Abstract

Singular-value decomposition (SVD) of a linear imaging system gives information on the null and measurement components of object and image and provides a method for object reconstruction from image data. We apply SVD to through-focus imaging systems that produce several two-dimensional images of a three-dimensional object. Analytical expressions for the singular functions are derived in the geometrical approximation for a telecentric, laterally shift-invariant system linear in intensity. The modes are evaluated numerically, and their accuracy confirmed. Similarly, the modes are derived and evaluated for a continuous image representing the limit of a large number of image planes.

© 2006 Optical Society of America

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  1. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  2. M. Bertero and P. Boccacci, Inverse Problems in Imaging (Institute of Physics, 1998).
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  3. H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).
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  7. B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, Vol. IX, E.Wolf, ed. (North-Holland, 1971), pp. 311-407.
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  8. C. Preza and J. Conchello, "Depth-variant maximum-likelihood restoration for three-dimensional fluorescence microscopy," J. Opt. Soc. Am. A 21, 1593-1601 (2004).
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  9. R. Pierri, A. Liseno, F. Soldovieri, and R. Solimene, "In-depth resolution for a strip source in the Fresnel zone," J. Opt. Soc. Am. A 18, 352-359 (2001).
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  10. A. Burvall, P. Martinsson, and A. T. Friberg, "Communication modes applied to axicons," Opt. Express 12, 377-383 (2004).
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  12. C. Lanczos, Linear Differential Operators (Van Nostrand, 1961).
  13. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43-63 (1961).
  14. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65-84 (1961).
  15. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).
  16. D. Slepian, "Some asymptotical expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99-140 (1965).

2004 (3)

2001 (1)

2000 (1)

1991 (1)

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).

1984 (1)

1978 (1)

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

1969 (1)

1965 (1)

D. Slepian, "Some asymptotical expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99-140 (1965).

1961 (2)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43-63 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65-84 (1961).

Aarsvold, J. N.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).

Agarwal, G. S.

Barrett, H. H.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Bertero, M.

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

Boccacci, P.

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

Burvall, A.

Conchello, J.

di Francia, G. Toraldo

Friberg, A. T.

Frieden, B. R.

B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, Vol. IX, E.Wolf, ed. (North-Holland, 1971), pp. 311-407.
[CrossRef]

Lanczos, C.

C. Lanczos, Linear Differential Operators (Van Nostrand, 1961).

Landau, H. J.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65-84 (1961).

Leone, G.

Liseno, A.

Martinsson, P.

Miller, D. A. B.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Pierri, R.

Pollak, H. O.

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65-84 (1961).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43-63 (1961).

Preza, C.

Roney, T. J.

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).

Slepian, D.

D. Slepian, "Some asymptotical expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99-140 (1965).

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43-63 (1961).

Soldovieri, F.

Solimene, R.

Wolf, E.

Appl. Opt. (1)

Bell Syst. Tech. J. (3)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—I," Bell Syst. Tech. J. 40, 43-63 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—II," Bell Syst. Tech. J. 40, 65-84 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis, and uncertainty—V: the discrete case," Bell Syst. Tech. J. 57, 1371-1430 (1978).

J. Math. Phys. (1)

D. Slepian, "Some asymptotical expansions for prolate spheroidal wave functions," J. Math. Phys. 44, 99-140 (1965).

J. Opt. Soc. Am. (1)

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

Lect. Notes Comput. Sci. (1)

H. H. Barrett, J. N. Aarsvold, and T. J. Roney, "Null functions and eigenfunctions: tools for the analysis of imaging systems," Lect. Notes Comput. Sci. 11, 211-226 (1991).

Opt. Express (1)

Other (4)

B. R. Frieden, "Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions," in Progress in Optics, Vol. IX, E.Wolf, ed. (North-Holland, 1971), pp. 311-407.
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

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

C. Lanczos, Linear Differential Operators (Van Nostrand, 1961).

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Figures (11)

Fig. 1
Fig. 1

(a) If image and object are expanded in two general orthogonal sets of functions or vectors, each object-space function contributes to each image-space function. (b) If SVD is used, the problem is diagonalized.

