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8 Field of view distortion correction

Field distortion in ISOCAM is mostly due to the off-axis mirror that directs the light beam toward each detector and to the fact that the ISOCAM field of view is an off-axis part of the full FOV of the ISO telescope. The field distortion was measured for the LW channel 6'' and 3'' lenses, using calibration observations of fields that contained many stars. No measurements have been made for the 1.5'' lens because they are difficult to perform as the amplitude of the satellite jitter is of the order of the quantities to be measured. Since the distortion with the 1.5'' lens is predicted to be negligible, no error would be made if it is not taken into account.

ISOCAM also suffers from lens wheel jitter. In order to avoid any mechanical blocking, the gear wheel has been designed with a small play. Therefore, the position at which the lens stops is not fixed.

It has been shown by the CAM Instrument Dedicated Team that there are only two broad families of positions that the lens wheel can take for a commanded position, and it is suspected that the wheel stops at either side of the play. This can be very easily detected by close inspection of the flat field derived from the data: the leftmost column of the detector receives very little light. This is called the "left'' position. This jitter results in an offset of about 1.2 pixels of the optical axis, thus $\approx$ 7'' with the 6'' lens. It also modifies the distortion pattern and therefore the latter has been measured for both positions. The measurement method is discussed in Aussel (1997).

  
\begin{figure}
\psfig {figure=velo.ps,bbllx=2cm,bblly=5.5cm,bburx=20cm,bbury=22.5cm,width=8.8cm}\end{figure} Figure 8: ISOCAM field of view distortion (6 arcsec lens)

Following the work done on the HST WFPC by Holtzman et al. (1995), each measurement is fitted with a general polynomial of degree 3, that is:
\begin{eqnarray}
x_{\rm c} &=& a_{0}+ a_{1}x + a_{2} y + a_{3}x^{2}+ a_{4}x y +
...
 ...nonumber \\ &&+ a_{6}x^{3}+ a_{7}x^{2}y + a_{8}xy^{2}+ a_{9}y^{3} \end{eqnarray}
(23)
\begin{eqnarray}
y_{\rm c} &=& b_{0}+ b_{1}x + b_{2} y + b_{3}x^{2}+ b_{4}x y +
...
 ...number \\  
&&+ b_{6}x^{3}+ b_{7}x^{2}y + b_{8}xy^{2}+ b_{9}y^{3} \end{eqnarray}
(24)
where $x_{\rm c}$ and $y_{\rm c}$ are the positions on the ISOCAM LW array in pixels, corrected for distortion, while x and y are the non-corrected ones. Figure 8 shows a map of the distortion of the LW channel of ISOCAM with the 6'' lens, where each vector starts from where the center of a pixel should fall were there no distortion and ends at its actual position. The length of the vectors are at the scale of the plot. At the lower corners of the array (lines 0 to 5), the effect is greater than one pixel.

Since the pixel size is not uniform on the sky, the pixels at the edges of the array cover a wider surface. Therefore, a new flat-field correction has to be applied in order to account for it. This flat-field is of the form:
\begin{displaymath}
F_{i,j} = \frac{S_{16,\, 16}}{S_{i,j}}\end{displaymath} (25)
with Si,j, the surface on the sky of pixel i,j. The pixel (16, 16), being the center pixel of ISOCAM LW array and therefore the less distorted, has been taken as reference.

  
\begin{figure}
\psfig {file=fig_nocor.ps,width=0.45\textwidth}
 \quad 
 
\psfig {file=fig_cor.ps,width=0.45\textwidth}\end{figure} Figure 9: Zoom on a part of the ISOCAM Hubble Deep Field observation at $15\
\mu$m. The same processing was applied to the two images, except for the correction of the distortion. Cuts on images are at the same level. Pixel size is 2''

This is where distortion corrections stop in the case of staring observations. In a raster mode, where an image has to be constructed from the coaddition of many smaller ones, the processing continues as follows: each raster sub-image is projected on the raster map, using a flux-conservative shift and add algorithm. The intersecting surface S(x,y,i,j) of each sky pixel of the raster map with each ISOCAM pixel is computed. The flux in the pixel (x,y) of the raster map is therefore:
\begin{displaymath}
R_{x,y} = \frac{\sum_{\rm pointings}S_{(x,y,i,j)}\sqrt{N_{i,...
 ..._{i,j}}
{\sum_{\rm pointings}S_{(x,y,i,j)}\sqrt{N_{i,j}}} \cdot\end{displaymath} (26)
Assuming Gaussian noise distributions for pixels, the noise map is:
\begin{displaymath}
\sigma_{x,y} = 
\sqrt{\frac{\sum_{\rm pointings}S^{2}_{(x,y,...
 ...2}_{i,j}}
{\sum_{\rm pointings}S^{2}_{(x,y,i,j)}N_{i,j}}} \cdot\end{displaymath} (27)
Where Rx,y is the value of the final raster map at (x,y), S(x,y,i,j) is the intercepted surface between ISOCAM pixel (i,j) and raster map pixel (x,y) and Ni,j is the number of readouts averaged together to produce Ii,j, the image of the raster pointing. The computation of S(x,y,i,j) is derived from the algorithm used by Fruchter et al. in their "drizzle'' IRAF task. The files containing the coefficients for the distortion correction are available from the authors under a format accepted by this task.

Figure 9 shows a zoom on a part of the ISOCAM Hubble Deep Field observation at 15$\mu$m. The same processing was applied to the two images, except for the correction of the distortion (pixel size is 2'').


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