Abstract

A theory for an integrated system is described that combines a logarithmic aspheric imaging lens with maximum-entropy digital processing to extend the depth of field ten times over that of a conventional lens and to provide near-diffraction-limited resolution. Two types of logarithmic aspheres are derived that are circularly symmetric lenses with controlled continuous radial variation of focal length. The details of an iterative maximum-entropy algorithm are also presented. The properties of convergence and speed of the algorithm are greatly improved by introducing a metric parameter to adjust the weight of different pixel values of the recovered picture in each loop properly.

© 2003 Optical Society of America

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References

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  1. J. Ojeda-Castaneda, L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994–997 (1990).
    [CrossRef] [PubMed]
  2. G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
    [CrossRef]
  3. E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  4. J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
    [CrossRef]
  5. W. Chi, N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
    [CrossRef]
  6. N. George, W. Chi, “Extended depth of field using the logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157–S163 (2003).
    [CrossRef]
  7. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [CrossRef] [PubMed]
  8. S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
    [CrossRef]
  9. X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
    [CrossRef]
  10. N. Wu, The Maximum Entropy Method (Springer-Verlag, Berlin, 1997), Chap. 3.
  11. Henry Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987), Chap. 5.
  12. R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Sect. 5.1.3.

2003

N. George, W. Chi, “Extended depth of field using the logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157–S163 (2003).
[CrossRef]

2001

1995

1990

1987

X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
[CrossRef]

1983

S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
[CrossRef]

1972

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
[CrossRef] [PubMed]

Berriel-Valdos, L. R.

Bowen, J. P.

J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
[CrossRef]

Burch, S. F.

S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
[CrossRef]

Cathey, W. T.

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

J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
[CrossRef]

Chi, W.

N. George, W. Chi, “Extended depth of field using the logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157–S163 (2003).
[CrossRef]

W. Chi, N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
[CrossRef]

Dowski, E. R.

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

J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
[CrossRef]

Frieden, B. R.

George, N.

N. George, W. Chi, “Extended depth of field using the logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157–S163 (2003).
[CrossRef]

W. Chi, N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875–877 (2001).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Sect. 5.1.3.

Gull, S. F.

S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
[CrossRef]

Haralick, R. M.

X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
[CrossRef]

Hausler, G.

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Ojeda-Castaneda, J.

OØstevold, E.

X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
[CrossRef]

Skilling, J.

S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
[CrossRef]

Stark, Henry

Henry Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987), Chap. 5.

van der Gracht, J.

J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
[CrossRef]

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Sect. 5.1.3.

Wu, N.

N. Wu, The Maximum Entropy Method (Springer-Verlag, Berlin, 1997), Chap. 3.

Zhuang, X.

X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
[CrossRef]

Appl. Opt.

Comput. Vision Graph. Image Process.

S. F. Burch, S. F. Gull, J. Skilling, “Image restoration by a powerful maximum entropy method,” Comput. Vision Graph. Image Process. 23, 113–128 (1983).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

X. Zhuang, E. OØstevold, R. M. Haralick, “A differential equation approach to maximum entropy image reconstruction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 208–218 (1987).
[CrossRef]

J. Opt. A Pure Appl. Opt.

N. George, W. Chi, “Extended depth of field using the logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157–S163 (2003).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Opt. Lett.

Other

J. van der Gracht, E. R. Dowski, W. T. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian ed., Proc. SPIE2537, 279–288 (1995).
[CrossRef]

N. Wu, The Maximum Entropy Method (Springer-Verlag, Berlin, 1997), Chap. 3.

Henry Stark, Image Recovery: Theory and Application (Academic, Orlando, Fla., 1987), Chap. 5.

R. C. Gonzalez, R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), Sect. 5.1.3.

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Figures (10)

Fig. 1
Fig. 1

Diagram of logarithmic asphere lens.

Fig. 2
Fig. 2

Object distance versus radius of the best-focus ring for the logarithmic aspheres. Dash curve, γ design; solid curve, β design; dotted curve, β0 design from ray optics. The f/# is 4, and the classical diffraction-limited depth of field is approximately ±8 mm.