Fig. 2
Fig. 2

Through-focus imaging system.

Fig. 3
Fig. 3

Unit-magnification telecentric imaging system. The principal rays are parallel to the optical axis in both image and object space, so the magnification is independent of axial position.

Fig. 4
Fig. 4

Object-domain eigenfunctions (a) U 1 ( ρ , z ) , (b) U 2 ( ρ , z ) , and (c) U 3 ( ρ , z ) for a system where f = 100 mm , D = 10 mm , z 1 = 20 mm , and z 2 = 20 mm . There are three image planes placed at ζ 1 = 10 mm , ζ 2 = 0 mm , and ζ 3 = 10 mm .

Fig. 5
Fig. 5

Singular values σ ρ , 1 (solid curve), σ ρ , 2 (dashed curve), and σ ρ , 3 (dotted curve) corresponding to the object-domain eigenfunctions shown in Fig. 4.

Fig. 6
Fig. 6

Absolute value of one object-domain eigenfunction, number 11, for the same system as in Fig. 4, except that D = 40 mm and there are 21 image planes linearly distributed between ζ 1 = 10 mm and ζ 21 = 10 mm . The image is oversaturated to improve visibility; its true maximum value is 175.

Fig. 7
Fig. 7

(a) Test object used in the subsequent analysis. (b) Measurement component of the object for three image planes, i.e., the object expansion in the eigenfunctions shown in Fig. 4. (c) Measurement component of the object for 21 image planes, using the same system parameters as in Fig. 6.

Fig. 8
Fig. 8

(a) Image produced by the object in Fig. 7a, using the same imaging system as in Fig. 7b. The dotted curve represents the intensity in image plane 1, the solid curve in image plane 2, and the dashed curve in image plane 3. (b) Difference between the images in (a) and the images produced by direct propagation of the object.

Fig. 9
Fig. 9

Singular functions and singular values (a) U ρ , 1 ( ρ , z ) , (b) σ ρ , 1 , (c) U ρ , 7 ( ρ , z ) , (d) σ ρ , 7 , (e) U ρ , 46 , and (f) σ ρ , 46 for the CC-CC model. Parameters are f = 100 mm , D = 10 mm , and z 0 = 20 mm .

Fig. 10
Fig. 10

(a) Measurement component of the object in Fig. 7a, for the CC-CC geometry in Fig. 9. (b) The same, except that D = 20 mm . (c) Image produced by the object in Fig. 7a for the same geometry as Fig. 9. (d) The same, except that D = 20 mm .

Fig. 11
Fig. 11

Difference between the image in Fig. 10d and the same image obtained through direct numerical propagation using Eq. (25). Before the difference was taken, the second image was normalized to the same maximum level as the first.

Equations (36)

