Linear Estimation of Time-Warped Signals

Abstract

We introduce a novel methodology for estimating the time-axis deformation between two observations on a time-warped signal. Since the problem of estimating the warping function is nonlinear, existing methods iteratively minimize some metric between the observation and a hypothesized deformed template. Assuming the family of possible deformations the signal may undergo admits a finite-dimensional representation, we show that there is a nonlinear mapping from the space of observations to a low-dimensional linear space, such that in this space the problem of estimating the parametric model of the warping function is solved by a linear system of equations. We call the family of estimators derived based on this representation, linear warping estimators (LWE). The new representation of the problem enables an analytic analysis of the behavior of the solution in the presence of model mismatches, which is prohibitive when iterative methods are employed. The ability to achieve this major simplification both in the solution and in analyzing its performance results from the representation of the problem in a new coordinate system which is natural to the properties of the problem, instead of representing it in the standard coordinate system imposed by the sampling mechanism. The proposed solution is unique and exact, as it provides a closed-form expression for evaluating each of the parameters of the warping model using only measurements of the amplitude information of the observed and reference signals. The solution is applicable to any elastic warping regardless of its magnitude. We analyze the behavior of the LWE in the presence of noise and obtain a minimum variance unbiased estimator for the model parameters, by finding an optimal set of nonlinear operators for mapping the original problem into a low-dimensional linear space.