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       RegBounds - Region Boundaries


       Describes how spatial region boundaries are handled.


       The golden rule for spatial region filtering was first enunciated by Leon VanSpeybroeck in

       Each photon will be counted once, and no photon will be counted more than once.

       This means that we must be careful about boundary conditions.  For example, if a circle is
       contained in an annulus such that the inner radius of the annulus is the same as the
       radius of the circle, then photons on that boundary must always be assigned to one or the
       other region. That is, the number of photons in both regions must equal the sum of the
       number of photons in each region taken separately.

       With this in mind, the rules for determining whether a boundary image pixel or table row
       are assigned to a region are defined below.

       Image boundaries  -  radially-symmetric shapes (circle, annuli, ellipse)

       For image filtering, pixels whose center is inside the boundary are included.  This also
       applies non-radially-symmetric shapes.  When a pixel center is exactly on the boundary,
       the pixel assignment rule is:

       ·   the outer boundary of a symmetric shape does not include such pixels

       ·   the inner boundary of a symmetric shape (annulus) includes such pixels

       In this way, an annulus with radius from 0 to 1, centered exactly on a pixel, includes the
       pixel on which it is centered, but none of its neighbors.

       These rules ensure that when defining concentric shapes, no pixels are omitted between
       concentric regions and no pixels are claimed by two regions.  When applied to small
       symmetric shapes, the shape is less likely to be skewed, as would happen with non-
       radially-symmetric rules.  These rules differ from the rules for box-like shapes, which
       are more likely to be positioned adjacent to one another.

       Image Boundaries: non-radially symmetric shapes (polygons, boxes)

       For image filtering, pixels whose center is inside the boundary are included. This also
       applies radially-symmetric shapes.  When a pixel center is exactly on the boundary of a
       non-radially symmetric region, the pixel is included in the right or upper region, but not
       the left or lower region.  This ensures that geometrically adjoining regions touch but
       don't overlap.

       Row Boundaries are Analytic

       When filtering table rows, the boundary rules are the same as for images, except that the
       calculation is not done on the center of a pixel, (since table rows, especially X-ray
       events rows, often have discrete, floating point positions) but are calculated exactly.
       That is, an row is inside the boundary without regard to its integerized pixel value.  For
       rows that are exactly on a region boundary, the above rules are applied to ensure that all
       rows are counted once and no row is counted more than once.

       Because row boundaries are calculated differently from image boundaries, certain programs
       will give different results when filtering the same region file. In particular,
       fundisp/funtable (which utilize analytic row filtering) perform differently from funcnts
       (which performs image filtering, even on tables).

       Image Boundaries vs. Row Boundaries: Practical Considerations

       You will sometimes notice a discrepancy between running funcnts on an binary table file
       and running fundisp on the same file with the same filter.  For example, consider the

         fundisp test1.fits"[box(4219,3887,6,6,0)]" ⎪ wc
         8893  320148 3752846

       Since fundisp has a 2-line header, there are actually 8891 photons that pass the filter.
       But then run funtable and select only the rows that pass this filter, placing them in a
       new file:

         ./funtable test1.fits"[box(4219,3887,6,6,0)]" test2.fits

       Now run funcnts using the original filter on the derived file:

         ./funcnts test2.fits "physical; box(4219,3887,6,6,0)"

         [... lot of processed output ...]

         # the following source and background components were used:
         source region(s)
         physical; box(4219,3887,6,6,0)

          reg       counts    pixels
         ---- ------------ ---------
            1     7847.000        36

       There are 1044 rows (events) that pass the row filter in fundisp (or funtable) but fail to
       make it through funcnts. Why?

       The reason can be traced to how analytic row filtering (fundisp, funtable) differs from
       integerized pixel filtering(funcnts, funimage). Consider the region:


       Analytically (i.e., using row filtering), positions will pass this filter successfully if:

         4216 <= x <= 4222
         3884 <= y <= 3890

       For example, photons with position values of x=4216.4 or y=3884.08 will pass.

       Integerized image filtering is different in that the pixels that will pass this filter
       have centers at:

         x = 4217, 4218, 4219, 4220, 4221, 4222
         y = 3885, 3886, 3887, 3888, 3889, 3890

       Note that there are 6 pixels in each direction, as specified by the region.  That means
       that positions will pass the filter successfully if:

         4217 <= (int)x <= 4222
         3885 <= (int)y <= 3890

       Photons with position values of x=4216.4 or y=3884.08 will NOT pass.

       Note that the position values are integerized, in effect, binned into image values.  This
       means that x=4222.4 will pass this filter, but not the analytic filter above. We do this
       to maintain the design goal that either all counts in a pixel are included in an
       integerized filter, or else none are included.

       [It could be argued that the correct photon limits for floating point row data really
       should be:

         4216.5 <= x <= 4222.5
         3884.5 <= y <= 3890.5

       since each pixel extends for .5 on either side of the center. We chose to the maintain
       integerized algorithm for all image-style filtering so that funcnts would give the exact
       same results regardless of whether a table or a derived non-blocked binned image is used.]


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