Fitting all parameters of the germ-grain Quermass-interaction model by the Takacs-Fiksel method

Existence of Quermass-Interaction Process for non locally stable interaction and non bounded convex grains

We prove the existence of infinite volume Quermass-Interaction Process in a general setting of non locally stable interaction and non bounded convex grains. No condition on the parameters of the linear combination of the Minkowski functionals is assumed. The only condition is that the square of the random radius of the grain admits exponential moments for all orders. Our methods are based on en...

Continuum percolation for quermass interaction model

Continuum percolation for Markov (or Gibbs) germ-grain models in dimension 2 is investigated. The grains are assumed circular with random radii on a compact support. The morphological interaction is the so-called Quermass interaction defined by a linear combination of the classical Minkowski functionals (area, perimeter and Euler-Poincaré characteristic). We show that percolation occurs for any...

Estimation of the intensity parameter of the germ-grain Quermass-interaction model when the number of germs is not observed

Towards optimal Takacs–Fiksel estimation

The Takacs–Fiksel method is a general approach to estimate the parameters of a spatial Gibbs point process. This method embraces standard procedures such as the pseudolikelihood and is defined via weight functions. In this paper we propose a general procedure to find weight functions which reduce the Godambe information and thus outperform pseudolikelihood in certain situations. The new procedu...

Quermass-interaction Processes: Conditions for Stability

We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of. If only the ...