Abstract

A new method of calculating the optical transfer function (OTF) for image blur caused by arbitrary motion is introduced. Previous methods, except for a few specific types of motion, were numerical rather than analytical. This new method makes it possible to obtain analytical expressions for the OTF deriving from any kind of motion by means of the statistical moments of the motion function. Analytical OTF expressions are derived for linear, quadratic, and exponential motion and for high- and low-frequency vibration. A comparison of image degradation is presented for linear and exponential motion and high-frequency vibration of the same blur extent. The method can be implemented in real-time restoration of images blurred by arbitrary motion and in image motion degradation analysis.

© 1997 Optical Society of America

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References

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  1. T. Trott, “The effects of motion in resolution,” Photogramm. Eng. 26, 819–827 (1960).
  2. N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.
  3. D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
    [CrossRef]
  4. O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
    [CrossRef]
  5. A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).
  6. O. Hadar, I. Dror, N. S. Kopeika, “Real-time restoration of images degraded by motion and vibration,” in Trends in Optical Engineering, J. Menon, ed. (Council of Scientific Research, Vilayil Gardens, Trivandrum, India, 1993), pp. 287–298.
  7. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.
  8. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).
  9. A. Zayezdny, D. Tabak, D. Wulich, Engineering Application of Stochastic Processes (Research Studies Press, London, 1989).
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 131, Eqs. TI226 and TI227.
  11. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 75–76, Eqs. 14.339, 14.347, and 17.366.
  12. Ref. 11, p. 136, Eq. 24.5.
  13. Ref. 11, p. 193, Eq. 35.8.
  14. Ref. 10, p. 15.

1994 (1)

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

1987 (1)

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

1960 (1)

T. Trott, “The effects of motion in resolution,” Photogramm. Eng. 26, 819–827 (1960).

Dror, I.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Real-time restoration of images degraded by motion and vibration,” in Trends in Optical Engineering, J. Menon, ed. (Council of Scientific Research, Vilayil Gardens, Trivandrum, India, 1993), pp. 287–298.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 131, Eqs. TI226 and TI227.

Hadar, O.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Real-time restoration of images degraded by motion and vibration,” in Trends in Optical Engineering, J. Menon, ed. (Council of Scientific Research, Vilayil Gardens, Trivandrum, India, 1993), pp. 287–298.

Jensen, N.

N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).

Kopeika, N. S.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Real-time restoration of images degraded by motion and vibration,” in Trends in Optical Engineering, J. Menon, ed. (Council of Scientific Research, Vilayil Gardens, Trivandrum, India, 1993), pp. 287–298.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 131, Eqs. TI226 and TI227.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 75–76, Eqs. 14.339, 14.347, and 17.366.

Tabak, D.

A. Zayezdny, D. Tabak, D. Wulich, Engineering Application of Stochastic Processes (Research Studies Press, London, 1989).

Trott, T.

T. Trott, “The effects of motion in resolution,” Photogramm. Eng. 26, 819–827 (1960).

Wulich, D.

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

A. Zayezdny, D. Tabak, D. Wulich, Engineering Application of Stochastic Processes (Research Studies Press, London, 1989).

Zayezdny, A.

A. Zayezdny, D. Tabak, D. Wulich, Engineering Application of Stochastic Processes (Research Studies Press, London, 1989).

Opt. Eng. (2)

D. Wulich, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration,” Opt. Eng. 26, 529–533 (1987).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibration. Part IV: Real time numerical calculation of optical transfer functions and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Photogramm. Eng. (1)

T. Trott, “The effects of motion in resolution,” Photogramm. Eng. 26, 819–827 (1960).

Other (11)

N. Jensen, Optical and Photographic Reconnaissance System (Wiley, New York, 1968), pp. 116–124.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1976).

O. Hadar, I. Dror, N. S. Kopeika, “Real-time restoration of images degraded by motion and vibration,” in Trends in Optical Engineering, J. Menon, ed. (Council of Scientific Research, Vilayil Gardens, Trivandrum, India, 1993), pp. 287–298.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), pp. 81–83.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1984).

