Transformation-invariant filtering using expectation maximization
Input signals can often be described by a combination of an underlying signal that is easy to adaptively model (e.g., using a Kalman filter or a hidden Markov model) and a highly nonlinear transformation that is randomly drawn from a known set of transformations. Examples include a video of an unknown object that randomly moves in the field of view; a speech signal that is corrupted by a selection from different types of office noise (chatter, keyboard typing, etc.); and a tomographic signal that is phase-wrapped by an unknown number of wavelengths. We show how transformations in the input, such as translation and shearing in images, can be incorporated into an iterative adaptive filter that uses the expectation maximization algorithm. The underlying system model is a hidden Markov model (HMM) and the iterative filter estimates the parameters of the HMM and performs inference in the HMM in a way that is invariant to transformations in the input. We illustrate the iterative filter on a toy example and on video sequences of people and trucks.