2. DEFINITIONS AND NOTATION 
Since we are interested in 
(mathematically) about 
CPs, we shall 
notation. 
saying something 
the spatial distribution of 
introduce some basic ideas and 
Our main tool will be the comparison of the spatial 
distribution of the (say "n") given CPs with 
respect to the uniform distribution of (say "n", 
again) points in the unit square (that we will 
write "S"); whenever data do not come in this 
scale, the transformation is obvious. 
The n CPs are said to be uniformly distributed in S 
iff their coordinates (x,, ys x y.) are 
the outcomes of 2n independent identically 
distributed random variables with the uniform 
distribution in the [0,1] interval (the reader 
should notice that these are not n evenly spaced 
points). We will test the (empirical) distribution 
of this set of n coordinates, against the 
hypothesis of uniform distribution over S, using 
simulation. The use of simulation is needed since 
there are no theoretical results about the tests we 
shall use. 
Let d, be the distance of the point j to its 
nearest neighbour; let d''-(d,,...,d ), with m-n(n- 
1)/2 be the n(n-1)/2 dimensional vector of the 
nearest neighbour distances among the n points. 
Denote d' the vector d'' with its elements sorted 
in ascending order, and d the vector formed by the 
first n elements of d'. We will compare the values 
of the n components of d with other values obtained 
by the sampling from the hypothesized distribution; 
this sampling is repeated a fixed number (M) of 
times ("iterations"), usually with M=19 or M=99. 
3. THE ALGORITHM 
The algorithm is as follows (Diggle, 1983): 
1) For j » 1 to n do 
min (j) =2 
max (j) = 0 
mean (j) = 0 
2) For k= 1 to M do 
2.1) Observe u,,...u 3 V,...V : 2n independent 
outcomes of a random variable uniformly 
distributed over [0,1]. 
2.2) Form the coordinates set for the current 
iteration CDAPPELEE (ub v). 
2.3) Calculate e - (e,,..., e ), where e is a 
n-dimensional vector defined in the same 
manner as d above, for the current set of 
coordinates, 
2.4) For j » 1 to n do 
mean (j) = mean (j) -* e, 
if e.€ min(j) then 
lét min(j) = e, 
if e.>max(j) then 
let max(j) = e 
3) For j » 1 to n do 
mean(j) = mean(j)/n 
4) Draw the plot 
mean(j), min(j), d., max(j) 
for every j between 1 dnd n. 
110 
5) End 
4. INTERPRETATION OF RESULTS 
For every j, min(j) and max(j) form an approximated 
confidence interval of level  M/(M-1) for d, 
obtained by simulation. mean(j) gives a (roughl 
though useful) approximation of the mathematical 
expectation of the random variable E., whose 
outcomes e, were observed. The plot “shows a 
confidence dnvelope for the input data: if the data 
lie in the region above the minima 8nd below the 
maxima we accept the hypothesis of uniform 
distribution over the sets. If the data lie in the 
region below the minima, there is evidence 
favouring an "attraction" model for the location of 
CPs, i.e. they are more spatially clustered than 
independent (uniformly distributed) points. If the 
data lie in the region above the maxima, there is 
evidence favouring a "repulsion" model for the 
location of CPs., i.e. they tended to be further 
appart from each other than independent (uniformly 
distributed) points. 
5. METHODOLOGY 
The algorithm above is very simple to implement, 
and we feel that it might be helpful in the 
diagnostic and identification of problems regarding 
the spatial distribution of CPs. (Listings in the 
CH programming language available from the authors 
upon request). 
Given n, the number of CPs, and once obtained the 
values mean(j), min(j), max(j) for every j, these 
envelopes could be understood as frontiers between 
the following regions: (i) the "acceptable" 
([min(j), max(j)]); (11) the "optimal" ar 
2 ,max(j))) and (iii) the "unacceptable" 
((0,min(j))). 
Regions (i), (ii) and (iii) correspond to the 
labels we would attach to the data sets whose 
vectors d fall into, with respect to their spatial 
quality. 
6. ANALYSIS OF THREE DATA SETS 
In the following we used M=19 obtaining, 
aproximated 957 confidence intervals for d 
thus, 
The first data set (W1) was used in the 
registration of an image of Tucurui, Para State, 
Brazil. The whole image matching (including the 
evaluation of its accuracy) involved seven CPs 
whose spatial distribution exhibits a strong 
tendency to the "attraction", as previously defined 
(see Fig, la). The plot 1b shows that most of these 
points are located in region (iii), considered as 
"unacceptable", 
(W2) comes from the area of 
Roncador, Mato Grosso State, Brazil. Ten 
used in this case, where an "atraction" 
model is also suited to describe their spatial 
distribution (weaker, though, than the Wl set; see 
Fig. 2a). The plot 2b shows that most of the points 
fall into region (i), considered as "acceptable", 
with a few shifted towards region (iii). 
The second data set 
Serra do 
CPs were 
The Wl and W2 data sets correspond to regions in 
the brazilian legal Amazonas, and are typical cases 
of the conditions found in that area: few or no 
features like  crossroads, bridges, etc. preclude 
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