Abstract

In this paper we use our derived approximate representation of the modulation transfer function to analytically solve the problem of the extension of the depth of field for two cases of interest: uniform quality imaging and task-based imaging. We derive the optimal result for each case as a function of the problem specifications. We also compare the two different imaging cases and discuss the advantages of using our optimization approach for each case. We also show how the analytical solutions given in this paper can be used as a convenient design tool as opposed to previous lengthy numerical optimizations.

© 2008 Optical Society of America

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References

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  1. S. Bagheri, P. E. X. Silveira, and D. P. de Farias, “Analytical optimal solution of the extension of the depth of field using cubic-phase wavefront coding. Part I. Reduced-complexity approximate representation of the modulation transfer function,” J. Opt. Soc. Am. A 25, 1051-1063 (2008).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1992).
  3. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  4. E. R. Dowski and G. E. Johnson, “Wavefront coding: A modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
    [CrossRef]
  5. W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41, 6080-6092 (2002).
    [CrossRef] [PubMed]
  6. G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
    [CrossRef]
  7. R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. 44, 701-712 (2005).
    [CrossRef] [PubMed]
  8. J. Daugman, “How iris recognition works,” IEEE Trans Circuits Systems Video Technol. 14, 21-30 (2004).
    [CrossRef]
  9. J. Daugman, “The importance of being random: statistical principles of iris recognition,” J Patt. Recogn Soc. 36, 279-291 (2003).
    [CrossRef]
  10. R. F. Stengel, Optimal Control and Estimation (Dover, 1994).

2008 (1)

2005 (2)

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

2004 (1)

J. Daugman, “How iris recognition works,” IEEE Trans Circuits Systems Video Technol. 14, 21-30 (2004).
[CrossRef]

2003 (1)

J. Daugman, “The importance of being random: statistical principles of iris recognition,” J Patt. Recogn Soc. 36, 279-291 (2003).
[CrossRef]

2002 (1)

1999 (1)

E. R. Dowski and G. E. Johnson, “Wavefront coding: A modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

1995 (1)

Bagheri, S.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1992).

Cathey, W. T.

Daugman, J.

J. Daugman, “How iris recognition works,” IEEE Trans Circuits Systems Video Technol. 14, 21-30 (2004).
[CrossRef]

J. Daugman, “The importance of being random: statistical principles of iris recognition,” J Patt. Recogn Soc. 36, 279-291 (2003).
[CrossRef]

de Farias, D. P.

Dowski, E. R.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

W. T. Cathey and E. R. Dowski, “New paradigm for imaging systems,” Appl. Opt. 41, 6080-6092 (2002).
[CrossRef] [PubMed]

E. R. Dowski and G. E. Johnson, “Wavefront coding: A modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

Johnson, G. E.

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

R. Narayanswamy, G. E. Johnson, P. E. X. Silveira, and H. B. Wach, “Extending the imaging volume for biometric iris recognition,” Appl. Opt. 44, 701-712 (2005).
[CrossRef] [PubMed]

E. R. Dowski and G. E. Johnson, “Wavefront coding: A modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

Narayanswamy, R.

Silveira, P. E. X.

Stengel, R. F.

R. F. Stengel, Optimal Control and Estimation (Dover, 1994).

Wach, H. B.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1992).

Appl. Opt. (3)

IEEE Trans Circuits Systems Video Technol. (1)

J. Daugman, “How iris recognition works,” IEEE Trans Circuits Systems Video Technol. 14, 21-30 (2004).
[CrossRef]

J Patt. Recogn Soc. (1)

J. Daugman, “The importance of being random: statistical principles of iris recognition,” J Patt. Recogn Soc. 36, 279-291 (2003).
[CrossRef]

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

Proc. SPIE (2)

E. R. Dowski and G. E. Johnson, “Wavefront coding: A modern method of achieving high performance and/or low cost imaging systems,” Proc. SPIE 3779, 137-145 (1999).
[CrossRef]

G. E. Johnson, P. E. X. Silveira, and E. R. Dowski, “Analysis tools for computational imaging systems,” Proc. SPIE 5817, 34-44 (2005).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1992).

R. F. Stengel, Optimal Control and Estimation (Dover, 1994).

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Figures (13)

Fig. 1
Fig. 1

Photos of a nonplanar object with (a) traditional imaging system and (b) imaging system with wavefront coding. Part (a) does not have a uniform image quality. Quality is good at best focus in the center of the photo but poor where out of focus. Part (b) has a uniform image quality; i.e., the image quality is the same both in and out of focus. By uniform image quality we mean the uniform transfer function of the imaging system at the spatial frequencies of interest over the depth of field.