A. Zayezdny, D. Tabak, D. Wulich, Engineering Application of Stochastic Processes (Research Studies Press, London, 1989).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), p. 131, Eqs. TI226 and TI227.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), pp. 75–76, Eqs. 14.339, 14.347, and 17.366.

Ref. 11, p. 136, Eq. 24.5.

Ref. 11, p. 193, Eq. 35.8.

Ref. 10, p. 15.

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Figures (7)

Fig. 1
Fig. 1

Motion function.

Fig. 2
Fig. 2

(a), (b) Same motion measured from different initial displacements; (c) centralized motion.

Fig. 3
Fig. 3

(a) Upper bound error for nth-order approximation at spatial frequency f=1/d, (b) upper bound error for the nth-order approximation with the use of central moments at spatial frequency f=1/d.

Fig. 4
Fig. 4

(a) Linear motion displacement, (b) exact MTF (solid curve) and Nth-order approximated MTF (other curves), (c) exact PTF (solid curve) and Nth-order approximated PTF (dashed and dotted lines).

Fig. 5
Fig. 5

Parabolic motion. (a) Four forms of parabolic motion with a 1-mm blur extent: A: a=1, V=10; B: a=1, V=1; C: a=1, V=0.1; D: a=1, V=0.01. (b) The solid curves refer to the exact MTF’s (calculated with the use of 40 moments), and the dashed curve refers to the eighth-order approximation. (c) PTF.

Fig. 6
Fig. 6

Low-frequency vibration: (a) displacement during four different exposures (te=0.15T0), (b) MTF’s for the four exposures.

Fig. 7
Fig. 7

(a) Exponential motion with three damping parameters a, (b) exponential motion MTF compared with linear motion (dashed curve) MTF and high-frequency vibration MTF (dotted curve) having similar blur extent (approximately 1 mm), (c) same as (b) but for the PTF comparison.

Equations (55)

