Left Invariant Evolution Equations on Gabor Transforms

Abstract

By means of the unitary Gabor transform one can relate operators on signals to operators on the space of Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H r . By using the left invariant vector fields on H r and the corresponding left-invariant vector fields on on phase space in the generators of our transport and diffusion equations on Gabor transforms we naturally employ the essential group structure on the domain of a Gabor transform. Here we mainly restrict ourselves to non-linear adaptive left-invariant convection (reassignment), while maintaining the original signal.

Mathematical Methods for Signal and Image Analysis and Representation

“…For details on phase covariant diffusions on Gabor transforms (see the right column in Fig. 8.1), we refer to [118,Chap. 7] and [236,Chap.…”

“…This however does not imply that it is a redundant direction in the group manifold H r itself, since clearly the s-axis represents the relevant phase and stores the noncommutative nature between position and frequency, [118,Chap. 1].…”

“…This gives a nice geometric interpretation to the phase variable s(t), since by the Stokes theorem it represents the net surface area between a straight line connection between (p(0), q(0), s(0)) and (p(t), q(t), s(t)) and the actual horizontal curve connection [0, t] τ → (p(τ ), q(τ ), s(τ )). For details, see [118].…”

“…Simply, because [∂ p , ∂ q ] = 0 whereas we should have (8.12). For further differential geometrical details see the appendices of [118], analogous to the differential geometry on orientation scores, [112], [118, Appendices D, C.1].…”

A variational approach to cardiac motion estimation based on covariant derivatives and multi-scale Helmholtz decomposition

“…[58,20], However, strain can also be directly computed from frequency fields using (enhanced) Gabor transforms, cf. [13].…”