Comparing filters for the detection of point sources

This paper considers filters (the Mexican hat wavelet, the matched and the scale-adaptive filters) that optimize the detection/separation of point sources on a background. We make a one-dimensional treatment; we assume that the sources have a Gaussian profile, i.e. tau(x) = e(-x2)/2R(2), and a background modelled by a homogeneous and isotropic Gaussian random field, characterized by a power spectrum P(q) proportional to q(-gamma), gamma greater than or equal to 0. Local peak detection is used after filtering. Then, the Neyman-Pearson criterion is used to define the confidence level for detections and a comparison of filters is performed based on the number of spurious and true detections. We have performed numerical simulations to test theoretical ideas and conclude that the results of the simulations agree with the analytical results.