  The log-Gaussian Cox process (LGCP) is a popular point process for modeling
non-interacting spatial point patterns. This paper extends the LGCP model to
handle data exhibiting fundamentally different behaviors in different
subregions of the spatial domain. The aim of the analyst might be either to
identify and classify these regions, to perform kriging, or to derive some
properties of the parameters driving the random field in one or several of the
subregions. The extension is based on replacing the latent Gaussian random
field in the LGCP by a latent spatial mixture model. The mixture model is
specified using a latent, categorically valued, random field induced by level
set operations on a Gaussian random field. Conditional on the classification,
the intensity surface for each class is modeled by a set of independent
Gaussian random fields. This allows for standard stationary covariance
structures, such as the Mat\'{e}rn family, to be used to model Gaussian random
fields with some degree of general smoothness but also occasional and
structured sharp discontinuities.
  A computationally efficient MCMC method is proposed for Bayesian inference
and we show consistency of finite dimensional approximations of the model.
Finally, the model is fitted to point pattern data derived from a tropical
rainforest on Barro Colorado island, Panama. We show that the proposed model is
able to capture behavior for which inference based on the standard LGCP is
biased.
