  The expected Euler characteristic (EEC) curve of excursion sets of a Gaussian
random field is used to approximate the distribution of its supremum for high
thresholds. Viewed as a function of the excursion threshold, the EEC is
expressed by the Gaussian kinematic formula (GKF) as a linear function of the
Lipschitz-Killing curvatures (LKCs) of the field, which solely depend on the
domain and covariance function of the field. So far its use for non-stationary
Gaussian fields over non-trivial domains has been limited because in this case
the LKCs are difficult to estimate. In this paper, consistent estimators of the
LKCs are proposed as linear projections of "pinned" observed Euler
characteristic curves and a linear parametric estimator of the EEC curve is
obtained, which is more efficient than its nonparametric counterpart for
repeated observations. A multiplier bootstrap modification reduces the variance
of the estimator, and allows estimation of LKCs and EEC of the limiting field
of non-Gaussian fields satisfying a functional CLT. The proposed methods are
evaluated using simulations of 2D fields and illustrated in thresholding of 3D
fMRI brain activation maps and cosmological simulations on the 2-sphere.
