A closed form solution to the Probability Hypothesis Density Smoother

A closed form Gaussian mixture solution to the forward-backward Probability Hypothesis Density smoothing recursion is proposed. The key to the closed form solutions is the use of an alternative form of the backward propagation, together with terse yet suggestive notations that have natural interpretation in terms of measurement predictions. The closed form backward propagation together with the Gaussian mixture PHD filter as the forward pass form the Gaussian mixture PHD smoother. Closed form solutions to smoothing for single target are also derived.