Statistical properties of local extrema in two-dimensional Gaussian random fields

This paper is concerned with the statistical properties of the local extrema and local maxima of two-dimensional (2D) Gaussian random fields (GRFs). A GRF may be represented by a linear filtering operation on a white noise field; the spatial properties of the GRF are then determined by the shape of the filter kernel function. New expressions are derived for the mean spatial density of local extrema and for the distribution of local extrema in a 2-D random field. The work is motivated by the problem of detecting known structures in images using 2D matched filters. The new results enable accurate performance predictions to be made of the reliability of such filters in the presence of noise. Case studies are presented for several well-known 2-D filter kernel functions.