Room geometry estimation from a single channel acoustic impulse response

For a 2D rectangular room of unknown dimensions and with unknown source and microphone positions, the times of arrival of reflections can be described in terms of image source positions. Adopting a microphone-centred coordinates system, it is shown that to satisfy certain combinations of arrival times imposes constraints on the possible room geometry: a second-order reflection from adjacent walls determines the source-microphone distance; a second-order reflection from opposite walls in a given dimension determines the source displacement in that dimension as a function of the source-receiver distance. Given a subset of time differences of arrival, the extent to which the geometry can be determined is related to these constraints. The geometry estimation is further posed as a least squares optimisation problem whose results verify the analytical results.