Abstract

We present several new results on the classic problem of estimating Gaussian profile parameters from a set of noisy data, showing that an exact solution of the maximum likelihood equations exists for additive Gaussian-distributed noise. Using the exact solution makes it possible to obtain analytic formulas for the variances of the estimated parameters. Finally, we show that the classic formulation of the problem is actually biased, but that the bias can be eliminated by a straightforward algorithm.

© 2007 Optical Society of America

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References

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  1. J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132–1139 (2005).
    [CrossRef]
  2. D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
    [CrossRef]
  3. D. D. Lenz, T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104–1106 (1992).
    [CrossRef]
  4. K. A. Winick, “Cramér-Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  5. N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57, 1152–1157 (1986).
    [CrossRef]
  6. B. E. A. Saleh, “Estimation of the location of an optical object with photodetectors limited by quantum noise,” Appl. Opt. 13, 1824–1827 (1974).
    [CrossRef] [PubMed]
  7. H. H. Barrett, K. Myers, Foundations of Image Science (Wiley, 2004).
  8. J. Dennis, J.R. E.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM), 1996).
    [CrossRef]
  9. H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1: Detection, Estimation, and Linear Modulation Theory (Wiley, 1968).
  10. M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Charles Griffin, 1969), Vol. 2.
  11. The Interactive Data Language (IDL) software is developed by ITT Visual Information Systems, http://www.ittvis.com/index.asp.

2005 (1)

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132–1139 (2005).
[CrossRef]

1992 (1)

D. D. Lenz, T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104–1106 (1992).
[CrossRef]

1986 (2)

K. A. Winick, “Cramér-Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
[CrossRef]

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57, 1152–1157 (1986).
[CrossRef]

1982 (1)

D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
[CrossRef]

1974 (1)

Ayres, T. R.

D. D. Lenz, T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104–1106 (1992).
[CrossRef]

Barrett, H. H.

H. H. Barrett, K. Myers, Foundations of Image Science (Wiley, 2004).

Bobroff, N.

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57, 1152–1157 (1986).
[CrossRef]

Dennis, J.

J. Dennis, J.R. E.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM), 1996).
[CrossRef]

Ireland, J.

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132–1139 (2005).
[CrossRef]

Kendall, M. G.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Charles Griffin, 1969), Vol. 2.

Landman, D. A.

D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
[CrossRef]

Lenz, D. D.

D. D. Lenz, T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104–1106 (1992).
[CrossRef]

Myers, K.

H. H. Barrett, K. Myers, Foundations of Image Science (Wiley, 2004).

Roussel-Dupré, R.

D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
[CrossRef]

Saleh, B. E. A.

Schnabel, J.R. E.B.

J. Dennis, J.R. E.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM), 1996).
[CrossRef]

Stuart, A.

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Charles Griffin, 1969), Vol. 2.

Tanigawa, G.

D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
[CrossRef]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1: Detection, Estimation, and Linear Modulation Theory (Wiley, 1968).

Winick, K. A.

Appl. Opt. (1)

Astrophys. J. (1)

J. Ireland, “Precision limits to emission-line profile measuring experiments,” Astrophys. J. 620, 1132–1139 (2005).
[CrossRef]

Astrophys. J. (1)

D. A. Landman, R. Roussel-Dupré, G. Tanigawa, “On the statistical uncertainties associated with line profile fitting,” Astrophys. J. 261, 732–735 (1982).
[CrossRef]

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

Publ. Astron. Soc. Pac. (1)

D. D. Lenz, T. R. Ayres, “Errors associated with fitting Gaussian profiles to noisy emission-line spectra,” Publ. Astron. Soc. Pac. 104, 1104–1106 (1992).
[CrossRef]

Rev. Sci. Instrum. (1)

N. Bobroff, “Position measurement with a resolution and noise-limited instrument,” Rev. Sci. Instrum. 57, 1152–1157 (1986).
[CrossRef]

Other (5)

H. H. Barrett, K. Myers, Foundations of Image Science (Wiley, 2004).

J. Dennis, J.R. E.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (SIAM), 1996).
[CrossRef]

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part 1: Detection, Estimation, and Linear Modulation Theory (Wiley, 1968).

