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.
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