Fig. 3
Fig. 3

Point-spread function of the β lens at various object distances. The designed depth-of-field range is the same as in Fig. 2, i.e., from 1400 to 1615 nm, and the wavelength of illumination is 0.5 µm.

Fig. 4
Fig. 4

OTFs of the lens corresponding to the point-spread functions shown in Fig. 3.

Fig. 5
Fig. 5

Point-spread functions of the β lenses with different center block radii at an object distance of 1420 mm.

Fig. 6
Fig. 6

OTFs corresponding to the point-spread functions shown in Fig. 5.

Fig. 7
Fig. 7

Diagram of the maximum-entropy method. Above the dashed line is the measurement process; below the dashed line is the recovery iterative process described in Subsections 3.B and 3.C.

Fig. 8
Fig. 8

Dependence of the algorithm loop number on the metric parameter γ. The big dot on the vertical axis is the value for the zebra image.

Fig. 9
Fig. 9

Computer simulations of the resolution of a two-point object separated by the diffraction limit. (a) Blurred image by an idealized lens for the object at 1500 mm; (b), (c), and (d) blurred image by the logarithmic asphere for object at distances 1450, 1500, and 1580 mm, respectively; (e), (f), (g), and (h) maximum entropy recovery of images in (a), (b), (c), and (d), respectively.

Fig. 10
Fig. 10

Computer simulations of logarithmic asphere blur and recovery. (a) Image blurred by the diffraction-limited lens, (b) image blurred by the logarithmic asphere, (c) maximum-entropy recovery from (b).

Equations (148)

