TOA sensor network calibration for receiver and transmitter spaces with difference in dimension

We study and solve the previously unstudied problem of finding both sender and receiver positions from time of arrival (TOA) measurements when there is a difference in dimensionality between the affine subspaces spanned by receivers and senders. Anchor-free TOA network calibration has uses both in sound, radio and radio strength applications. Using linear techniques and requiring only a minimal number of receivers and senders, an algorithm is constructed for general dimension p for the lower dimensional subspace. Degenerate cases are determined and partially characterized as when receivers or senders inhabits a lower dimensional affine subspace than was given as input. The algorithm is further extended to the overdetermined case in a straightforward manner. Simulated experiments show good accuracy for the minimal solver and good performance under noisy measurements. An indoor environment experiment using microphones and speakers gives a RMSE of 2.35cm on receiver and sender positions compared to computer vision reconstruction.