In this paper, we look at the types of errors that are introduced when using
the Euler characteristic integral approach to count the number of targets in a
discrete sensor field. In contrast to other theoretical analyses, this paper
examines discrete sensor fields rather than continuous ones. The probabilities
of first and second order errors are worked out combinatorially, and a general
formula is found which is discovered to be proportional to much higher order
errors. Asymptotic results are also derived. Thus, this work gives us the bias
of the Euler characteristic integral as a point estimator for the number of
targets in a sensor field, and a general understanding of how the Euler
characteristic integral fails, or more often succeeds.Comment: 20 pages, 12 figures, 2 table