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Eθ=iλ0Aρ2+s2exp-i 2πλ0ρ2+s2,
I(ρ2;s)=δRRρdρ02πdϕ iλ0Aρ2+s2iλ0×t[(ρ cos ϕ-ρ2 cos ϕ2)2+(ρ sin ϕ-ρ2 sin ϕ2)2+t2]1/2×exp-i 2πλ0ρ2+s2-iϕ(ρ)-i 2πλ0[(ρ cos ϕ-ρ2 cos ϕ2)2+(ρ sin ϕ-ρ2 sin ϕ2)2+t2]1/22,
(ρ cos ϕ-ρ2 cos ϕ2)2+(ρ sin ϕ-ρ2 sin ϕ2)2+t2ρ2+t2,
[(ρ cos ϕ-ρ2 cos ϕ2)2+(ρ sin ϕ-ρ2 sin ϕ2)2+t2]1/2ρ2+t2-ρρ2 cos(ϕ-ϕ2)ρ2+t2.
I(ρ2;s)δRR 2πAλ02ρ 1sexp-i 2πλ0ρ2+s2-iϕ(ρ)-i 2πλ0ρ2+t2J02π ρρ2λ0ρ2+t2dρ2.
I(ρ2=0; s)=4π2t2A2λ04s2δRR ρρ2+t2exp-i 2πλ0ρ2+s2-iϕ(ρ)-i 2πλ0ρ2+t2dρ2.
I(ρ2=0; s)s2=4π2t2A2λ04δRR ρρ2+t2exp-i 2πλ0ρ2+s2-iϕ(ρ)-i 2πλ0ρ2+t2dρ2
ϕP(ρ)ϕ(ρ)+2πλ0(ρ2+t2-t).
GδRR ρρ2+t2×exp-i 2πλ0λ02πϕP(ρ)+ρ2+s2dρ2
λ02πϕP(ρ0)=-ρ0/ρ02+s2.
Gλ0ρ02λ02πϕP(ρ0)+1ρ02+s2-ρ02(ρ02+s2)3/2×1ρ02+t2.
G=2πρ02ϕP(ρ0)-ϕP(ρ0)ρ0+λ024π2[ϕP(ρ0)]3ρ01ρ02+t2.
(ρ2+t2)ϕP(ρ)-ϕP(ρ)ρ+λ024π2[ϕP(ρ)]3ρ=2aρ2,
ϕP(ρ)-ϕP(ρ)ρ+λ024π2[ϕP(ρ)]3ρ0.
ϕPβ(ρ)=aβ t2+ρ22{ln[Aβ(t2+ρ2)]-1}+λ028π2116aβ3(t2+ρ2)2[45-42 ln[Aβ(t2+ρ2)]+18{ln[Aβ(t2+ρ2)]}2-4{ln[Aβ(t2+ρ2)]}3]-Bβρ22+const.,
s=s1  ρ0=δR,
s=s2  ρ0=R,
aβ=2πλ01/δR2+s12-1/R2+s22lnR2+t2δR2+t2,
Aβ=1δR2+t2exp-R2+s22R2+s22-δR2+s12×lnR2+t2δR2+t2.
ϕPβ(R)=ddρaβ t2+ρ22{ln[Aβ(t2+ρ2)]-1}ρ=R,
Bβ=-aβ3(R2+t2)(-6+6 ln[Aβ(t2+ρ2)]-3{ln[Aβ(t2+ρ2)]}2+{ln[Aβ(t2+ρ2)]}3).
ϕP(ρ)-ϕP(ρ)ρ+λ024π2[ϕP(ρ)]3ρ0,
ϕPγ(ρ)=-aγ t2+ρ22{ln[Aγ(t2+ρ2)]-1}+λ028π2116aγ3(t2+ρ2)2[-45+42 ln[Aγ(t2+ρ2)]-18{ln[Aγ(t2+ρ2)]}2+4{ln[Aγ(t2+ρ2)]}3]+Bγρ22+const.,
aγ=2πλ01/R2+s12-1/δR2+s22lnR2+t2δR2+t2,
Aγ=1δR2+t2expR2+s12δR2+s22-R2+s12×lnR2+t2δR2+t2,
Bγ=-aγ3(R2+t2)(-6+6 ln[Aγ(t2+ρ2)]-3{ln[Aγ(t2+ρ2)]}2+{ln[Aγ(t2+ρ2)]}3).
ϕPβ(ρ)=aβ t2+ρ22{ln[Aβ(t2+ρ2)]-1}+const.,
aβ=2πλ01/δR2+s12-1/R2+s22lnR2+t2δR2+t2,
Aβ=1δR2+t2exp-R2+s22R2+s22-δR2+s12×lnR2+t2δR2+t2,
ϕPγ(ρ)=-aγ t2+ρ22{ln[Aγ(t2+ρ2)]-1}+const.,
aγ=2πλ01/R2+s12-1/δR2+s22lnR2+t2δR2+t2,
Aγ=1δR2+t2expR2+s12δR2+s22-R2+s12×lnR2+t2δR2+t2.
sβ,γ=4π2λ2aβ,γ2{ln[Aβ,γ(t2+ρ2)]}-2-ρ21/2.
