  This paper considers the problem of compact source detection on a Gaussian
background in 1D. Two aspects of this problem are considered: the design of the
detector and the filtering of the data. Our detection scheme is based on local
maxima and it takes into account not only the amplitude but also the curvature
of the maxima. A Neyman-Pearson test is used to define the region of
acceptance, that is given by a sufficient linear detector that is independent
on the amplitude distribution of the sources. We study how detection can be
enhanced by means of linear filters with a scaling parameter and compare some
of them (the Mexican Hat wavelet, the matched and the scale-adaptive filters).
We introduce a new filter, that depends on two free parameters (biparametric
scale-adaptive filter). The value of these two parameters can be determined,
given the a priori pdf of the amplitudes of the sources, such that the filter
optimizes the performance of the detector in the sense that it gives the
maximum number of real detections once fixed the number density of spurious
sources. The combination of a detection scheme that includes information on the
curvature and a flexible filter that incorporates two free parameters (one of
them a scaling) improves significantly the number of detections in some
interesting cases. In particular, for the case of weak sources embedded in
white noise the improvement with respect to the standard matched filter is of
the order of 40%. Finally, an estimation of the amplitude of the source is
introduced and it is proven that such an estimator is unbiased and it has
maximum efficiency. We perform numerical simulations to test these theoretical
ideas and conclude that the results of the simulations agree with the
analytical ones.
