Localising Diffusion Sources from Samples Taken Along Unknown Parametric Trajectories

In a recent paper we showed that it is possible to localise diffusion sources observed with a mobile sensor whilst simultaneously estimating the piecewise linear trajectory of the sensor. Here we address the case in which the sensor moves along an arbitrary unknown parametric trajectory and we show that by solving a linear system of equations, we can retrieve the inner products between the parameters of the trajectory. From these inner products we then retrieve the curve parameters up to an orthogonal transformation, which allows us to also perfectly estimate the amplitudes of the sources and find their locations up to an orthogonal transformation.