variation V of the image function as a function of t. . For sufficiently small 
changes At in the parameter t the total variation is given by 
V = ÿ | 1(x(n.At)) = I{xin.40 + ^t)) || (371) 
n= 
(where N=1/At-1 is an integer value). Consider the line segment 
x(t) s (tr, b) 0 stes (3-2) / 
which is a line parallel to the x-axis along y-b, extending a single raster 
unit. The total variation of the quantized image function I for a 
spatially invariant entity H is 
V = ee rone To x D.ac»d 
ne : 
+) 2 IHR - n.At, j - b) - BG-(nrl)*ac, j - 5)| 
3; 1 n=0 
zT [H(n.at,4 - b) - É(n*D-at, 4 - b)] 
j n 
uS UR. (3-3) 
3 J 
where R; is the variation in H(x, y) along the line y = j+b. 
The form of the total variation is particularly simple for entities which are 
unimodal along all cross-sections. A unimodal function is simply one which 
increases monotonically to a maximum value then decreases monotonically 
again. The gaussian function is a good example. For such entities, the 
variation along the line segment 
x). s. (cb) 0.« t Si (3-4) 
is given by 
V2 2-* f MAX (H(x, b * 95] 
j x 
=2 +) M, (3-3) 
j m 
where 
M, = MAX (À (x, bd+j)} 
= x n : 
Since this bounds the number of locales crossed along the line segment, the 
number of locales is bounded by N,: 
D 
- e " 
N. 5 2 «5 M. *-*1 (3-6) 
Figure 1 illustrates the partition of a unit square by the locales defined 
by the entity function 
63-7)