Abstract

The Taylor expansion of the incoherent optical transfer function with respect to defocus is a valuable tool in the design and analysis of computational imaging systems. It efficiently describes the behavior of the system near best focus and beyond. Formulas for computing the coefficients in this expansion are derived and shown to be amenable to efficient digital calculation. Their application to the design of phase masks for systems insensitive to defocus aberrations and for systems that estimate object range are explored.

© 2008 Optical Society of America

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References

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2008

2007

2006

2005

P. Favaro and S. Soatto, “A geometric approach to shape from defocus” IEEE Trans. Pattern Anal. Mach. Intell. 27, 406-417 (2005).
[CrossRef] [PubMed]

2003

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

2000

1995

1988

1984

J. O. Castañeda, “Bilinear optical systems Wigner distribution function and ambiguity function representations,” Opt. Acta 1, 255-260 (1984).

1983

K. Brenner, A. Lohmann, and J. O. Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Barwick, S.

Brenner, K.

K. Brenner, A. Lohmann, and J. O. Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Castañeda, J. O.

Casteñeda, J. O.

K. Brenner, A. Lohmann, and J. O. Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Cathey, W.

Chen, Y.

Dowski, E.

Favaro, P.

P. Favaro and S. Soatto, “A geometric approach to shape from defocus” IEEE Trans. Pattern Anal. Mach. Intell. 27, 406-417 (2005).
[CrossRef] [PubMed]

Goodman, J. W.

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

Greengard, A.

Isgleas, A. N.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Liu, L.

Lohmann, A.

K. Brenner, A. Lohmann, and J. O. Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Lu, W.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

Montes, E.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Pauca, V.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Piestun, R.

Plemmons, R.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Prasad, S.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Ramos, R.

Schechner, Y. Y.

Shamir, J.

Soatto, S.

P. Favaro and S. Soatto, “A geometric approach to shape from defocus” IEEE Trans. Pattern Anal. Mach. Intell. 27, 406-417 (2005).
[CrossRef] [PubMed]

Sun, J.

Torgersen, T.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Valdos, L. R. B.

van der Gracht, J.

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Yang, Q.

Ye, Zi

Yu, F.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhang, W.

Zhao, T.

Zhu, Y.

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

P. Favaro and S. Soatto, “A geometric approach to shape from defocus” IEEE Trans. Pattern Anal. Mach. Intell. 27, 406-417 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Opt. Acta

J. O. Castañeda, “Bilinear optical systems Wigner distribution function and ambiguity function representations,” Opt. Acta 1, 255-260 (1984).

Opt. Commun.

K. Brenner, A. Lohmann, and J. O. Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

S. Prasad, T. Torgersen, V. Pauca, R. Plemmons, and J. van der Gracht, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1-12 (2003).
[CrossRef]

Other

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1998).

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

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Figures (7)

Fig. 1
Fig. 1

H ( u , v , τ ) H ( u , v , 0 ) E versus misfocus τ plotted for three phase masks that have been optimized for defocus insensitivity by using the quadratic Taylor coefficient as a metric. In terms of the phase function defined in [10] the mask coefficients are (mask 1) [ 0.5 , 33 , 76 , 50 , 22 ] , (mask 2) [ 66 , 87 , 75 , 7 , 80 ] , and (mask 3) [ 39 , 10 , 48 , 35 , 28 ] .

Fig. 2
Fig. 2

In the black-and-white mappings (a)–(c) white indicates that the real part of the quadratic Taylor coefficient has an opposite sign to the real part of the quartic Taylor coefficient for (a) mask 1, (b) mask 3, and (c) mask 2. A gray-scale plot of the magnitude of the quartic Taylor coefficient for mask 2 is shown in (d). All plots are versus normalized spatial frequencies u and v.

Fig. 3
Fig. 3

Normalized Fisher information J versus misfocus τ is plotted for design parameters a 1 , a 2 , and ES. Plots were normalized by a common factor so that the maximum possible FI equals 1.

Fig. 4
Fig. 4

H ( u , v , τ ) H ( u , v , 0 ) E versus misfocus τ plotted for the systems with design parameters a 1 , a 2 , and ES with the E norm calculated (a) over the entire frequency plane and (b) within a restricted band near the spatial frequency origin.

