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Geometric Distortion

Geometric distortion can be separated into three components: optical distortion, detector distortion, and format-dependent distortion. The optical component of the distortion can be described as any distortion due to the HST+FOC optical configuration, i.e., everything which originates upstream from the intensifier tube. It is determined by ray-tracing models of the optical system. The format-dependent distortions include those aspects which differ from format to format, such as the scan distortion discussed above. Most of these distortions result from events which take place downstream from the intensifier tube. Finally, the detector distortions are those which occur within the tube itself.

The accuracy of the geometric distortion correction of FOC data is currently limited by the coarseness and size of the reseau grid (60 pixel spacing). Nominally, the accuracy of the geometric correction is very good - about 0.2%. That is, the true relative separation between points after geometric correction is good to about 0.5 pixels over a separation of about 250 pixels. However, there are several conditions on this which must be fulfilled.

The geometric distortion is only explicitly defined at the positions of the reseau marks present in the format. The actual geometric correction is the result of a 2-D polynomial fit between the observed reseau marks and a reference grid, hence the pixel-by-pixel distortion is the result of interpolation between these known points on the assumption that the variation of the distortion has a relatively long scalelength and is both smooth and well-behaved.

(Note: As discussed in the last section, the scan component of the distortion is not modelled by the fitting procedure because it varies on scale lengths approximately equal to the reseau spacing and so, after geometric correction, this component remains.) The accuracy of the fit to the distortion at any given pixel depends only on the order of the fitted function and how well it is constrained. The order of the 2-D fitting function is determined by the dimensions of the reseau grid that is present in the format being addressed, so that if there are columns and rows then it is normal to use a polynomial which is order in , and order in (for both and ).

If we think about the situation (see Fig. 2), it is clear that the observed reseau grid marks the limit of the constrained fit and that any areas of the image which are outside the observed grid can have no reliable geometric correction based on the reseau grid (light shading in Fig. 2.). Immediately within the reseau grid boundaries we have a region which is only constrained by a single point on the outside (dark shading in Fig. 2.) and in this region the geometric correction can only be considered fair, i.e., position residuals may be as much as 1-1.5 pixels. After taking these considerations into account we see that the geometrically reliable region, where the position residuals will be 0.5 pixels or less, is only the unshaded area in the center, representing about 40%of the 512512 (normal) format. (Figures similar to Fig. 2 can be derived for other formats simply by applying the same rules.) These considerations also highlight the significance of the missing top row of reseau marks in the full 5121024 (zoomed) format since there can be no reliable geometric correction in the region above the reseau grid which, on its own, accounts for a full 15%of the image.

The significance of this discussion is that

  1. The measured distance between two points will have a larger than nominal error if one or both of the points lies in the shaded areas in Fig. 2, and
  2. Structures (such as PSF haloes) which overlap the shaded areas may not be reliably corrected and may retain a residual distortion.

The advent of COSTAR itself, will make no difference to these limitations on the quality of the geometric correction. However, we are investigating a method which should produce a general improvement for most formats, with the exception of the largest (the 5121024 zoomed). Basically, observations will be taken of a crowded star field (in 47 Tucanæ), using all of the necessary formats and, after geometrically correcting the large format, all smaller formats will be corrected using a transformation which aligns the stars with the larger format. In other words, the star positions in the geometrically corrected 5121024 (zoomed) image will be used as the reference for all smaller formats. The standard geometric correction procedure using the reseau grid will only be used for for the large format. Since most of the smaller formats lie within the geometrically reliable area of the large format, and since the stars provide a much finer grid (a random distribution with an average separation of 10-15 pixels), it should be possible to correct these formats almost right to their edges with no significant increase in the geometric residuals. This will mean that the geometrically reliable imaging area of the smaller formats will, in effect, be virtually the whole image.

One bonus of this method is that the camera platescales will, by default, be identical in each format. A somewhat less important advantage will be that images in different formats will be automatically co-aligned.



Next: Non-Linearity Up: Restoration of FOC Imaging Previous: Scan Distortion


rlw@
Thu Jun 2 15:22:28 EDT 1994