This paper presents a new method that addresses measurement origin uncertainty. Measurement origin uncertainty occurs when the object a measurement originated from is not clear. The systems considered contain multiple bodies which are dynamically indistinguishable other than initial conditions. Each measurement originates from one of the bodies in the system. In the past, recursive data association methods have been used to address problems of this nature. A new technique is presented which treats the measurement association problem as a batch post-processing problem. Reformulating the problem as such, it is possible to transform the data association problem into a trajectory optimization problem. From this point of view it is then possible to solve the measurement association problem using first- and second-order optimization algorithms that rely on having first- and second-order derivatives for cost functions that depend on impulsive trajectories.