Fig. 2
Fig. 2

MTF e ( u , 0 ) of traditional [(a), (b), and (c)] and wavefront coded [(d), (e), and (f)] imaging systems (uniform image quality imaging problem; optical system specifications are from Tables 1, 2). The region between dashed vertical lines represents the range of spatial frequencies of interest. Note how the transfer function of the wavefront coded imaging system at the spatial frequencies of interest is uniform over the depth of field. (a) Traditional imaging system (near field, W 20 λ = + 7 ). (b) Traditional imaging system (in focus, W 20 λ = 0 ). (c) Traditional imaging system (far field, W 20 λ = 5 ). (d) Optimized imaging system (near field, W 20 λ = + 6 ). (e) Optimized imaging system (in focus, W 20 λ = 0 ). (f) Optimized imaging system (far field, W 20 λ = 6 ).

Fig. 3
Fig. 3

Defocus of the (a) traditional imaging system and (b) optimized imaging system in the uniform quality imaging problem (optical system specifications are from Tables 1, 2). Note that in the optimized imaging system the best focus has been moved toward the lens to reduce the maximum absolute defocus from 7 λ to 6 λ .

Fig. 4
Fig. 4

Optimum cubic-phase coefficient ( α * ) for the uniform image quality problem. Using given problem specifications one can find the corresponding u max and W 20 [Eq. (13)], and then α * can be directly read from this figure [Eq. (12)].

Fig. 5
Fig. 5

Iris recognition images as an example of task-based imaging. (a) Far-field image ( d o = 800 mm ) . (b) Near field image ( d o = 200 mm ) . Although part (a) appears to be a higher quality image, parts (a) and (b) have equal amounts of usable iris information (i.e., equal number of effective line-pairs across the iris for iris recognition).

Fig. 6
Fig. 6

Contour plot of the MTF e with respect to partial defocus W 20 and depth of field d o . The numbers in the boxed labels represent the value of the MTF at the corresponding contours. The region between the vertical dashed lines represents the depth of field of interest. The goal is to have maximum MTF e [ u max ( d o ) , 0 ] = MTF e ( 2 π S f o d o k D , 0 ) in this region. To do so we find the W 20 for which MTF a 2 ( 2 π S f o d o k D , 0 ) is the same at both ends of this region of interest [see Eq. (20)]. In this figure we have used k α = 10 , k D 2 8 = 10 4 mm and 2 π S f o ( k D ) = 10 3 mm 1 .

Fig. 7
Fig. 7

MTF e ( u , 0 ) of traditional [(a), (b) and (c)] and wavefront coded [(d), (e) and (f)] imaging systems (task-based imaging problem; optical system specifications are from Tables 3, 4). The region between vertical dashed lines represents the range of spatial frequencies of interest for that particular depth of field. Note how this range of spatial frequencies of interest gets smaller as the object gets closer to the imaging system. (a) Traditional imaging system (near field, W 20 λ = + 7 ). (b) Traditional imaging system (in focus, W 20 λ = 0 ). (c) Traditional imaging system (far field, W 20 λ = 5 ). (d) Optimized imaging system (near field, W 20 λ = + 8 ). (e) Optimized imaging system (in focus, W 20 λ = 0 ). (f) Optimized imaging system (far field, W 20 λ = 4 ).

Fig. 8
Fig. 8

Contour plot of the MTF e as a function of partial defocus and depth of field for the optimum system (task-based imaging problem; imaging system specified in Tables 3, 4). The numbers in the boxed labels represent the value of the MTF at the corresponding contours. The region between the vertical dashed lines represents the depth of field of interest. The horizontal solid line represents the optimum value of W 20 . Note how MTF e ( 2 π S f o d o 1 k D , 0 ) MTF e ( 2 π S f o d o 2 k D , 0 ) as expected from Eq. (20).

Fig. 9
Fig. 9

Defocus of the (a) traditional imaging system and (b) optimized imaging system in the task-based imaging problem (optical system specifications are from Tables 3, 4). Note in the optimized imaging system how the best focus is moved far from the imaging system to balance the modulation at the highest spatial frequency of interest over the entire depth of field.

