  Topological measurements are increasingly being accepted as an important tool
for quantifying complex structures. In many applications, these structures can
be expressed as nodal domains of real-valued functions and are obtained only
through experimental observation or numerical simulations. In both cases, the
data on which the topological measurements are based are derived via some form
of finite sampling or discretization. In this paper, we present a probabilistic
approach to quantifying the number of components of generalized nodal domains
of nonhomogeneous random processes on the real line via finite discretizations,
that is, we consider excursion sets of a random process relative to a
nonconstant deterministic threshold function. Our results furnish explicit
probabilistic a priori bounds for the suitability of certain discretization
sizes and also provide information for the choice of location of the sampling
points in order to minimize the error probability. We illustrate our results
for a variety of random processes, demonstrate how they can be used to sample
the classical nodal domains of deterministic functions perturbed by additive
noise and discuss their relation to the density of zeros.
