Abstract

The Fourier-space statistical properties of one-dimensional or two-dimensional focal plane array data subject only to photon-counting noise are investigated theoretically by using the discrete Fourier transform. Signal-to-noise ratios and probability density functions for the noise and for the components of the Fourier transform are presented for two cases: when the Fourier transform itself is considered to be the signal and when the power spectrum is considered to be the signal.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. C. Dainty, A. H. Greenaway, “Estimation of spatial power spectra in speckle interferometry,” J. Opt. Soc. Am. 69, 786–790 (1979).
    [CrossRef]
  2. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  3. F. Roddier, “Pupil plane versus image plane in Michelson stellar interferometry,” J. Opt. Soc. Am. A 3, 2160–2166 (1986).
    [CrossRef]
  4. J. F. Walkup, J. W. Goodman, “Limitations of fringe-parameter estimation at low light levels,” J. Opt. Soc. Am. 63, 399–407 (1973).
    [CrossRef]
  5. J. F. Belsher, V. L. Gamiz, P. H. Roberts, “Phase-dependent Poisson noise when estimating fringe phase from patterns of multiple fringes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 30–41 (1999).
    [CrossRef]
  6. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 89 and Chap. 11.
  7. G. Zelniker, F. J. Taylor, Advanced Digital Signal Processing (Marcel Dekker, New York, 1994), Chap. 11.
  8. N. A. J. Hastings, J. B. Peacock, Statistical Distributions (Butterworth, London, 1975), pp. 96 and 108.
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 297, Eq. (7.1.1) and p. 376, Eq. (9.6.16).

1986

1979

1973

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 297, Eq. (7.1.1) and p. 376, Eq. (9.6.16).

Belsher, J. F.

J. F. Belsher, V. L. Gamiz, P. H. Roberts, “Phase-dependent Poisson noise when estimating fringe phase from patterns of multiple fringes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 30–41 (1999).
[CrossRef]

Dainty, J. C.

Gamiz, V. L.

J. F. Belsher, V. L. Gamiz, P. H. Roberts, “Phase-dependent Poisson noise when estimating fringe phase from patterns of multiple fringes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 30–41 (1999).
[CrossRef]

Goodman, J. W.

Greenaway, A. H.

Hastings, N. A. J.

N. A. J. Hastings, J. B. Peacock, Statistical Distributions (Butterworth, London, 1975), pp. 96 and 108.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 89 and Chap. 11.

Peacock, J. B.

N. A. J. Hastings, J. B. Peacock, Statistical Distributions (Butterworth, London, 1975), pp. 96 and 108.

Roberts, P. H.

J. F. Belsher, V. L. Gamiz, P. H. Roberts, “Phase-dependent Poisson noise when estimating fringe phase from patterns of multiple fringes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 30–41 (1999).
[CrossRef]

Roddier, F.

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 89 and Chap. 11.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 297, Eq. (7.1.1) and p. 376, Eq. (9.6.16).

Taylor, F. J.

G. Zelniker, F. J. Taylor, Advanced Digital Signal Processing (Marcel Dekker, New York, 1994), Chap. 11.

Walkup, J. F.

Zelniker, G.

G. Zelniker, F. J. Taylor, Advanced Digital Signal Processing (Marcel Dekker, New York, 1994), Chap. 11.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. F. Belsher, V. L. Gamiz, P. H. Roberts, “Phase-dependent Poisson noise when estimating fringe phase from patterns of multiple fringes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 30–41 (1999).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 89 and Chap. 11.

G. Zelniker, F. J. Taylor, Advanced Digital Signal Processing (Marcel Dekker, New York, 1994), Chap. 11.

N. A. J. Hastings, J. B. Peacock, Statistical Distributions (Butterworth, London, 1975), pp. 96 and 108.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), p. 297, Eq. (7.1.1) and p. 376, Eq. (9.6.16).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Error figure, with signal and noise normalized by p, for Re S2k=Im S2k=0.5, showing constant-probability contours that are exp(-1/2) (inner curve) and exp(-2) (outer curve) below the peak value. The error ellipse is displaced from the origin by a normalized signal, represented by the arrow, having an SNR of 2: ζFT,k=|Xk|/p=2. Dotted lines show the Cartesian coordinates used to approximate polar coordinates (see the text).

Fig. 2
Fig. 2

Comparison of PDF’s obtained from relation (42) and from numerical integration of Eq. (25), using parameters ζFT,k=2, Re S2k=Im S2k=0 (see the text).

