<title>Time-frequency and time-scale vector fields for deforming time-frequency and time-scale representations</title>

Abstract

We study local deformations of time-frequency and timescale representations, in the framework of the so-called reassignment methods, which aim at "deblurring" time-frequency representations. We focus on deformations generated by appropriate vector fields defined on time-frequency or time scale plane, and constructed on the basis of geometric and group-theoretical arguments. Such vector fields may be used as such for signal analysis (as quantities generalizing instantaneous frequency or group delay) in the framework of reassignment algorithms.

Left Invariant Evolution Equations on Gabor Transforms

“…For many such applications, the phase of the Gabor transform is of secondary impor-tance (see, e.g., the characterization of function spaces via Gabor coefficient decay [20]). However, since the Gabor transform uses highly oscillatory complex-valued functions, its phase information is often crucial, a fact that has been specifically acknowledged in the context of reassignment for Gabor transforms [7]. For this aspect of Gabor transform, as for many others, the group-theoretic viewpoint becomes particularly beneficial.…”

“…Reassignment will be seen to be a special case of left-invariant convection. A useful source of ideas specific to Gabor analysis and reassignment was the paper [7].…”

“…Since the phase of the Gabor transform is neglected, the original signal is not easily recovered from the reassigned spectrogram. Since then, various authors developed reassignment methods that were intended to allow (approximate) signal recovery [2,5,7]. We claim that a proper treatment of phase may be understood as phase-covariance, rather than phase-invariance, as advocated previously.…”

“…Thus if Φ is left-invariant, and A : L 2 (H r ) → R(W n ψ ) ⊥ an arbitrary operator, then Φ + A cannot be expected to be left-invariant, but the resulting operator on the signal side will be the same as for Φ, thus covariant with respect to time-frequency shifts. The authors [7] studied reassignment procedures that leave the phase invariant, whereas we shall put emphasis on phase-covariance. Note however that the two properties are not mutually exclusive; convection along equiphase lines fulfills both.…”

“…An illustration of this claim is contained in Figure 1. (2) processed Gabor transform Φ t (W ψ f ) where Φ t denotes a phase invariant shift (for more elaborate adaptive convection/reassignment operators see Section 6 where we operationalize the theory in [7]) using a discrete Heisenberg group, where l represents discrete spatial shift and m denotes discrete local frequency, (3) processed Gabor transform Φ t (W ψ f ) where Φ t denotes a phase covariant diffusion operator on Gabor transforms with stopping time t > 0. For details on phase covariant diffusions on Gabor transforms, see [14, ch:7] and [25, ch:6].…”