Abstract

A rapid focus-detection technique based directly on the spectral content of digital holograms is developed. It differs from previous approaches in that it does not need a full reconstruction of the image. The technique uses l1 norms of object spectral components associated with the real and imaginary parts of the reconstruction kernel. Further, the l1 norms can be computed efficiently in the spatial frequency domain using a polar coordinate system, yielding a drastic speedup of 2 orders of magnitude compared with image-based focus detection. Significant computational savings are achieved when subsequent image reconstructions are done selectively over the detected focus distances. Focus-detection results from holograms of plankton are demonstrated that show the technique is both accurate and robust.

© 2007 Optical Society of America

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  1. D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
    [CrossRef]
  2. P. R. Hobson and J. Watson, "The principles and practice of holographic recording of plankton," J. Opt. A, Pure Appl. Opt. 4, 34-49 (2002).
    [CrossRef]
  3. S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
    [CrossRef]
  4. W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
    [CrossRef] [PubMed]
  5. E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
    [CrossRef] [PubMed]
  6. J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
    [CrossRef]
  7. G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
    [CrossRef] [PubMed]
  8. J. H. Milgram and W. Li, "Computational reconstruction of images from holograms," Appl. Opt. 45, 853-864 (2002).
    [CrossRef]
  9. U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
    [CrossRef]
  10. F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, "Focus plane detection criteria in digital holography microscopy by amplitude analysis," Opt. Express 14, 5895-5908 (2006).
    [CrossRef] [PubMed]
  11. M. Liebling and M. Unser, "Autofocus for digital Fresnel holograms by use of a Fresnelet-sparsity criterion," J. Opt. Soc. Am. A 21, 2424-2430 (2004).
    [CrossRef]
  12. A. Thelen, J. Bongartz, G. Dominik, S. Frey, and P. Hering, "Iterative focus detection in hologram tomography," J. Opt. Soc. Am. A 22, 1176-1180 (2005).
    [CrossRef]
  13. L. Yu and L. Cai, "Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram," J. Opt. Soc. Am. A 18, 1033-1045 (2001).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  15. A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
    [CrossRef]

2006 (4)

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
[CrossRef]

F. Dubois, C. Schockaert, N. Callens, and C. Yourassowsky, "Focus plane detection criteria in digital holography microscopy by amplitude analysis," Opt. Express 14, 5895-5908 (2006).
[CrossRef] [PubMed]

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

2005 (1)

2004 (2)

2003 (2)

G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
[CrossRef] [PubMed]

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

2002 (3)

J. H. Milgram and W. Li, "Computational reconstruction of images from holograms," Appl. Opt. 45, 853-864 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

P. R. Hobson and J. Watson, "The principles and practice of holographic recording of plankton," J. Opt. A, Pure Appl. Opt. 4, 34-49 (2002).
[CrossRef]

2001 (2)

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

L. Yu and L. Cai, "Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram," J. Opt. Soc. Am. A 18, 1033-1045 (2001).
[CrossRef]

Averbuch, A.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Bongartz, J.

Cai, L.

Callens, N.

Carl, D.

Coifman, R. R.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Dominik, G.

Donoho, D. L.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Dubois, F.

Elad, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Frey, S.

Garcia-Sucerquia, J.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Goodman, J. W.

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

Hering, P.

Hobson, P. R.

P. R. Hobson and J. Watson, "The principles and practice of holographic recording of plankton," J. Opt. A, Pure Appl. Opt. 4, 34-49 (2002).
[CrossRef]

Israeli, M.

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Jericho, M. H.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Jericho, S. K.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Katz, J.

J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
[CrossRef]

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Kemper, B.

Kreuzer, H. J.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Li, W.

J. H. Milgram and W. Li, "Computational reconstruction of images from holograms," Appl. Opt. 45, 853-864 (2002).
[CrossRef]

Liebling, M.

Malkiel, E.

J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
[CrossRef]

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Meinertzhagen, I. A.

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Meng, H.

Milgram, J. H.

J. H. Milgram and W. Li, "Computational reconstruction of images from holograms," Appl. Opt. 45, 853-864 (2002).
[CrossRef]

Pan, G.

Schnars, U.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Schockaert, C.

Sheng, J.

J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
[CrossRef]

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Strickler, J.

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

Thelen, A.

Unser, M.

Von Bally, G.

Watson, J.

P. R. Hobson and J. Watson, "The principles and practice of holographic recording of plankton," J. Opt. A, Pure Appl. Opt. 4, 34-49 (2002).
[CrossRef]

Wernicke, G.

Xu, W.

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Xu, W. B.

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Yourassowsky, C.

Yu, L.