Equations (86)

Equations on this page are rendered with MathJax. Learn more.

ndn exp-i 2πNnkn1=0N1-1n2=0N2-1dn1n2 exp-i 2πN1n1k1-i 2πN2n2k2,
ζPS,k=0=p2(4p3+2p2)1/2p2q2,
Dk=Xk+Yk=ndn exp-i 2πNnk=n(xn+yn)exp-i 2πNnk,
Yk=nyn exp-i 2πNnk=nynexp-i 2πNnk=0,
|Yk|2=YkYk*=n, mynymexp-i2πN(n-m)k=nxn=p,
ζFT,k|Xk||Yk|2=|Xk|p=p|Sk||Dk|q,
(Re Yk)2=n,mynymcos2πNnkcos2πNmk=nxn cos22πNnk=12 nxn1+cos2πNn2k=12(p+Re X2k)=p2(1+Re S2k).
(Im Yk)2=p2(1-Re S2k),
(Re Yk)(Im Yk)=p2 Im S2k.
ζRe FT,k=|Re Xk|p2(1+Re S2k)1/2,
ζIm FT,k=|Im Xk|p2(1-Re S2k)1/2.
Yk2=p2[1+|S2k|cos(arg S2k-2α)],
Yk2=p2[1-|S2k|cos(arg S2k-2α)].
signal=|Xk|2=p2|Sk|2,
Nk|Dk|2-q-|Xk|2=YkXk*+Yk*Xk+|Yk|2-q=2p Re(Sk*Yk)+|Yk|2-q.
Var(Nk)=2p|Xk|2+|X2k|2+p2+2 Re(X2kXk*2)=2p3[|Sk|2+Re(S2kSk*2)]+p2(1+|S2k|2).
ζPS,k˜=p2|Sk|2{2p3[|Sk|2+Re(S2kSk*2)]+p2(1+|S2k|2)}1/2,
ζNI,PS,k˜p2|Sk|2(2p3|Sk|2+p2)1/2=p|Sk|2(2p|Sk|2+1)1/2p21/2|Sk||Dk|2q,
ζPS,k=0,N/2=p2|Sk|2(4p3|Sk|2+2p2)1/2p2|Sk||Dk|2q.
ζPS,k>0=W0pR/2(1+W0pR)1/2=4|Sk|2p/4(1+4|Sk|2p/2)1/2=p|Sk|2(1+2p|Sk|2)1/2,
|Dk|4=(|Xk|2+YkXk*+Yk*Xk+|Yk|2)2=|Xk|4+4|Xk|2|Yk|2+Yk2Xk*2+Yk*2Xk2+2YkXk*|Yk|2+2Yk*Xk|Yk|2+|Yk|4=|Xk|4+4|Xk|2|Yk|2+|Yk|4+2 Re(Xk*2Yk2)+4 Re(Xk*Yk|Yk|2)=|Xk|4+|X2k|2+2 Re(X2kXk*2)+4(1+p)|Xk|2+2p2+p,
Re Yk=nyn cos2πNnk,
Var(Yk=0,N/2)=nVar(±yn)=nxn=p,
PY,k=0,N/2(Yk)=12πp exp-Yk22p.
PY,k˜(Yk)=12πσ1σ2 exp-(Re Yk cos α+Im Yk sin α)22σ12-(-Re Yk sin α+Im Yk cos α)22σ22,
σ12=p2(1+|S2k|),
σ22=p2(1-|S2k|),
α=12 arg S2k.
PNI,Y,k˜(YK)=1πp exp-|Yk|2p.
PRe Y, k˜(Re Yk)=12πσR2 exp-(Re Yk)22σR2,
PIm Y, k˜(Im Yk)=12πσI2 exp-(Im Yk)22σI2,
Parg Y, k˜(arg Yk)=σ1σ22π[σ22 cos2(arg Yk-α)+σ12 sin2(arg Yk-α)].
P|Y|, k˜(|Yk|)=|Yk|σ1σ2 exp-|Yk|24 1σ12+1σ22×I0|Yk|24 1σ12-1σ22,
PNI,|Y|,k˜(|Yk|)=2|Yk|p exp-|Yk|2p,
Parg D,k˜(θ)=σ1σ22πA(θ) exp-C2σ12σ22+B(θ)8π[A(θ)]3/2 exp-C-B2(θ)/A(θ)2σ12σ22×1+erfB(θ)σ1σ22A(θ),
PNI,argD,k˜(θ)=12π exp-|Xk|22σ2+|Xk|cos θ8πσ2 exp-|Xk|2 sin2 θ2σ2×1+erf|Xk|cos θ2σ2,
Parg D,k˜(θ)|Xk|2πσarg Dk2 exp-|Xk|2θ22σarg Dk2,
σ|Dk|2Var(|Dk|)=|Dk|2-|Dk|2,
|Dk|2=|Xk|2+p,
|Dk|2=|Xk|2+p-σ|Dk|2,
|Dk|=(|Xk|2+p-σ|Dk|2)1/2|Xk|1+p-σ|Dk|22|Xk|2.
P|D|,k˜(|Dk|)12πσ|Dk|2 exp-(|Dk|-|Dk|)22σ|Dk|2.
PNI,N,k˜(Nk)=1ep exp-Nkp,Nk-p=0, Nk<-p,
PY2,N/2(YN/22)=12|YN/2|2PY,N/2(YN/2)=1(2πpYN/22)1/2 exp-YN/222p,
Nk=|Dk|2-q-|Xk|2|Dk|2-p-|Xk|2=(|Dk|+|Xk|2+p)(|Dk|-|Xk|2+p).
