Jorge Brito
Interval *a;" Interval “b;” Pr (G | ai, bi) Pr (2G | ai, bj)
ài bi 1.0 0.0
a b 1.0 0.0
21 b; 1.0 0.0
a, bi MC' (abi) 1.0 - MC” (a.,b1)
ap b, 1.0 0.0
ar bi 1.0 0.0
a; bi MC' (a,b) 1.0 - MC' (as,b;)
a; b. MC' (as,b;) 1.0 - MC' (as,b;)
a; b; 1.0 0.0
MC indicates that it is necessary to compute the conditional probability of agreement between a;
and bj, e.g. through a Monte Carlo simulation.
Tablel. Conditional probability matrix for occlusion analysis on the point pair shown in figure 6
2.3 Computing the Probabilities
2.3. Conditional Probabilities Pr(G | ai, by)
The computation of the conditional probability matrix of agreement between x’, and x',, given that x', belongs to the
class interval “a;”, and x’, belongs to “b;”, respectively, depends on the relative position of x', and x',. These positions
are random variables, since “A” and “B” can be located anywhere in the object space. This characteristic renders the
dynamic aspect of the case being discussed. In practical terms, it means that the conditional probability matrix cannot
be pre-computed and stored in memory for future use. Conversely, it needs to be evaluated for every pair of points
being tested. More specifically, it is necessary to compute the probability of agreement of ab with AB given that x’, is
in range a; and x”, is in range by, for i, j = 1,...,3. In other words, it is necessary to determine Pr (G | a;, bi), Pr (G | ài,
b») ..., Pr (CG | a5, b3). See figure 6 and table 1 above. As shown earlier in figure 3, ab has the same orientation af AB
i.e., they agree whenever x’, < x’, ,and, conversely, they disagree when x’, = x’,. The analysis of figure 6 supports
the conclusion that one can be certain of agreement (i.e., Pr = 1) given ranges where x’, < x’, throughout the whole
domain. For example, Pr (G | a;, by) = 1.0, since x’, < x’, throughout a, and by, respectively. Similarly, Pr (G | a, D») =
1.0, Pr(G | ay, bs) = 1.0, and so forth. Therefore, it is necessary to investigate further the cases where there is an overlap
between x’, and x’, ranges, i.e., to investigate Pr (G | à», bı), Pr (G | az, bı), and Pr (G | az, by). In these cases x’, could be
less or greater than. x'y. Therefore, it is necessary to estimate the probability of x', being less than x^. In other words,
it is necessary to compute the conditional probability of agreement between ab and AB given that x', belongs to the
class interval “a,” and x’, belongs to “b;”, respectively. This computation can be done either numerically or analytically.
A numerical technique, namely the Monte Carlo simulation, was adopted for estimating the probabilities mentioned
above. For instance, to estimate the first probability Pr (G | ay, by), using the Monte Carlo simulation, random points for
x’, and x^, are drawn uniformly within their corresponding ranges a, and by, respectively. The “ x’, points that are less
than x’, are then counted and divided by the total number of points drawn, providing an estimate of the conditional
probability.
2.3.2 Unconditional Probabilities Pr(a;) and Pr(b;)
Since there are uncertainties associated with the coordinates of the points being tested, it is necessary to compute the
probability of x’, and x’, belonging to each of their respective class intervals “a;” and “b;”. This requires choosing a
probability distribution. It is also necessary to integrate this cumulative probability function within the limits of each
interval. The central limit theorem provides mathematical support for choosing the Gaussian Distribution for the density
probability function. This can be justified by the fact that the projections of ground points "A" and "B" onto the digital
image space, "x,", and "x", respectively, are random variables that can be expressed as a combination of "n^
independent variables v;, v», ...v.. The choice of interval amplitudes is another issue, directly related to the
probabilities to be assigned to each interval. The following criteria were used to choose those intervals:
(1) the intervals should be symmetrically distributed around the expected values for x", and x’, (see figure 8 below);
(2) a certain confidence level must be set. (This issue is discussed below) and;
(3) the integration of the Gaussian should be truncated so the probabilities (the areas under the Gaussian limited by each
interval) add up to 1.0.
106 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