Fig. 10
Fig. 10

Minimum value of MTF e in the range of spatial frequencies of interest {namely MTF e [ u max ( d o ) , 0 ] = MTF e ( 2 π S f o d o k D , 0 ) } versus depth of field. The solid curve represents the optimized task-based imaging system, the dashed curve the optimized uniform quality imaging system. This figure shows how the optimized uniform quality imaging system is not efficient for task specific imaging. Note how the suboptimization of Eq. (20) has increased MTF e ( 2 π S f o d o k D , 0 ) over the depth of field of interest as shown by the solid-curve graph. Optical system specifications are from Tables 2, 4.

Fig. 11
Fig. 11

Optimum cubic-phase coefficient ( α * ) for task-based imaging. Using the range of interest of object distances ( d o 1 and d o 2 ), one can find α * from this figure [Eq. (27)]. In this figure we have used λ = 0.55 × 10 3 mm , D = 8 mm and S f o = 7   line - pair mm .

Fig. 12
Fig. 12

Optimum image plane and exit pupil distance ( d i * ) for task-based imaging system. Using the range of interest of object distances ( d o 1 and d o 2 ), one can find d i * from this figure [Eq. (27)]. In this figure we have used λ = 0.55 × 10 3 mm , D = 8 mm , f = 50 mm , and S f o = 7   line - pair mm .

Fig. 13
Fig. 13

Graphical representation of the optimum cubic coefficient (uniform quality imaging problem). This figure is plotted using the optical system specifications in Table 1. It shows how numerical optimization is in accordance with our analytical optimization. The optimum α from the figure is 4.65 λ , yielding a worst-case MTF e of 0.10 (or 0.9927 with four digit accuracy), whereas analytical optimization has shown α * = 4.60 λ , yielding a worst-case MTF e of 0.10 (or 0.9915 with four digit accuracy).

Tables (4)

Tables Icon

Table 1 Problem Specifications for the Sample Uniform Image Quality Imaging Problem

Tables Icon

Table 2 Optimized Design Parameters for the Sample Uniform Image Quality Imaging Problem

Tables Icon

Table 3 Problem Specifications for the Sample Task-Based Imaging Problem

Tables Icon

Table 4 Optimized Design Parameters for the Sample Task-Based Imaging Problem

Equations (46)