M. G. Kendall, A. Stuart, The Advanced Theory of Statistics, 3rd ed. (Charles Griffin, 1969), Vol. 2.

The Interactive Data Language (IDL) software is developed by ITT Visual Information Systems, http://www.ittvis.com/index.asp.

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Figures (6)

Fig. 1
Fig. 1

Example data for a 1D Gaussian profile, generated from the signal ( A = 100.0 , x ¯ = 23.5 , w = 8.0 , σ = 10.0 , δ x = 1 , and M = 101 ). The solid line represents the object f ( x ) (true underlying Gaussian profile); the filled circles represent the measured data values, g.

Fig. 2
Fig. 2

Comparison between the parameter variances as measured by a Monte Carlo simulation (filled circles) and the approximate parameter variance formulas (16) (curves). The model function used for the Monte Carlo simulation is that of Fig. 1, with one of the model parameters allowed to vary, as shown in the figure abscissa. The data for var ( U ^ ) has been rescaled to fit within the plotting range. (Note that the data here was simulated with δ-sampling, so that the bias discussed in Section 6 below is not present in this simulation.)

Fig. 3
Fig. 3

Illustration of constant, linear, and quadratic functions across an idealized pixel. A δ-sampling measurement model selects the function value at the midpoint of the pixel whereas a rect-model selects the mean of the function across the pixel as the result. (The dotted line represents the mean of the parabolic function.)

Fig. 4
Fig. 4

(a) Illustration of the three functions involved in the bias discussion: (1) the underlying continuous function f ( x ) (grey fill), rescaled by 1 / δ x to match the dimensions of g; (2) the piecewise continuous pixel model (stairstep), and (3) the discrete δ-sampled model (spikes). (b) The bias in f ^ ( x ) resulting from using data measured from an array detector while estimating with a δ-sampled model. The curve is calculated from Eq. (28); the filled circles represent data obtained by measuring the bias directly in a simulation. The model f ( x ) used for both (a) and (b) is that of Fig. 1 but measured with 13 pixels rather than 101.

Fig. 5
Fig. 5

Comparison of var ( x ¯ ^ ) determined from Eq. (7) with a Monte Carlo simulation of a Poisson-noise-dominated measurement. In descending order, from top to bottom, the lines and data points correspond to U = 20 000 , 10 000 , 5000 , and 2000 (in units of mean number of photoelectrons).

Fig. 6
Fig. 6

Example data for a 1D Gaussian profile with a quadratic background, generated from the signal ( b 0 = 25.0 , b 1 = 0.15 , b 2 = 0.0008 , A = 100.0 , x ¯ = 23.5 , w = 4.0 , σ = 5.0 , δ x = 1 , and M = 101 ). The solid line represents the object f ( x ) (true underlying Gaussian profile), the filled circles represent the measured data values, g.

Equations (89)