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g m ( r d ) = [ H f ] m ( r d ) = d 2 r z 1 z 2 d z h m ( r d r ; z ) f ( r , z ) ,
[ H g ] ( r , z ) = m = 1 M d 2 r d h m * ( r d r ; z ) g m ( r d ) ,
[ H H f ] ( r , z ) = d 2 r z 1 z 2 d z p ( r r ; z , z ) f ( r , z ) ,
p ( r r ; z , z ) = m = 1 M d 2 r d h m * ( r d r ; z ) h m ( r d r ; z ) .
[ H H g ] m ( r d ) = m = 1 M d 2 r d k m m ( r d r d ) g m ( r d ) ,
k m m ( r d r d ) = d 2 r z 1 z 2 d z h m ( r d r ; z ) h m * ( r d r ; z ) .
[ H H v ρ , j ] m ( r d ) = μ ρ , j v ρ , j ( r d ) ,
v ρ , j ( r d ) = V j ( ρ ) exp ( 2 π i ρ r d ) .
[ H H v ρ , j ] m ( r d ) = exp ( 2 π i ρ r d ) m = 1 M K m m ( ρ ) [ V j ( ρ ) ] m ,
K m m ( ρ ) = d 2 r d k m m ( r d ) exp ( 2 π i ρ r d ) .
K ( ρ ) V j ( ρ ) = μ ρ , j V j ( ρ ) ,
[ H u ρ , j ] = σ ρ , j v ρ , j ,
[ H v ρ , j ] = σ ρ , j u ρ , j
μ ρ , j u ρ , j ( r , z ) = exp ( 2 π i ρ r ) m = 1 M [ V j ( ρ ) ] m d 2 r d h m * ( r d ) exp ( 2 π i ρ r d ) ,
h m ( x x d ; z ) = 1 d m ( z ) rect [ x x d d m ( z ) ] ,
v ρ , j ( x d ) = V j ( ρ ) exp ( 2 π i ρ x d ) ,
K m m ( ρ ) = z 1 z 2 d z sin [ π ρ d m ( z ) ] π ρ d m ( z ) sin [ π ρ d m ( z ) ] π ρ d m ( z ) .
u ρ , j ( x , z ) = U j ( ρ , z ) exp ( 2 π i ρ x ) ,
U j ( ρ , z ) = m = 1 M [ V j ( ρ ) ] m sin [ π ρ d m ( z ) ] π ρ d m ( z ) .
f meas ( x , z ) = d ρ j = 1 M A j ( ρ ) U j ( ρ , z ) exp ( 2 π i ρ x ) ,
A j ( ρ ) = z 1 z 2 d z U j ( ρ , z ) F ( ρ , z ) .
B j ( ρ ) = σ j , ρ A j ( ρ ) ,
g m ( x d ) = d ρ j = 1 M B j ( ρ ) [ V j ( ρ ) ] m exp ( 2 π i ρ x d ) .
u ρ , j ( x , z ) = U j ( ρ , z ) exp ( 2 π i ρ x ) .
f ( x , z ) = d x d z 0 z 0 d ζ h ( x d x ; ζ z ) g ( x d , ζ ) ,
g ( x d , ζ ) = d x z 0 z 0 d z h * ( x d x ; ζ z ) f ( x , z ) ,
μ ρ , j u ρ , j ( x , z ) = d x d z 0 z 0 d ζ d x z 0 z 0 d z h ( x d x ; ζ z ) h * ( x d x ; ζ z ) u ρ , j ( x , z ) .
μ ρ , j U j ( ρ , z ) = z 0 z 0 d z z 0 z 0 d ζ sin [ π ρ D f ( ζ z ) ] π ρ D f ( ζ z ) sin [ π ρ D f ( ζ z ) ] π ρ D f ( ζ z ) U ( ρ , z ) ,
U j ( ρ , z ) = α j 1 ( π ρ D z 0 f , z ) ,
x 0 x 0 d x sin [ Ω ( y x ) ] π ( y x ) α j ( x 0 Ω , x ) = λ j α j ( x 0 Ω , y ) ,
σ ρ , j = f λ j 1 D ρ ,
N = 2 ρ D z 0 f .
μ 0 , j U j ( 0 , z ) = 2 z 0 z 0 z 0 d z U j ( 0 , z ) .
λ j = 2 π [ 2 2 j ( j ! ) 3 ( 2 j ) ! ( 2 j + 1 ) ! ] 2 c 2 j + 1 exp [ ( 2 j + 1 ) c 2 ( 2 j 1 ) 2 ( 2 j + 3 ) 2 ] .
σ ρ , j + 1 = 2 z 0 [ 2 2 j ( j ! ) 3 ( 2 j ) ! ( 2 j + 1 ) ! ] 2 c 2 j exp [ ( 2 j + 1 ) c 2 ( 2 j 1 ) 2 ( 2 j + 3 ) 2 ] ,
σ ρ , 1 2 z 0 ,

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