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 non-linear, existing methods iteratively minimize some metric be-

Universal Manifold Embedding for Geometrically Deformed Functions

“…It provides an exact description of the manifold despite using as few as a single observation, and hence the need for using large numbers of observations in order to learn the manifold or a corresponding dictionary, is eliminated. The results in this work generalize and extend the results of [11], and [12], where the fundamental problems of estimating the parametric models of 1-D elastic and 2-D affine deformations of a single object were analyzed, to problems that require joint detection, recognition and deformation estimation of multiple and deformable objects.…”

“…. , n. Moreover, using (11) we have that since T g,1 is a linear operator from to R M it admits an M × (n + 1) matrix representation, given by T g,1 . Thus, T g,1 is invertible if and only if there exists a set of linearly independent functions {w k } M k=1 ∈ W , where M ≥ n + 1, such that T g,1 is of rank n + 1.…”

“…We note that only subsets of the finite dimensional space R N actually describe physically meaningful geometric deformations. (A detailed analysis on the reasoning for adopting this type of model can be found in [11]). …”

“…Lemma 1 [11]: Let h, g ∈ S g be two observations on the same object such that h = g • ϕ and such that (1) holds. Then, every w ∈ W provides a single linear constraint…”

“…However, there are clearly pathological examples in which it is impossible to reconstruct the geometric deformation, for example when g is constant on its domain. The exact conditions such that g ∈ S admits a mapping to a full rank T g are considered in [11].…”