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f ( x ) = A e ( x x ¯ ) 2 / 2 w 2 ,
g ¯ = f ( x ) .
g ¯ m = δ x f ( x ) δ ( x x m ) d x = δ x A e ( x m x ¯ ) 2 / 2 w 2 ,
g = g ¯ + n = f ( x ) + n ,
pr ( g m ) = 1 2 π σ m 2 e ( g m g ¯ m ) 2 / 2 σ m 2 ,
L = pr ( g | θ ) = m = 1 M ( 1 2 π σ m 2 ) 1 / 2 exp [ ( g m g ¯ m ) 2 2 σ m 2 ]
= [ ( 2 π σ m 2 ) 1 / 2 ] exp ( m = 1 M ( g m g ¯ m ) 2 2 σ m 2 ) .
= ln L = [ 1 2 ln ( 2 π σ m 2 ) ] m = 1 M ( g m g ¯ m ) 2 2 σ m 2 .
m = 1 M 1 2 σ m 2 ( g m g ¯ m ) 2 1 σ m 2 ( g m g ¯ m 1 2 g ¯ m 2 ) .
θ i = 1 σ m 2 ( g m g ¯ m ) g ¯ m θ i .
θ ( k + 1 ) = θ ( k ) ( H ( k ) ) 1 ( k ) .
[ ] 1 = A = δ x E m γ x [ ] 2 = x ¯ = δ x A w 2 E m ξ m γ x [ ] 3 = w = δ x A w 3 E m ξ m 2 γ x } ,
H θ 2 = ( 2 A 2 2 A x ¯ 2 A w 2 x ¯ A 2 x ¯ 2 2 x ¯ w 2 w A 2 w x ¯ 2 w 2 ) .
H 11 = δ x 2 1 σ m 2 E m 2 H 12 = δ x w 2 E m ξ m Γ m H 13 = δ x w 3 E m ξ m 2 Γ m H 22 = δ x A w 4 E m [ ξ m 2 Γ m w 2 γ m ] H 23 = δ x A w 5 E m [ ξ m 3 Γ m 2 ξ m w 2 γ m ] H 33 = δ x A w 6 E m [ ξ m 4 Γ m 3 ξ m 2 w 2 γ m ] } ,
K F 1 .
F i j = E { H i j } = ( θ ) ( 2 θ i θ j ) d M g .
F 11 = δ x 2 1 σ m 2 E m 2 F 12 = δ x 2 A w 2 1 σ m 2 E m 2 ξ m F 13 = δ x 2 A w 3 1 σ m 2 E m 2 ξ m 2 F 22 = δ x 2 A 2 w 4 1 σ m 2 E m 2 ξ m 2 F 23 = δ x 2 A 2 w 5 1 σ m 2 E m 2 ξ m 3 F 33 = δ x 2 A 2 w 6 1 σ m 2 E m 2 ξ m 4 } .
var ( A ^ ) = K 11 = F 22 F 33 F 23 2 D ,
var ( x ¯ ^ ) = K 22 = F 11 F 33 F 13 2 D ,
var ( w ^ ) = K 33 = F 11 F 22 F 12 2 D .
E m 2 = 1 δ x e ξ m 2 / w 2 δ x
1 δ x e ξ 2 / w 2 d ξ = π w δ x .
E m 2 ξ m 0 , E m 2 ξ m 2 π w 3 2 δ x ,
E m 2 ξ m 3 0 , E m 4 ξ m 4 3 π w 5 4 δ x ,
F δ x π σ 2 ( w 0 A 2 0 A 2 2 w 0 A 2 0 3 A 2 4 w ) .
K σ 2 π δ x ( 3 2 w 0 1 A 0 2 w A 2 0 1 A 0 2 w A 2 ) ,
var ( A ^ ) 3 σ 2 2 w δ x π ,
var ( x ¯ ^ ) var ( w ^ ) 2 σ 2 w A 2 δ x π .
U = A e ( x x ¯ ) 2 / 2 w 2 d x = A w 2 π .
var ( U ^ ) = U θ T K θ ^ U θ ,
var ( U ^ ) = ( w 2 π 0 A 2 π ) T ( K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 ) ( w 2 π 0 A 2 π )
= 2 π ( w 2 K 11 + 2 A w K 13 + A 2 K 33 )
= 3 π σ 2 w / δ x ,
E m 2 σ m 2 2 π w A δ x 2 ,
E m 2 ξ m σ m 2 0 ,
E m 2 ξ m 2 σ m 2 2 π w 3 A δ x 2 ,
E m 2 ξ m 3 σ m 2 0 ,
E m 2 ξ m 4 σ m 2 3 2 π w 5 A δ x 2 ,
F 2 π ( w A 0 1 0 A w 0 1 0 3 A w ) ,
