  We introduce Lipschitz-Killing curvature (LKC) regression, a new method to
produce $(1-\alpha)$ thresholds for signal detection in random fields that does
not require knowledge of the spatial correlation structure. The idea is to fit
observed empirical Euler characteristics to the Gaussian kinematic formula via
generalized least squares, which quickly and easily provides statistical
estimates of the LKCs --- complex topological quantities that can be extremely
challenging to compute, both theoretically and numerically. With these
estimates, we can then make use of a powerful parametric approximation via
Euler characteristics for Gaussian random fields to generate accurate
$(1-\alpha)$ thresholds and $p$-values. The main features of our proposed LKC
regression method are easy implementation, conceptual simplicity, and
facilitated diagnostics, which we demonstrate in a variety of simulations and
applications.