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MTF a 2 ( u , 0 ) = 2 3 π 2 k u α [ Arcsin ( u ) + 3 π k u α 2 1 u 2 × ( Arctan { 2 π k u 3 α [ W 20 + 3 α ( 1 u ) ] } Arctan { 2 π k u 3 α [ W 20 + 3 α ( 1 + u ) ] } ) ] ,
W 20 = D 2 8 ( 1 d i + 1 d o 1 f ) ,
max α , d i { min d o , u { MTF a 2 ( u , 0 ) } } α R , d i R ;
d o [ d o 1 , d o 2 ] , u [ 0 , u max ] .
u max = S f i 2 f o ,
u max = 2 π d i S f i k D .
max α , d i { min d o { MTF a 2 ( 2 π d i S f i k D , 0 ) } }
α R , d i R , d o [ d o 1 , d o 2 ] .
min d i { max d o { W 20 } } d i R , d o [ d o 1 , d o 2 ] .
d i * = 2 f d o 1 d o 2 2 d o 1 d o 2 f ( d o 1 + d o 2 ) ,
d o * = d o 1 or d 0 2 .
max α { MTF a 2 ( u max , 0 ) } α R ,
MTF a 2 ( α ) α = 0 .
α * λ = 1 24 u max ( 1 u max ) 2 [ 1 + 8 u max W 20 λ ( 1 u max ) + 1 + 16 u max W 20 λ ( 1 u max ) ] ,
α * λ = 1 + 8 u max W 20 λ ( 1 u max ) + 1 + 16 u max W 20 λ ( 1 u max ) 24 u max ( 1 u max ) 2 ,
d i * = 2 f d o 1 d o 2 2 d o 1 d o 2 f ( d o 1 + d o 2 ) ,
u max = 2 π d i S f i k D ,
W 20 = D 2 8 ( 1 d i + 1 d o 1 1 f ) .
max α , d i { min d o , u { MTF a 2 ( u , 0 ) } } α R , d i R ,
d o [ d o 1 , d o 2 ] , u [ 0 , u max ( d o ) ] .
u max = S f i 2 f o ,
S f i = S f o M ,
u max = 2 π S f o d o k D .
max α , d i { min d o { MTF a 2 ( 2 π S f o d o k D , 0 ) } }
α R , d i R , d o [ d o 1 , d o 2 ] .
W 20 = D 2 8 ( 1 d i 1 f ) .
MTF a 2 ( 2 π S f o d o 1 k D , 0 ) = MTF a 2 ( 2 π S f o d o 2 k D , 0 ) .
W 20 * λ = 48 π α λ d o 1 d o 2 ( u max 2 u max 1 ) k D 2 ( d o 1 + d o 2 ) 32 π d o 1 d o 2 .
max α { MTF w a 2 ( 2 π S f o d o 1 k D , 0 ) } α R ,
MTF w a 2 ( α ) α = 0 .
MTF w a 2 ( α ) = c 1 α + c 2 α [ Arctan ( c 3 α + c 4 α + c 5 α ) Arctan ( c 3 α c 4 α + c 5 α ) ] ,
α * = 1 + 2 c 3 c 4 + c 3 c 5 + 1 + 4 c 3 c 4 + 4 c 3 2 c 5 2 2 ( c 5 2 c 4 2 ) ,
α * λ = u 2 + 2 C Δ u ( Δ u + 1 u 1 ) 12 π u 1 u 2 ( 2 u 1 u 2 ) [ Δ u + 2 ( 1 u 1 ) ] + u 2 2 + 4 C u 2 Δ u 2 + 16 C 2 ( 1 u 1 ) 2 Δ u 2 12 π u 1 u 2 ( 2 u 1 u 2 ) [ Δ u + 2 ( 1 u 1 ) ] ,
α * λ = u 2 + 2 C Δ u ( Δ u + 1 u 1 ) + u 2 2 + 4 C u 2 Δ u 2 + 16 C 2 ( 1 u 1 ) 2 Δ u 2 12 π u 1 u 2 ( 2 u 1 u 2 ) [ Δ u + 2 ( 1 u 1 ) ] ,
1 d i * = 1 f + 48 π α * λ d o 1 d o 2 ( u 2 u 1 ) k D 2 ( d o 1 + d o 2 ) 2 d o 1 d o 2 k D 2 ,
u 1 = 2 π S f o d o 1 k D , u 2 = 2 π S f o d o 2 k D ,
Δ u = u 2 u 1 , C = π S f o D 4 .
MTF a 2 ( α ̂ ) = c 1 α ̂ + c 2 α ̂ [ Arctan ( c 3 α ̂ + c 4 α ̂ ) Arctan ( c 3 α ̂ c 4 α ̂ ) ] ,
c 1 = Arcsin ( u max ) 3 π 3 u max ,
c 2 = 1 u max 2 π 2 3 u max ,
c 3 = 2 π u max 3 W 20 λ ,
c 4 = 2 π 3 u max ( 1 u max ) ,
c 1 α ̂ 2 + c 2 c 4 ( 3 c 3 2 + α ̂ + c 4 2 α ̂ 2 ) [ α ̂ + ( c 3 c 4 α ̂ ) 2 ] [ α ̂ + ( c 3 + c 4 α ̂ ) 2 ] + c 2 2 α ̂ 3 2 Arctan ( 2 c 4 α ̂ 3 2 c 3 2 + α ̂ c 4 2 α ̂ 2 ) = 0 .
2 c 1 ( c 3 2 + α ̂ c 4 2 α ̂ 2 ) [ α ̂ + ( c 3 c 4 α ̂ ) 2 ] [ α ̂ + ( c 3 + c 4 α ̂ ) 2 ] + c 2 α ̂ { c 3 6 π + c 3 4 α ̂ ( 3 π + 4 c 4 α ̂ + 3 c 4 2 π α ̂ ) + α ̂ 3 ( 1 + c 4 2 α ̂ ) ( π 4 c 4 3 α ̂ 3 2 + c 4 4 π α ̂ 2 ) + c 3 2 α ̂ 2 [ 4 c 4 α ̂ + π ( 3 + 2 c 4 2 α ̂ 3 c 4 4 α ̂ 2 ) ] } = 0 .
α ̂ * 1 + 2 c 3 c 4 + 1 + 4 c 3 c 4 2 c 4 2 .
α ̂ * 1 + 8 u max W 20 λ ( 1 u max ) 24 u max ( 1 u max ) 2 + 1 + 16 u max W 20 λ ( 1 u max ) 24 u max ( 1 u max ) 2 .

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