ua 
function" I(x,y) as 
H(x, y) = Bo if Ix | < 1/2 and ly| « 1/2, 
0 otherwise. (2-5) 
If the pixel centers (x4 9344? form a unit grid, then the locales of H are 
the unit square centered'on the pixel (excluding the perimeter) and the 
unusual locale consisting of the boundary of the square. The latter locale 
results from the fact that, for these positions of the object, all pixels 
values are zero. 
It should be noted that, for a uniform square grid and a spatially invariant 
entity, the partition of the plane into locales has the same spatial 
periodicity as the grid. 
For a quantized entity Hi 063), we define the (ordered) set-valued "image 
I(x, y) = (5G "x, j.7 y)} (2-6) 
where the indicies i, j are ordered in some manner. Thus for a given 
position, the image function is the ordered set of all pixel values for the 
entity. Since the entity has finite support (i.e. it is greater than zero 
only on a set of finite area) only a finite (and usually only a few) pixel 
values are not zero. The "image distance" between two points (x, >Y1) and 
(x5,y5) is defined here as 
| I(xi,yj) ^ I(x», y2) I = ls | Ho 2313.J3:7 yj? 
- Ha = X5, $7 yo). (277) 
According to this definition, two points (x;,y;) and (x5,y5) belong to the 
same locale if and only if 
MI Ce,» 71) (25 95) = 0. (2-8) 
Furthermore, if the unit of quantization of pixel values is q, then if the 
two points are in different locales 
lI (xp yp-I(G. y 7 ka (2-9) 
for some positive integer k. Usually q is taken as unity. 
In some sense each locale corresponds to a unit of available geometric 8 
precision in that the larger the locale the greater the lack of geometric 
precision. The variability of the size and shape of locales, and their 
dependence upon the character of the entity makes it awkward to use locales 
as a direct measure of precision. As an approach to establishing a measure 
of available geometric precision in a digital image we will estimate an 
upper bound on the number of locales along a unit line segment. It should 
be noted that locales may span pixel boundaries and that they may be 
disconnected. The size of a locale must always be less than or equal to 
the size of the unit raster square since a shift by one sampling unit will 
change any non-zero image function. 
3. BOUND ON THE NUMBER OF LOCALES 
As we move along a parametrized line segment x (t) = (x(t), y(t)), an upper 
bound on the number of locale boundaries crossed is given by the total