|Dk|+|Xk|2|Xk|=2p|Sk|
Nk2p|Sk|(|Dk|-|Xk|).
PN,k˜(Nk)1(8πp2|Sk|2σ|Dk|2)1/2 exp-Nk28p2|Sk|2σ|Dk|2,
PN,k=0,N/2(Nk)1(8πp3Sk2)1/2 exp-Nk28p3Sk2,
Pr(d)=exp(-x) xdd!,
d=x,d2=x2+x,
d3=x3+3x2+x,d4=x4+6x3+7x2+x,
y=0,
y2=y3=x,
y4=3x2+x.
Pd(d)=12πx exp[-(d-x)2/2x].
d=x,d2=x2+x,
d3=x3+3x2,d4=x4+6x3+3x2
y=0,y2=x,
y3=0,y4=3x2.
Nk2=(YkXk*+Yk*Xk+|Yk|2-q)2=Yk2Xk*2+2|Xk|2|Yk|2+2YkXk*|Yk|2-2YkXk*q+Yk*2Xk2+2Yk*Xk|Yk|2-2Yk*Xkq+|Yk|4-2|Yk|2q+q2=2|Xk|2|Yk|2+|Yk|4-2|Yk|2q+q2+2 Re(Xk*2Yk2)+4 Re(Xk*Yk|Yk|2)-4 Re(Xk*Ykq).
Yk2=nmynymexp-i 2πN(n+m)k=nxn exp-i 2πN2nk=X2k,
Ykq=nlyn(xl+yl)exp-i 2πNnk=nxn exp-i 2πNnk=Xk,
|Yk|2q=n,m,lynym(xl+yl)exp-i 2πN(n-m)k=n,lxnxl+nxn=nxnlxl+nxn=p2+p,
Yk|Yk|2=n,m,lynymylexp-i 2πN(n+m-l)k=nxn exp-i 2πNnk=Xk,
|Yk|4=n,m,l,jynymylyj×exp-i2πN(n-m+l-j)k.
nyn4=n(3xn2+xn)=3nxn2+p,
nllnyn2yl2=nlxnxl-lxl2=p2-lxl2,
nmmnyn2ym2exp-i 2πN2(n-m)k=nmxnxm exp-i 2πN2(n-m)k-mxm2=X2kX2k*-mxm2.
|Yk|4=|X2k|2+2p2+p,
Nk2=2p|Xk|2+|X2k|2+p2+2 Re(X2kXk*2).
Nk2=(2p-4)|Xk|2+|X2k|2+p2+p+2 Re(X2kXk*2).
u=u cos α+v sin α,v=-u sin α+v cos α,
u=u cos α-v sin α,v=u sin α+v cos α.
Pu,v(u,v)=12πσ1σ2 exp-u22σ12-v22σ22,
u2=12πσ1σ2 (u cos α-v sin α)2×exp-u22σ12-v22σ22dudv=σ12 cos2 α+σ22 sin2 α,
v2=12πσ1σ2 (u sin α+v cos α)2×exp-u22σ12-v22σ22dudv=σ12 sin2 α+σ22 cos2 α,
uv=12πσ1σ2 (u cos α-v sin α)(u sin α+v cos α)exp-u22σ12-v22σ22dudv=(σ12-σ22)sin α cos α=12(σ12-σ22)sin 2α.
Pu(u)=12πσ1σ2  exp-(u cos α+v sin α)22σ12-(-u sin α+v cos α)22σ22dv=12πσ1σ2  exp-u22σR2-σR22σ12σ22(v-Eu)2dv=12πσR2 exp-u22σR2,
σR2=σ12 cos2 α+σ22 sin2 α=p2(1+Re S2k).
Pθ,SNR=0(θ)=12πσ1σ2 0 exp-r22 cos2(θ-α)σ12+sin2(θ-α)σ22r dr=σ1σ22π[σ22 cos2(θ-α)+σ12 sin2(θ-α)].
Pr,SNR=0(r)=12πσ1σ2 02π exp-r22 cos2 θσ12+sin2 θσ22r dθ=r2πσ1σ2 exp-r24 1σ12+1σ22×02π exp-r24 1σ12-1σ22cos 2θdθ=rσ1σ2 exp-r24 1σ12+1σ22I0r24 1σ12-1σ22,
Pθ(θ)=12πσ1σ2 0 exp-[(r cos θ-u0)cos α+r sin θ sin α]22σ12-[-(r cos θ-u0)sin α+r sin θ cos α]22σ22r dr=12πσ1σ2 0 exp-12σ12σ22 A(θ)r-B(θ)A(θ)2+C-B2(θ)/A(θ)r dr=12π exp-C-B2(θ)/A(θ)2σ12σ22σ1σ2A(θ) exp-B2(θ)2σ12σ22A(θ)+B(θ)[A(θ)]3/2 π21/21+erfB(θ)σ1σ22A(θ),
 A(θ)=σ22 cos2(θ-α)+σ12 sin2(θ-α)=p2[1-|S2k|cos 2(θ-α)],
B(θ)=u0[σ22 cos(θ-α)cos α-σ12 sin(θ-α)sin α]=u0 p2[cos θ-|S2k|cos(θ-2α)],
C=u02(σ22 cos2α+σ12 sin2α)=u02 p2(1-|S2k|cos 2α).

Metrics