sβ=δR2+s12 lnt2+R2t2+δR2δR2+s12R2+s22-1lnt2+ρ2t2+R2+δR2+s12R2+s22lnt2+R2t2+δR22-ρ21/2,
sγ=δR2+s22 lnt2+R2t2+δR2δR2+s22R2+s12-1lnt2+ρ2t2+R2+δR2+s22R2+s12lnt2+R2t2+δR22-ρ21/2.
ϕPβ(ρ)=aβ t2+ρ22{ln[Aβ(t2+ρ2)]-1}+const.,
aβ=2πλ01/s1-1/R2+s22ln(1+R2/t2),
Aβ=1t2exp-R2+s22R2+s22-s1ln(1+R2/t2).
s=s1 ln(1+R2/t2)s1R2+s22-1lnt2+ρ2t2+R2+s1R2+s22ln(1+R2/t2)2-ρ21/2.
ϕPβ(ρ)=aβ t2+ρ22{ln[Aβ(t2+ρ2)]-1}+const.,
aβ=2πλ01s1 ln(1+R2/t2),
Aβ=1t2+R2,
ϕPβ(ρ)=πλ0t2+ρ2s1 ln(1+R2/t2)lnt2+ρ2t2+R2-1+const.,
s=s1 ln(1+R2/t2)lnt2+R2t2+ρ2.
d(x, y)=-f(ξ, η)h(x-ξ, y-η)dξdη+θ(x, y),
f(m, n)=f(a, b)1aA;1bB01+AmM;1+BnN,
h(m, n)=h(a, b)1aC;1bD01+CmM;1+DnN,
θ(m, n)=θ(a, b)1aC;1bD01+CmM;1+DnN.
d(m, n)=i=1Mj=1Nf(i, j)h(m-i, n-j)+θ(m, n).
f(m-1)N+n=f(m, n),
d(m-1)N+n=d(m, n),
θ(m-1)N+n=θ(m, n).
H=H1HMHM-1H2H2H1HMH3H3H2H1H4HMHM-1HM-2H1,
Hjh(j, 1)h(j, N)h(j, 2)h(j, 2)h(j, 1)h(j, 3)h(j, N)h(j, 1).
d=Hf+θ.
Ω=W!i=1nfi!,
S=ln Ω=ln (W!i=1nfi!)W ln W-i=1nfi ln fi.
S1=-i=1nfi ln fi,
S2=-i=1npi ln pi,pi=fij=1nfj,
S3=-i=1nfi ln fif-1,fisthemeanoftheimage.
d=Hf+θ.
C=i=1n 1σi2di-k=1nHikfk2.
S=-i=1nfi ln fif-1,
Q=S-λC,
eA=f.γ .×Q.
Q=0.
eB=f.γ .×[Q·(f.γ .×Q)].
eA=f.γ .×S-λf.γ .×C,
eB=λ2f.γ .×[C·(f.γ .×C)]-λf.γ .×[C·(f.γ .×S)]+f.γ .×[S·(f.γ .×S)]-λf.γ .×[S·(f.γ .×C)].
e1=f.γ .×S,
e2=f.γ .×C,
e3=f.γ .×[C·(f.γ .×C)],
e4=f.γ .×[C·(f.γ .×S)],
e5=f.γ .×[S·(f.γ .×S)],
e6=f.γ .×[S·(f.γ .×C)].
(S)i=-ln fif,
(S)ij=-1fiδij,
(C)i=k=1nj=1n 2σk2(Hkjfj-dk)Hki
(C)ij=k=1n 2σk2HkjHki.
(e1)i=-fiγ ln fif,
(e2)i=fiγk=1nj=1n 2σk2(Hkjfj-dk)Hki,
(e3)i=fiγj=1nk=1nl=1nm=1nfmγ 4σk2σl2×(Hkjfj-dk)HkmHliHlm,
(e4)i=-fiγj=1nk=1nfjγ 2σk2HkiHkj ln fjf,
(e5)i=fi2γ-1 ln fif,
(e6)i=-fi2γ-1k=1nj=1n 2σk2(Hkjfj-dk)Hki.
λ=i=1nfiγ(S/fi)2i=1nfiγ(C/fi)21/2.
f(n+1)=f(n)+δf,
δf=x1e1+x2e2+x3e3.
S˜=S(f(n))+S·δf+12δf·S·δf,
C˜=C(f(n))+C·δf+12δf·C·δf.
S˜=S0+AX-12XTBX,
C˜=C0+MX+12XTNX.
S0=S(f(n)),