K 1 2 π ( 3 A 2 w 0 1 2 0 w A 0 1 2 0 w 2 A ) ,
var ( U ^ ) 2 π A w .
g ¯ m = δ x f ( x ) δ ( x x m ) d x .
g ¯ m = f ( x ) rect ( x x m δ x ) d x ,
r e c t ( x / L ) = { 1 : | x | < L / 2 0 : | x | > L / 2 .
θ ^ unbiased = θ ^ biased θ ^ correction .
f ^ unbiased ( x ) = f ^ biased ( x ) b ( x ) .
f ( x ) = [ f ( x ) ] x = x m + ( x x m ) [ f ( x ) ] x = x m mod e l + ( x x m ) 2 2 [ f ( x ) ] x = x m + b i a s e r r o r t e r m , b ( x )
f ( x ) = A w 2 [ ( x x ¯ ) 2 w 2 1 ] e ( x x ¯ ) 2 / 2 w 2 ,
b ( x m ) f ( x m ) 1 2 δ x x m 1 2 δ x x m + 1 2 δ x ( x x m ) 2 d x = A δ x 2 24 w 2 [ ( x m x ¯ ) 2 w 2 1 ] e ( x m x ¯ ) 2 / 2 w 2 .
Uniform: var ( x ¯ ^ ) = 2 U σ 2 π A 3 δ x ,
P o i s s o n : var ( x ¯ ^ ) = U 2 π A 2 .
f ( x ) = δ x [ b 0 + b 1 x + b 2 x 2 + A e ( x x ¯ ) 2 / 2 w 2 ] .
[ ] 1 = b 0 = δ x γ m ,
[ ] 2 = b 1 = δ x x m γ m ,
[ ] 3 = b 2 = δ x x m 2 γ m ,
[ ] 4 = A = δ x E m γ m ,
[ ] 5 = x ¯ = δ x A w 2 E m ξ m γ m ,
[ ] 6 = w = δ x A w 3 E m ξ m 2 γ m ,
H = ( 2 b 0 2 2 b 1 b 0 2 b 2 b 0 2 A b 0 2 x ¯ b 0 2 w b 0 2 b 1 2 2 b 2 b 1 2 A b 1 2 x ¯ b 1 2 w b 1 2 b 2 2 2 A b 2 2 x ¯ b 2 2 w b 2 2 A 2 2 x ¯ A 2 w A 2 x ¯ 2 2 w x ¯ 2 w 2 )
H 11 = δ x 2 S m ,
H 12 = δ x 2 S m x m ,
H 13 = δ x 2 S m x m 2 ,
H 14 = δ x 2 S m E m ,
H 15 = δ x 2 A w 2 S m E m ξ m ,
H 16 = δ x 2 A w 3 S m E m ξ m 2 ,
H 22 = δ x 2 S m x m 2 ,
H 23 = δ x 2 S m x m 3 ,
H 24 = δ x 2 S m E m x m ,
H 25 = δ x 2 A w 2 S m E m x m ξ m ,
H 26 = δ x 2 A w 3 S m E m x m ξ m 2 ,
H 33 = δ x 2 S m x m 4 ,
H 34 = δ x 2 S m E m x m 2 ,
H 35 = δ x 2 A w 2 S m E m x m 2 ξ m ,
H 36 = δ x 2 A w 3 S m E m x m 2 ξ m 2 ,
H 44 = δ x 2 S m E m 2 ,
H 45 = δ x w 2 E m ξ m Γ m ,
H 46 = δ x w 3 E m ξ m 2 Γ m ,
H 55 = δ x A w 4 E m [ ξ m 2 Γ m w 2 γ m ] ,
H 56 = δ x A w 5 E m [ ξ m 3 Γ m 2 ξ m w 2 γ m ] ,
H 66 = δ x A w 6 E m [ ξ m 4 Γ m 3 ξ m 2 w 2 γ m ] ,
F 1 ,: = H 1 ,:
F 2 ,: = H 2 ,:
F 3 ,: = H 3 ,:
F 44 = H 44
F 45 = E { H 45 } = δ x 2 A w 2 S m E m 2 ξ m ,
F 46 = E { H 46 } = δ x 2 A w 3 S m E m 2 ξ m 2 ,
F 55 = E { H 55 } = δ x 2 A 2 w 4 S m E m 2 ξ m 2 ,
F 56 = E { H 56 } = δ x 2 A 2 w 5 S m E m 2 ξ m 3 ,
F 66 = E { H 66 } = δ x 2 A 2 w 6 S m E m 2 ξ m 4 .

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