Gaussian Function - Gaussian Profile Estimation

Gaussian Profile Estimation

A number of fields such as stellar photometry, Gaussian beam characterization, and emission/absorption line spectroscopy work with sampled Gaussian functions and need to accurately estimate the height, position, and width parameters of the function. These are, and for a 1D Gaussian function, and for a 2D Gaussian function. These most common method for estimating the profile parameters is to take the logarithm of the data and fit a parabola to the resulting data set. While this provides a simple least squares fitting procedure, the resulting algorithm is biased by excessively weighting small data values, and this can produce large errors in the profile estimate. One can partially compensate for this through weighted least squares estimation, in which the small data values are given small weights, but this too can be biased by allowing the tail of the Gaussian to dominate the fit. In order to remove the bias, one can instead use an iterative procedure in which the weights are updated at each iteration (see Iteratively reweighted least squares).

Once one has an algorithm for estimating the Gaussian function parameters, it is also important to know how accurate those estimates are. While an estimation algorithm can provide numerical estimates for the variance of each parameter (i.e. the variance of the estimated height, position, and width of the function), one can use Cramer-Rao bound theory to obtain an analytical expression for the lower bound on the parameter variances, given some assumptions about the data.

  1. The noise in the measured profile is either i.i.d. Gaussian, or the noise is Poisson-distributed.
  2. The spacing between each sampling (i.e. the distance between pixels measuring the data) is uniform.
  3. The peak is "well-sampled", so that less than 10% of the area or volume under the peak (area if a 1D Gaussian, volume if a 2D Gaussian) lies outside the measurement region.
  4. The width of the peak is much larger than the distance between sample locations (i.e. the detector pixels must be at least 5 times smaller than the Gaussian FWHM).

When these assumptions are satisfied, the following covariance matrix K applies for the 1D profile parameters, and under i.i.d. Gaussian noise and under Poisson noise:

where is the width of the pixels used to sample the function, is the quantum efficiency of the detector, and indicates the standard deviation of the measurement noise. Thus, the individual variances for the parameters are, in the Gaussian noise case,

and in the Poisson noise case,

For the 2D profile parameters giving the amplitude, position, and width of the profile, the following covariance matrices apply:

\mathbf{K}_{\text{Gauss}} = \frac{\sigma^2}{\pi \delta_x \delta_y Q^2} \begin{pmatrix} \frac{2}{\sigma_x \sigma_y} &0 &0 &\frac{-1}{A \sigma_y} &\frac{-1}{A \sigma_x} \\ 0 &\frac{2 \sigma_x}{A^2 \sigma_y} &0 &0 &0 \\ 0 &0 &\frac{2 \sigma_y}{A^2 \sigma_x} &0 &0 \\ \frac{-1}{A \sigma_y} &0 &0 &\frac{2 \sigma_x}{A^2 \sigma_y} &0 \\ \frac{-1}{A \sigma_x} &0 &0 &0 &\frac{2 \sigma_y}{A^2 \sigma_x} \end{pmatrix} \, \qquad \mathbf{K}_{\text{Poiss}} = \frac{1}{2 \pi} \begin{pmatrix} \frac{3A}{\sigma_x \sigma_y} &0 &0 &\frac{-1}{\sigma_y} &\frac{-1}{\sigma_x} \\ 0 &\frac{\sigma_x}{A \sigma_y} &0 &0 &0 \\ 0 &0 &\frac{\sigma_y}{A \sigma_x} &0 &0 \\ \frac{-1}{\sigma_y} &0 &0 &\frac{2 \sigma_x}{3A \sigma_y} &\frac{1}{3A} \\ \frac{-1}{\sigma_x} &0 &0 &\frac{1}{3A} &\frac{2 \sigma_y}{3A \sigma_x} \end{pmatrix} \ .

where the individual parameter variances are given by the diagonal elements of the covariance matrix.

Read more about this topic:  Gaussian Function

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