C0=C(f(n)),
X=[x1x2x3]T,
A=[S·e1S·e2S·e3],
B=-e1·S·e1e1·S·e2e1·S·e3e2·S·e1e2·S·e2e2·S·e3e3·S·e1e3·S·e2e3·S·e3,
M=[C·e1C·e2C·e3],
N=e1·C·e1e1·C·e2e1·C·e3e2·C·e1e2·C·e2e2·C·e3e3·C·e1e3·C·e2e3·C·e3.
RBRT=diag(λ1, λ2, λ3),
Y=RX.
S˜=S0+ARTY-12YT diag(λ1, λ2, λ3)Y,
C˜=C0+MRTY+12YTRNRTY.
Z=diag(λ1, λ2, λ3)Y.
S˜=S0+ART diag1λ1, 1λ2, 1λ3Z-12ZTZ,
C˜=C0+MRT diag1λ1, 1λ2, 1λ3Z+12ZTPZ,
P=diag1λ1, 1λ2, 1λ3RNRT diag1λ1, 1λ2, 1λ3.
VPVT=diag(μ1, μ2, μ3),
U=VZ.
S˜=S0+ART diag1λ1, 1λ2, 1λ3VTU-12UTU,
C˜=C0+MRT diag1λ1, 1λ2, 1λ3V T U+12UT diag(μ1, μ2, μ3)U.
X=RT diag1λ1, 1λ2, 1λ3VTU.
z1z2z3=0=λ1000λ20000y1y2y3=0.
S˜=S0+ART diag1λ1, 1λ2, 0Z-12ZTZ,
C˜=C0+MRT diag1λ1, 1λ2, 0Z+12ZTPZ,
P=diag1λ1, 1λ2, 0RNRT diag1λ1, 1λ2, 0.
VPVT=μ1000μ2000μ3=0,
V=ν11ν120ν21ν220000.
U=VZ,
S˜=S0+ART diag1λ1, 1λ2, 0VTU-12UTU,
C˜=C0+MRT diag1λ1, 1λ2, 0VTU+12UT diag(μ1,μ2,0)U.
X=RT diag1λ1, 1λ2, 0VTU.
S˜=S0+siui-12ui2,
C˜=C0+ciui+12μiui2.
C˜min=C0-12 ci2μi.
C˜=C˜0maxC0-13 ci2μi, Caim.
Q˜=αS˜-C˜,α>0,
ui=αsi-ciμi+α.
C0+ciαsi-ciμi+α+12μiαsi-ciμi+α2=C˜0, α>0.
test=S·CS·SC·C.
(ρ2+t2)ϕP(ρ)-ϕP(ρ)ρ+λ024π2[ϕP(ρ)]3ρ=2aβρ2.
(ρ2+t2)ϕP(ρ)-ϕP(ρ)ρ+2 [ϕP(ρ)]3ρ=2aβρ2.
φ(ρ)=ϕP(ρ);
(ρ2+t2)φ(ρ)-φ(ρ)ρ+2 [φ(ρ)]3ρ=2aβρ2.
(ρ2+t2)ϕ˜(ρ)-ϕ˜(ρ)ρ=2aβρ2.
ϕ˜(ρ)=aβρ ln[Aβ(t2+ρ2)],
φ(ρ)=ϕ˜(ρ)+ψ(ρ).
ρψ-ψ+2{aβ ln[Aβ(t2+ρ2)]}3ρ3=0,
ψ=aβ3ρ(t2+ρ2)(6-6 ln[Aβ(t2+ρ2)]+3{ln[Aβ(t2+ρ2)]}2-{ln[Aβ(t2+ρ2)]}3)-Bβρ,
ϕP(ρ)=aβt2+ρ22{ln[Aβ(t2+ρ2)]-1}+116aβ3(t2+ρ2)2(45-42 ln[Aβ(t2+ρ2)]+18{ln[Aβ(t2+ρ2)]}2-4{ln[Aβ(t2+ρ2)]}3)-Bβρ22+const.
zn=s1+(s2-s1)n/N
n=zn-s1s2-s1N.
an=c[s1+(s2-s1)n/N]-2,
πrn2=cn=1[(xn-s1)/(s2-s1)]N[s1+(s2-s1)n/N]-2,
π(R2-δR2)=cn=1N[s1+(s2-s1)n/N]-2.
rn2=(R2-δR2) n=1[(xn-s1)/(s2-s1)]N[s1+(s2-s1)n/N]-2n=1N[s1+(s2-s1)n/N]-2.
r02=(R2-δR2) 0(s-s1)/(s2-s1)[s1+(s2-s1)ξ]-2dξ01[s1+(s2-s1)ξ]-2dξ.
s=s1s2(R2-δR2)s2(R2-δR2)-(s2-s1)r2,δRrR.

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