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OTF(ω)=F[LSF(x)]=- LSF(x)exp(-jωxx)dx,
OTF(ω)=n=01n!nOTF(ω)ωnω=0ωn.
nOTF(ω)ωnω=0=nωn- LSF(x)exp(-jωx)dxω=0=(-j)n - xnLSF(x)dx.
- xnLSF(x)=E(xn)mn,
mn=E(xn)=- xn(t)ft(t)dt=1tetxtx+te xn(t)dt,
OTF(ω)=n=0mnn!(-jω)n,
mn=1tetxtx+te xn(t)dt.
mn=1S xin.
OTF(ω)n=0Nmnn!(-jω)n.
|Err(N)|ωN+1(N+1)!μN+1,
|Err(N)|ωN+1(N+1)!mN+1,Nodd,
|Err(N)|ωN+1(N+1)!(XM)N+1,
μk=1tetxtx+te |x(t)|k dt,
XM=max|x(t)|,txttx+te.
Err(N)(2π)N+1(N+1)!dN+1fN+1,
xc(t)x(t)-m,
OTF(ω)=OTFc(ω)exp(-jωm),
OTFc(ω)=n=0Mnn!ωn,
MnE([x(t)-m]n)=1tetxtx+te [x(t)-m]n dt.
Mn=k=0n nkmk(-m)n-k,
Err(N)dN+12N+1(N+1)!(2π)N+1fN+1.
x(t)=Vt,tx<t<tx+te,
mn(tx; V, te)=1tetxtx+te (Vt)n dt=Vntn+1te(n+1)txtx+te.
OTF(ω)=n=0 [V(tx+te)]n+1-(Vt)n+1Vte(n+1)!(-jω)n=exp(-jVtxω)-jVteω[exp(-jVteω)-1]=exp[-jV(tx+te/2)ω]-jVteω×[exp(-jVteω/2)-exp(jVteω/2)]=sincVte2ωexp[-jV(tx+te/2)ω].
MTF(ω)=|OTF(ω)|=sincd2ω=|sinc(πdf)|
PTF(ω)=phase(OTF)=-(2πVtx+πd)f,2nd<f<2n+1d,n=0, 1, -(2πVtx+πd)f+π,2n-1d<f<2nd,n=1, 2, ,
x(t)=Vt+at2,tx<t<tx+te.
mn(ω; a, V, tx, te)=1tetxtx+te (at2+Vt)n dt=1tetxtx+tek=0nnk(at)n-k(Vt)k dt=1tenkan-kvkt2n-k+12n-k+1t=txt=tx+te.
OTF(ω)=n=0(-jωa)ntek=0n(V/a)kt2n-k+1(2n-k+1)(n-k)!k!t=txt=tx+te.
x(t)=D sin(ω0t),tx<t<tx+te,
mn(tx; D, ω0, ts, te)=1tetxtx+te [D sin(ω0t)]n dt;
mn=l=0(n-1)/2 an,l sin[(n-2l)ω0tx+ϕn,l],n=1, 3, 5, Dnn!2n[(n/2)!]2+l=0n/2-1 bn,l cos[(n-2l)ω0tx+ϕn,l],n=0, 2, 4, ,
an,l=Dnω0te(-1)(n-1+2l)/22n-1nlsinn2-lω0ten2-l,
bn,l=Dn(-1)n/2+lω0te2n-1nlsinn2-2ω0ten2-l,
ϕn,l=n2-lteω0,
m1=1tetxtx+te D sin(ω0t)dt=-D cos(ω0t)teω0txtx+te,
m2=1tetxtx+te D2 sin2(ω0t)dt=D2tet2-sin(2ω0t)4ω0txtx+te,
mn=1tetxtx+te D sin(ω0t)n dt=Dnte-sinn-1(ω0t)cos(ω0t)nω0txtx+te-nn-1temn-2Dn-2.
mn=0,n=1, 2, 3, Dnn!2n[(n/2)!]2,n=0, 2, 4, .
OTFHF(ω)=k=0(-jωD)2k22k(k!)2J0(ωD),
MTFHF(ω)=|J0(ωD)|=|J0(2πDf)|,
PTFHF(ω)=0,Z2n/2πD<f<Z2n+1/2πD,n=0, 1, 2, π,Z2n-1/2πD<f<Z2n/2πD,n=1, 2, 3, ,
x(t)=C exp(-at)tx<t<tx+te,
mn(tx; C, ts, te)=1tetxtx+te [C exp(-at)]n dt,
mn=1,-[C exp(-at)]nnteat=txt=tx+te=Cn exp(-antx)[1-exp(-ante)]ntea,n=0 n=1, 2, 3, .
OTF(ω)=1-1aten=1 [-jωC exp(-at)]nnn!t=txt=tx+te=1+1aten=1×[-jωC exp(-atx)]n[1-exp(-ante)]nn!,
OTF(ω)=1ate{Ei[jωx(tx+te)]-Ei[jωx(tx)]},
Err(N)=ωN+1(N+1)!N+1 OTF(ω)ωN+1ω=ζ,0ζω.
n OTF(ω)ωnω=ζ=nωn- LSF(x)exp(-jζx)dx=(-j)n+1 - xn+1LSF(x)exp(-jζx)dx.
|Err(N)|=ωN+1(N+1)!- xN+1LSF(x)exp(-jζx)dxωN+1(N+1)!- |xN+1LSF(x)|dx.
|Err(N)|ωN+1(N+1)!- |xN+1|LSF(x)dxωN+1(N+1)!- |x|N+1LSF(x)dxωN+1(N+1)!μN+1,
μk- |x|kLSF(x)dx=1tetxtx+te |x(t)|k dt.
|Err(N)|ωN+1(N+1)!mN+1,Nodd.
|Err(N)|ωN+1(N+1)!(XM)N+1,

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