Error Estimation in Elliptical Isophote Fitting

I. C. Busko

Astrophysics Division, National Institute for Space Research, Brazil

Abstract:

1. The Problem

The algorithm described by Jedrzejewski (1987) for fitting elliptical isophotes to galaxy images works by applying iterative corrections to the geometrical parameters of a trial ellipse. It does not compute directly the parameters (center coordinates, ellipticity, position angle), but only their increments. Thus, there is no way of directly estimating parameter errors from a covariance matrix and error propagation.

In this work, a simple statistical approach was used to assess the ``true'' parameter distributions: simulated observations of a ``perfect'' elliptical object were built; from sets of repeated observations of the same object, properties of the statistical distribution of each parameter were estimated, and from them a sensible computation for the parameter errors was devised. Important parameters whose errors can be computed directly by error propagation (isophotal intensity/magnitude, and shape parameter) were also included in the study.

  
Figure 1: Bias and accuracy of error computation, for geometrical parameters (top to bottom) ellipticity, position angle, and center coordinate. Ordinate is measured minus ``true'' error (units are degrees for P.A. and pixels for ). Measured errors were computed with empirical formulae. Each symbol depicts the average over the 30 isophotes with fixed average signal-to-noise ratio across one set of simulated observations. Error bars in this plot depict the standard deviation in the 30-isophote set. Solid symbols are from flat object (), representative of the range . Open symbols are from almost round object (). For small gradient errors, estimators are unbiased, but for gradient errors larger than 50 % they systematically underestimate the true error, besides loosing accuracy. Outliers at small gradient error are due to divergency in solutions for round objects.
Figure 1: PS 181 Kb

  
Figure 2: Bias and accuracy of error computation, for (top to bottom) isophotal intensity, magnitude, and shape parameter . Errors were computed with standard formulae. See Figure 1 caption for explanation. Intensity scale unit is one sky standard deviation (per pixel). Only solid symbols for object are shown. Intensity and magnitude errors are accurate but biased: small and roughly constant intensity bias generates trend in magnitude bias, due to correlation between gradient error and isophotal intensity (larger gradient errors are associated with fainter isophotes). errors also show small underestimation bias at small gradient errors, and evidence of overestimation bias at large gradient errors, loosing accuracy for gradient errors large than 40%.
Figure 2: PS 148 Kb

2. Simulations

The simulated observations are -pixel frames depicting elliptical objects with deVaucouleurs profiles and encompassing a range of brightnesses and ellipticities (). Noise from either pure Gaussian or Gaussian + Poisson distributions was added in such a way as to make the (combined) range of pixel signal-to-noise ratio to span from several hundreds down to S/N 0.01. Thirty such observations of each ``object'' were built and measured by ellipse, each time using a different empirical formulation for the error computations.

Each measured isophotal level was scanned across the 30 observations of a given set. Thus, the scanning included isophotes observed with a fixed average S/N. From those 30-isophote samples, three statistics were constructed: (i) the local radial gradient relative error, , averaged over the 30 isophotes. It directly correlates with pixel S/N at the given isophotal level, which in turn is constant (on average) across the 30-isophote set¹; (ii) for each isophote parameter under study, the average of its computed error; (iii) again for each of the parameters, the standard deviation of its 30 measured values.

If the error computation is correct, that is, if it reflects the population true standard deviation, the difference

will distribute itself around zero. If any bias is present, the average of this distribution will depart from zero. Also, the distribution's width is a measure of the error estimator's accuracy.

3. Results

The main results from this study are summarized in Figures 1 and 2. They depict the dependency of the differences with the radial gradient relative error, . The particular error computation used for getting these results is now implemented in a new version of ellipse.

Conclusions from this study can be summarized as follows.

A single parameter, the local radial gradient relative error , suffices to describe the precision to which a given isophote can be fitted.

Geometry errors are unbiased and accurate for smaller than 50%. Larger gradient errors induce decreased accuracy, and non-zero bias in the sense that estimators underestimate the true error. A practical upper limit of 50% for the radial gradient relative error is suggested by the present results.

Intensity and magnitude errors, computed with standard formulae, show good accuracy. Intensity errors slightly underestimate the true error, and this causes magnitude errors to show an underestimation trend that depends on the particular magnitude scale chosen.

errors, also computed with standard formulation, show more complex behavior: small underestimation bias at , and evidence of overestimation bias above .

Too ``round'' isophotes cannot be fitted with the same precision as flatter ones, for a given signal-to-noise ratio, as expected. Results suggest that is a practical limit for reliable position angle determination at any S/N.

Results showed absolutely no dependency with the tested noise models, either pure Gaussian or Gaussian + Poisson.

Acknowledgments:

The author whishes to acknowledge the Program Organizing Committee for providing support to attend the Conference.

References:

¹Other S/N-related quantities were also tested, but was the one that gave best results.