Thinking of heatmap software for multiple indistinguishable targets and multiple transmitter locations. I work on inverse problems and I am interested in calculating heat maps of likelihood, eg that each grid square has at least one transmitter. Has anyone found any literature on the theory of this? I have only found “ad hoc” methods where the processed signal along each direction line is added into the vale at each grid square, but no just justification for this method (and I think I have counterexamples where it produces false local maxima).
I emailed the Kracken team but their reply so far is that the software in the Android app is propitiatory.
The heatmap will already sort of give you a likelihood, if there are more than two transmitters. The heatmap should end up with two or more hotspots.
Yes it is the “sort of” that I am worried about! First of all likelihood of what? One idea is “at least one transmitted in that square”. If we are addiding and the measurements from each RX position are assumed independent then we should be calculating “log likelihood of there being at least one transmitter in each direction”. Actually we need to normalize so we have log probability densities before we add them, while this is a constant in the log case it has to be the right constant. To normalize we also need some finiteness condition along lines, for example an a pirori assumption that the transmitters are no more than a certain distance away.
We have a few ways to process the directional signals, eg MUSIC. If we are just looking for a maximum in the most likely direction then it doesnt matter much, but to combine them by adding we need the angular probability density (of something well defined). If there is one
transmitter it is just the pdf of the DoA, but if there are more than one transmitter one has to be careful what that means.
I really expect someone has actually done this theory before, before I spend a lot of time working it out for myself I need to check I am not reinventing the wheel!