Filter Design for the Detection/Estimation of the Modulus of a Vector. Application to Polarization Data

We consider a set of M images, whose pixel intensities at a common point can be treated as the components of a M-dimensional vector. We are interested in the estimation of the modulus of such a vector associated to a compact source. For instance, the detection/estimation of the polarized signal of compact sources immersed in a noisy background is relevant in some fields like Astrophysics. We develop two different techniques, one based on the Maximum Likelihood Estimator (MLE) applied to the modulus distribution, the modulus filter (ModF) and other based on prefiltering the components before fusion, the filtered fusion (FF), to deal with this problem. We present both methods in the general case of M images and apply them to the particular case of three images (linear plus circular polarization). Numerical simulations have been performed to test these filters considering polarized compact sources immersed in stationary noise. The FF performs better than the ModF in terms of errors in the estimated amplitude and position of the source, especially in the low signal-to-noise case. We also compare both methods with the direct application of a matched filter (MF) on the polarization data. This last technique is clearly outperformed by the new methods.