Models for point processes observed with noise

Consider a pair of point processes, X and Y, where X is regarded as a 'true' point process and Y is an imperfect observation of X. For the transformation from X to Y, we consider a number of disturbance mechanisms covering random thinning, displacement, censoring of the displaced points and superposition of extra points. We present the conditional likelihood of Y given X. When both point processes are observed the likelihood may be used for inference about the disturbance mechanisms. The likelihood is a sum, typically with very many terms, and we discuss an approximation with a small number of terms. The results are applied to an example, where X denotes a set of 'true' positions of tree tops, and Y denotes tree-top positions estimated by template matching in a digital image obtained by high-resolution aerial photography. The parameters governing the various disturbance mechanisms are estimated from the conditional likelihood.