Appl. Comput. Harmon. Anal. (1)

A. Averbuch, R. R. Coifman, D. L. Donoho, M. Elad, and M. Israeli, "Fast and accurate polar Fourier transform," Appl. Comput. Harmon. Anal. 21, 146-167 (2006).
[CrossRef]

Appl. Opt. (4)

D. Carl, B. Kemper, G. Wernicke, and G. Von Bally, "Parameter-optimized digital holographic microscope for high-resolution living-cell analysis," Appl. Opt. 43, 6536-6544 (2004).
[CrossRef]

J. Sheng, E. Malkiel, and J. Katz, "Digital holographic microscope for measuring three-dimensional particle distributions and motions," Appl. Opt. 41, 3893-3901 (2006).
[CrossRef]

G. Pan and H. Meng, "Digital holography of particle fields: reconstruction by use of complex amplitude," Appl. Opt. 42, 827-833 (2003).
[CrossRef] [PubMed]

J. H. Milgram and W. Li, "Computational reconstruction of images from holograms," Appl. Opt. 45, 853-864 (2002).
[CrossRef]

J. Exp. Biol. (1)

E. Malkiel, J. Sheng, J. Katz, and J. Strickler, "The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography," J. Exp. Biol. 206, 3657-3666 (2003).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt. (1)

P. R. Hobson and J. Watson, "The principles and practice of holographic recording of plankton," J. Opt. A, Pure Appl. Opt. 4, 34-49 (2002).
[CrossRef]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Express (1)

Proc. Natl. Acad. Sci. U.S.A. (1)

W. B. Xu, M. H. Jericho, I. A. Meinertzhagen, and H. J. Kreuzer, "Digital in-line holography for biological applications," Proc. Natl. Acad. Sci. U.S.A. 98, 11301-11305 (2001).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

S. K. Jericho, J. Garcia-Sucerquia, W. Xu, M. H. Jericho, and H. J. Kreuzer, "Submersible digital in-line holographic microscope," Rev. Sci. Instrum. 77, 043706 (2006).
[CrossRef]

Other (1)

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

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Figures (6)

Fig. 1
Fig. 1

Object wavefront propagation and hologram recording: (a) object plane; (b) propagated object field; (c) the hologram.

Fig. 2
Fig. 2

Spatial convolution kernels sinc ( k r 1 ) (first column), sinc ( k r 2 ) (second column), and c ( x , y ; Δ z , z ) = sinc ( k r 1 ) + sinc ( k r 2 ) (third column), with z = 200,000 λ and Δ z = 0 (first row), 50 λ (second row), and 500 λ (third row).

Fig. 3
Fig. 3

Given a point object and a hologram of size 256 × 256 , pixel size 7.9375 μ m , and wavelength 633 nm , (a) F s ( Δ z ; z ) (upper panels) and (b) F c ( Δ z ; z ) (lower panels) for Δ z [ z , 100 ] and z = 10 , 55 , 100 mm .

Fig. 4
Fig. 4

Focus detection of a single object at z = 25 mm . The lower left panel shows a portion of the hologram containing the object; clockwise starting from the upper left are images reconstructed at several distances as labeled. The plot shows consistent response from the spatial focus metric F s p a (dotted curve with circle), the focus metric product in Cartesian system F c a r t (dashed curve with dot), and the focus metric product in polar system F p o l (solid curve with cross).

Fig. 5
Fig. 5

Focus detection of two objects located at z = 61 and 163 mm . A portion of the hologram containing both objects is shown in the lower left panel; object images reconstructed at several distances are shown clockwise starting from the upper left. Focus responses of F s p a , F c a r t , and F p o l are consistent and predict the focus distances. F c a r t and F p o l as shown have much sharper response than F s p a .

Fig. 6
Fig. 6

Focus detection of three objects located at z = 45 , 64 , and 72.5 mm , respectively. Lower left panel shows a portion of the hologram containing the objects; clockwise from the upper left are object images reconstructed at various distances. F s p a , F c a r t , and F p o l all show peak responses accurately predicting the focus distances of the reconstructed images. F c a r t , and F p o l have an improved focus response compared with F s p a .

Tables (2)

Tables Icon

Table 1 Algorithm for Spectral l 1 Norm Focus Detection in Polar Coordinate System and Selective Reconstruction

Tables Icon

Table 2 Statistics from Experimental Results

Equations (50)

