Qdi.j) = [Ae (9 69^ 
where A is the amplitude of the dot, (x,y) is its 
position, (i,j) are pixel indices, and the peculiar 
brackets indicate integer truncation. A commonly used 
estimate of the x-axis of the centroid is given by the 
"center of mass” calculation: 
2iQ(,j) 
ZQ(,.j) 
with the summation being over all i and j between 
some values (-1 and +1, say). It is obvious that if the 
object's position is constrained to a square (pixel) 
such as Ixl<1/2, lyl<1/2, then the x-position estimate 
can have only a finite number of values. In 
particular, if 1<A<2 then these values are { 0, +1/2, 
+1/3 }. Due to symmetry, the same is true of the 
estimate of the y-axis of the centroid. The possible 
combinations of (x,y) estimates are restricted to only 
13 values, which are shown as dots in figure 2. The 
regions delineated in the figure are the locales which 
correspond to each of the estimate values. Note that 
ANY position estimator will have no more than 13 
possible values, (with (x,y) constrained to the unit 
square, 1<A<2, and no noise present), since there are 
only 13 possible digital representation for the object. 
^ 
x = 
  
  
FIGURE 2 
LOCALES WITHIN A UNIT SQUARE AND 
LOCATIONS OF THE CORRESPONDING 
POSITION ESTIMATES 
The presence of noise will complicate the situation. 
Detailed analysis of locales in the presence of noise 
is beyond the scope of this presentation, but it 
should be realized that noise can be incorporated into 
the basic theory in an approximate manner by 
defining the number of effective quantization levels 
(dynamic range) to be the number of digital levels 
divided by the number of levels spanned by the 
additive noise. The primary applications of the theory 
are the estimation of quantization uncertainty and 
optimal position estimation. Both will be seen to be 
robust to noise. À more detailed discussion of locales 
can be found in [10]. 
57 
The concept of locales can be easily extended from 
regions of object position to higher dimensional or 
more abstract parameter spaces. For example, the 
position of an edge (ignoring end-points or assuming 
it is infinitely long) can be expressed in terms of its 
slope and distance from the origin. These two 
parameters can be used to construct a locale pattern 
for a straight binary or grey level edge. It is then 
possible to establish bounds on the position and 
orientation of the line as well as an optimal estimate 
of the two parameters. Further elaboration on this 
example will not be presented but the reader is 
invited (challenged) to construct the locale pattern 
based on the discussion in the following section. 
5. GENERATION OF LOCALE 
PATTERNS 
The locale patterns are generated from the contours of 
the target, as explained in [11]. The Gaussian dot 
discussed in the previous section can be represented 
by contours with unit intervals which form 
concentric circles. Using the center of a pixel as the 
reference origin, the contours are drawn concentric to 
the origin to represent the target at position (0,0). 
Displaced versions of this contour pattern are then 
overlaid on the original one to get the locale map. 
The displaced versions are generated by moving the 
contour pattern so that it is concentric with each of 
the other pixels in the analysis window. For a 3 by 3 
window, 9 copies of the basic contour pattern are 
overlaid to get the final locale map. This is how 
figure 2 was generated, except that the resulting 
overlaid contours were truncated at the boundary of 
the unit pixel. 
In the case of higher dimensional position spaces 
(three dimensions) the same procedure is used to 
generate a multi-dimensional mesh of locale volume 
elements. In the case of three dimensional position 
(x,y,z) for the Gaussian dot, with z along the optical 
axis of the imaging camera, the basic contour pattern 
is a set of concentric cones. This pattern is replicated 
by translation in (x,y), then the replicas are all 
merged to form the locale pattern of volume 
elements. 
If the coordinates are parameters (such as orientation 
or size) rather than object position then the 
translation of the basic "contour" pattern when 
constructing the replicas is based on the position of 
the centers of the image pixels in the selected 
parameter space. Detailed or formal discussion of the 
more abstract representation of locales is beyond the 
scope of this paper, but the generality of the concept 
of locales should be noted. 
The basic method of generating locale patterns is 
very simple. It provides an easy method of 
appreciating the distribution of quantization induced 
position uncertainty for any target, no matter how 
complex the target is and no matter how many gray