Subject: Update of note concerning construction of mosaic images


Hello:

I have updated my note on the construction of mosaic images from
JEM-X. The updated note is attached here.

niels

PS: I may possibly issue this in latex later.

****


    How to construct sensible mosaics from JEM-X skymap data

                   SDAST Technical Note no. 12 rev. 1

                     Niels Lund. September 8, 2003


1) What is the statistical uncertainty on a pixel value in one of the
JEM-X skymaps?

   In all of our methods the skymap pixel values are obtained by adding
   statistically independent counts from a number of points in the
   shadowgram. Assuming Poisson statistics (and many photons) the 
   variance on the total number is just the number itself. 
   In those routines which uses "balanced reconstruction" counts from 
   the non-illuminated detector pixels are subtracted from the skymap 
   pixel value. In midisky8 I subtract counts taken from an empty 
   field exposure. In any cases there are normally many more background 
   pixels than signal pixels so some scaling is applied before the 
   subtraction is done. The variance coming from the background 
   subtraction is therefore normally less than the variance on the 
   signal part.- but it must not be forgotten.
   In any case, we can calculate a variance for each pixel of a skymap
   coming from a particular science window.

2) How much real source exposure are associated to a particular skymap
   pixel for a given science window.

   When we move off axis the signal in the detector from the source 
   diminishes due to the vignetting, first by the collimator alone, 
   later by both the collimator and the mask vignetting. The background 
   also diminishes as we collect our source signal from fewer detector 
   pixels, but even so, as a minimum, we will loose sensitivity by the 
   squareroot of the vignetting factor. 
   
   But, in fact, we loose much more than that: Because of the limited 
   position resolution in the detector source photons being absorbed in 
   a particular pixel will be assigned a "random" position within a 
   cluster of 10 to 30 pixels around the interaction point. For on-axis 
   sources the illuminated pixels comes in clumps corresponding to the 
   mask holes (about 10 pixels) or clusters of mask holes. The source 
   signals from one pixel therefore have a good chance of being picked 
   up in another illuminated pixel - without adding extra background. 
   But when we move off axis the collimator vignetting will cut down 
   the typical size of our illuminated pixel clusters, and more signal
   will be lost to the non-illuminated regions. Of course we can 
   expand the region from which we collect the source signal, but then 
   we add background and although we can increase the sensitivity some,
   we can never recover the on-axis value. 
   
   Using our source illumination function we must derive a generic map 
   of how the source "recovery fraction"  varies as function of sky
   position. The recovery fraction expresses what fraction of the 
   source flux actually is projected back to the position by our 
   reconstruction method - or collected by our light curve generation
   tool.

   This recovered fraction must be multiplied with the number of pixels
   used to collect the signal and by the exposure time. We now have a
   quantity expressing how much additional exposure this map will 
   provide to a combined map.

3) When building a mosaic from several skymaps - how do we assign
   weights to the individual contributions?

   We will then have available for our combined skymap a number of
   potential contributions coming from skymaps from individual science 
   windows.  Each pixel in each contributing skymap is characterized 
   by three numbers: a pixel signal (I assume that all contributions 
   are balanced, i.e. the sum of all pixel signals across the skymap
   is zero), a variance on the pixel signal and a pixel exposure. 

   When adding a new contribution to the combined map we want to 
   maximize the S/N ratio in each pixel. I have a come to the following 
   result:

   If we want to include a pixel characterized by:
                                 signal_x, variance_x and exposure_x

   into the combined image, where the corresponding pixel has:
                                 signal_c, variance_c and exposure_c

   then we calculate a weight factor, w, as:

     if ((exposure_c > 0) && (variance_x > 0))
       w = (exposure_x/variance_x) / (exposure_c/variance_c);
     else w = 1.0;

   and we update the c-quatities by:

       signal_c += w * signal_x;
       exposure_c += w * exposure_x;
       variance_c += w * w * variance_x;

   NOTE: One important point here: When we start building our mosaic
   map we will initially just have data from one science window and
   both the exposure and the variance will be very small at the edge
   of the field of view. Later, other science windows may be added
   with much better exposure to this pixel. This would lead to a weight
   factor larger than one - sometimes much larger than one. This is
   not desirable. So we should modify the prescription above:

      if (w <= 1.0) do as above;
      else {
        w = 1/w; 
        signal_c = signal_x + w * signal_c;
        exposure_c = exposure_x + w * exposure_c;
        variance_c = variance_x + w * w * variance_c;
      }

   In this way we will assure that it is always those science windows
   which have the best exposure to a given pixel which decides the 
   normalization in that pixel.

4) The quantities to plot are: 
       signal / exposure            as showing the source signal
   and:
       sqrt(variance) / exposure    as giving the noise in each pixel.