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o ( x , y ) = h ( x , y ; z ) * * a ( x , y ) ,
O ( f x , f y ) = H ( f x , f y ; z ) A ( f x , f y ) ,
h ( x , y ; z ) = 1 j λ exp [ j k ( x 2 + y 2 + z 2 ) 1 2 ] ( x 2 + y 2 + z 2 ) 1 2 ,
H ( f x , f y ; z ) = { exp { j k z [ 1 ( λ f x ) 2 ( λ f y ) 2 ] 1 2 } ( f x 2 + f y 2 ) 1 2 < 1 λ 0 , otherwise ,
i ( x , y ) = 1 + o ( x , y ) 2 = o ( x , y ) + o * ( x , y ) + o ( x , y ) 2 + 1 .
i ̃ ( x , y ) = o ( x , y ) + o * ( x , y ) .
I ̃ ( f x , f y ) = O ( f x , f y ) + O * ( f x , f y ) .
a ̂ ( x , y ; z ̂ ) = h * ( x , y ; z ̂ ) * * i ̃ ( x , y ) ,
A ̂ ( f x , f y ; z ̂ ) = H * ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) ,
z focus = { arg min z ̂ x , y a ̂ ( x , y ; z ̂ ) d x d y , if a ( x , y ) is real and positive arg max z ̂ x , y a ̂ ( x , y ; z ̂ ) d x d y if a ( x , y ) is a pure phase object .
a ̂ ( x , y ; z ̂ ) 1 = x , y a ̂ ( x , y ; z ̂ ) d x d y .
H c ( f x , f y ; z ) = cos ( k r f z ) ,
H s ( f x , f y ; z ) = sin ( k r f z ) ,
A ̂ c ( f x , f y ; z ̂ ) H c ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) ,
A ̂ s ( f x , f y ; z ̂ ) H s ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) .
F c ( Δ z ; z ) A ̂ c ( f x , f y ; z ̂ ) 1 ,
F s ( Δ z ; z ) A ̂ s ( f x , f y ; z ̂ ) 1 .
F c ( Δ z ; z ) = A ̂ c ( f x , f y ; z ̂ ) d f x d f y ,
= H c ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) d f x d f y ,
F s ( Δ z ; z ) = A ̂ s ( f x , f y ; z ̂ ) d f x d f y ,
= H s ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) d f x d f y .
I ̃ ( f x , f y ) = 2 H c ( f x , f y ; z ) A ( f x , f y ) .
A ̂ c ( f x , f y ; z ̂ ) = A ( f x , f y ) C ( f x , f y ; Δ z , z ) ,
C ( f x , f y ; Δ z , z ) [ cos ( k r f Δ z ) + cos ( k r f ( 2 z + Δ z ) ) ] .
A ̂ s ( f x , f y ; z ̂ ) = H s ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) = A ( f x , f y ) S ( f x , f y ; Δ z , z ) ,
S ( f x , f y ; Δ z , z ) [ sin ( k r f Δ z ) + sin ( k r f ( 2 z + Δ z ) ) ] .
a ̂ c ( x , y ; z ̂ ) = a ( x , y ) c ( x , y ; Δ z , z ) ,
a ̂ s ( x , y ; z ̂ ) = a ( x , y ) s ( x , y ; Δ z , z ) ,
a ̂ c ( x , y ; z ̂ ) F 1 ( A ̂ c ( f x , f y ; z ̂ ) )
a ̂ s ( x , y ; z ̂ ) F 1 ( A ̂ s ( f x , f y ; z ̂ ) ) ,
c ( x , y ; Δ z , z ) = F 1 ( C ( f x , f y ; Δ z , z ) ) = 2 π λ 2 [ sinc ( k r 1 ) + sinc ( k r 2 ) ] ,
s ( x , y ; Δ z , z ) = F 1 ( S ( f x , f y ; Δ z , z ) ) = j 2 π λ 2 [ cinc ( k r 1 ) + cinc ( k r 2 ) ] ,
a ̂ c ( x , y ; Δ z ) a ( x , y ) * * 2 π λ 2 sinc ( k r 1 ) ,
a ̂ s ( x , y ; Δ z ) a ( x , y ) * * 2 π λ 2 cinc ( k r 1 ) ,
A ̂ c ( f x , f y ; z ̂ ) A ( f x , f y ) cos ( k r f Δ z ) ,
A ̂ s ( f x , f y ; z ̂ ) A ( f x , f y ) sin ( k r f Δ z ) .
F c ( Δ z ; z ) = A ̂ c ( f x , f y ; z ̂ ) d f x d f y A ( f x , f y ) cos ( k r f Δ z ) d f x d f y A ( f x , f y ) d f x d f y ,
F s ( Δ z ; z ) = A ̂ s ( f x , f y ; z ̂ ) d f x d f y A ( f x , f y ) sin ( k r f Δ z ) d f x d f y 0 ,
F c ( Δ z ; z ) point source = C ( f x , f y ; Δ z , z ) d f x d f y F p , c ( Δ z ; z ) ,
F s ( Δ z ; z ) point source = S ( f x , f y ; Δ z , z ) d f x d f y F p , s ( Δ z ; z ) .
F p , c cos ( k r f Δ z ) d f x d f y ,
F p , s sin ( k r f Δ z ) d f x d f y .
A ̂ c ( f x , f y ; z ̂ ) = H c ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) = H c ( f x , f y ; z ̂ ) I ̃ ( f x , f y ) .
H c ( f x , f y ; z ̂ ) = H c ( f r ; z ̂ ) ,
F c ( Δ z ; z ) = f r = 0 f θ = 0 2 π H c ( f r ; z ̂ ) I ̃ ( f r , f θ ) f r d f r d f θ = f r = 0 H c ( f r ; z ̂ ) [ f θ = 0 2 π I ̃ ( f r , f θ ) d f θ ] f r d f r .
o p spatial = 2 N 2 log 2 N + ( 2 log 2 N + k + 1 ) N 2 S .
o p f r e q = 2 N 2 log 2 N + ( k + 1 ) N 2 S .
o p spatial o p f r e q = 2 k + 1 log 2 N + 1 .
o p p o l = 2 N 2 log 2 N + ( 6 + k ) R θ + ( k + 1 ) R S .
o p spatial o p p o l = ( 2 log 2 N + k + 1 ) N 2 S ( k + 6 ) N 2 + 2 ( k + 1 ) N S .

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