ( x , y ) .
𝓢 = { C i ( z , j ) = c i , j ( z ) , i = 1 , , N ; j = 1 , , B 1 } .
C i ( z , j ) = k = 1 B 2 c i , k P k ( z , j ) ,
S i ( z ) ( x , y ) = S ̄ ( z ) ( x , y ) + k = 1 B 2 c i , k Q k ( x , y , z ) ,
Q k ( x , y , z ) = j = 1 B 1 P k ( z , j ) P j ( z ) ( x , y )
S i ( x , y , z ) = S ̄ ( x , y , z ) + k = 1 B 2 c i , k Q k ( x , y , z ) ,
g ( x , y , z ) = x , y , z h ( x x , y y , z , z ) f ( x , y , z ) dx dy dz ,
h ( x , y , z , z ) = h ( x , y , z z , 0 ) ,
g ( x , y , z ) = h 0 ( x , y , z ) * f ( x , y , z ) ,
h ( x , y , z , z ) = h ( x , y , z + az , z ) .
g ( x , y , z ) = x , y , z h ( x x , y y , z az , z ) f ( x , y , z ) dx dy dz ,
g ( x , y , z / a ) = x , y , z h ( x x , y y , z / a z , z ) f ( x , y , z ) dx dy dz ,
g ( x , y , z ) = x , y , z h ( x x , y y , z z , z ) f ( x , y , z ) ,
𝓢 = { h z ( x , y , z ) = h ( x , y , z , z ) , z = 0 , , N z 1 }
h ( x , y , z . z ) = h ̄ ( x , y , z ) + j = 1 B c j ( z ) P j ( x , y , z ) ,
c j ( z ) = x , y , z ( h ( x , y , z , z ) h ̄ ( x , y , z ) ) ) P j ( x , y , z )
g ( x , y , z ) = h ̄ ( x , y , z ) * f ( x , y , z ) + j = 1 B P j ( x , y , z ) * [ c j ( z ) f ( x , y , z ) ] ,
h a ( B ) ( x , y , z , z ) = h ̄ ( x , y , z ) + j = 1 B P j ( x , y , z ) c j ( z ) .
E = h h a ( B ) 2 h 2 ,

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