Quantization is accomplished by integer truncation. The entity function 
has five levels of quantization (counting zero) and has support radius of 
about 1.2, that is; it is non-zero only within a radius of 1.2 of the 
origin. The image function is obtained by sampling the quantized entity 
function H on the set A; 
A = 162,1) : fut + ]3«2] (3-8) 
Explicitly, the image function is as follows: | 
(x,y) = {H(x,y), H(x+l,y), H(x-1,y), 
H(x,y+1), H(x,y-1)} (3-9) 
Along the line segment {(t,b) : 1 <t <0 ] the number of locales is 
bounded by 
z 
li 
2 My FHM) +] 
2 « (8(0, b-1) + H(0, b) + HCO, b+1)) + 1 (3-10) 
The value of N, gives us the maximum number of (distinguishable) positions 
for the entity along such a unit line segment. On the average then, we can 
only know the position of the object to within a distance no smaller than 
1/N, . 
Using equation (3-10), for b=0.0, b=0.25, and b=0.5 the bounds are 13, 11, 
and 13 respectively. The number of locales along the above line segments 
is 11, 9, and 7 respectively (see figure 1). Several of the boundaries 
between locales along the lines happen to be the intersection of prominent 
arcs in the partition diagram. These intersections are responsible for the 
numerical difference between the upper bound and the number of locales 
along the line segment. For other entities, such as in figure 2 a locale 
boundary may correspond to a change in the image function I(x,y) of several 
units. 
The relationship (3-6) between the variation of the entity function and the 
number of locales is similar in principle to the observation by Forstner 
(1982) that geometric precision in image correlation will depend upon the 
variance of the first derivative. The variation is like the variance of the 
first derivative in that both quantities are measures of the 'texture' of the 
function. A similar principle is used by Ryan, Gray and Hunt (1980) in 
defining indicators of correlation errors (see table VII of their paper). 
4, DATA STORAGE CONSIDERATIONS 
Large data sets consume computer resources during storage, retrieval and 
computation as well as by virtue of the volume of their required storage 
medium. In order to make the data handling as efficient as possible, the 
'natural' word size of the machine is often taken into account when deciding 
on the number of data bits to use per pixel. We will not be constrained here 
by computer architecture; we will presume that the number of pits per pixel 
can be freely selected, and may be fully utilized. For a given number of 
bits we will establish a bound on the number of locales across an entity of a 
given size. If an entity is non-zero at only one sampling point, then the 
summation in equation 3-6 is trivial. The bound on the number of locales N, 
across a unit raster square for such an entity is just 
N, = 9b. 4 1 (4-1)