  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. $\tau (x) = e^{- x^2/2R^2}$, and a
background modelled by an homogeneous and isotropic Gaussian random field,
characterised by a power spectrum $P(q)\propto q^{-\gamma}, \gamma \geq 0$.
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 done 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.
