Summary
Describes how spatial region boundaries are handled.
Description
The golden rule for spatial region filtering was first enunciated by
Leon VanSpeybroeck in 1986:
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)
or 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.
Other pages of interest:
Last updated: August 7, 2015