Fig. 5
Fig. 5

Simulations of the three systems were run on the image of coins (quarters) lying on a flat surface shown in (a). The log 10 of the mse between the actual and the estimated defocus values are plotted versus SNR over the following ranges of misfocus: (b)  1 < τ < 1 , (c)  4 < τ < 4 , and (d)  10 < τ < 10 .

Fig. 6
Fig. 6

Gray-scale contour map of | H ( u , v , τ = 0 ) | for the system with design parameters a 1 plotted versus normalized spatial frequency coordinates u and v.

Fig. 7
Fig. 7

Gray-scale contour map of | H ( u , v , τ = 7 ) | for the system with design parameters a 1 plotted versus normalized spatial frequency coordinates u and v.

Tables (2)

Tables Icon

Table 1 Taylor Coefficient E Norms

Tables Icon

Table 2 Design Metric Values

Equations (21)

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τ = π R 2 λ ( 1 f 1 d i 1 d o ) ,
A f ( α , β , a , b ) = f ( x + α 2 , y + β 2 ) f * ( x α 2 , y β 2 ) exp ( j 2 π [ a x + b y ] ) d x d y ] ,
H ( u , v , τ ) = A P ( u , v , u τ π , v τ π ) ,
A f ( α , β , a , b ) = f ^ ( r + a 2 , s + b 2 ) f ^ * ( r a 2 , s b 2 ) exp ( j 2 π [ α r + β s ] ) d r d s ,
H ( u , v , τ ) = h ( r + u τ 2 π , s + v τ 2 π ) h * ( r u τ 2 π , s + v τ 2 π ) exp ( j 2 π [ u r + v s ] ) d r d s .
H ( u , v , τ ) = k = 0 k H ( u , v , 0 ) τ k τ k k ! .
k H ( u , v , 0 ) τ k = 1 ( 2 π ) k m = 0 k u m v k m F 1 { l = 0 k a = min A max A ( 1 ) k 1 C k , a , l , m l h r a s l a k l h * r m a s k m l + a } ,
C k , a , l , m = k ! a ! ( l a ) ! ( m a ) ! ( k m l + a ) ! ,
k H ( u , 0 ) τ k = u k ( 2 π ) k F 1 { l = 0 k ( 1 ) k l k ! ( k l ) ! l ! d l h d r l d k l h * d r k l } .
2 H ( u , 0 ) τ 2 = u 2 ( 2 π ) 2 F 1 { 2 } , 4 H ( u , 0 ) τ 4 = u 4 ( 2 π ) 4 F 1 { 8 } .
H ( u , v , τ ) H ( u , v , 0 ) E 2 H ( u , v , 0 ) τ 2 E τ 2 2 .
h ˜ = a 1 1 , 1 + a 2 5 , 3 + a 3 9 , 5 + a 4 13 , 7 + a 5 17 , 9 + a 6 21 , 11 ,
I ( θ , τ ) = ρ l ρ u | H ( ρ , θ , τ ) | d ρ ,
a 1 = [ 0.15 , 0.419 , 0.976 , 1.0 , 0.236 , 0.029 ] ,
a 2 = [ 0.098 , 0.04 , 0.496 , 1.0 , 0.703 , 0.079 ] .
J ( τ ) = E [ { d d τ ln ( p ( x , y ) [ K ( x , y , τ ) ] } 2 ] d x d y ,
J ( τ ) ( K ( x , y , τ ) τ ) 2 d x d y .
k H ( u , v , 0 ) τ k = F 1 { d k d τ k [ h ( r ˜ ( τ ) , s ˜ ( τ ) ) h * ( r ̑ ( τ ) , s ̑ ( τ ) ) ] | τ = 0 } ,
d d τ = u 2 π r ˜ + v 2 π s ˜ or d d τ = u 2 π r ̑ + v 2 π s ̑ ,
u q + γ v w + η ( 2 π ) k ( 1 ) γ + η ( q + w ) h r q s w ( γ + η ) h * r γ s η ,
C = ( q + w + γ + η ) ! q ! w ! γ ! n ! = k ! q ! w ! γ ! η ! .

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