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 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.
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Notes
- 1.
As we explain in [118, Appendices B and C] the Gabor domain is a principal fiber bundle
equipped with the Cartan connection form \(\omega_{g}(X_{g})= \langle\mathrm{d}s + \frac{1}{2}(p\,\mathrm{d}q -q\, \mathrm{d}p) , X_{g}\rangle\), or equivalently, it is a contact manifold, cf. [54, p. 6], [118, Appendix B, Definition B.14], 
. - 2.
The metric tensor is degenerate on H r, but we consider a contact manifold
where tangent vectors along horizontal curves do not have an 
-component. - 3.
equipped with the Cartan connection form 
.
where tangent vectors along horizontal curves do not have an 
-component.References
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The Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged for financial support.
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Duits, R., Führ, H., Janssen, B. (2012). Left Invariant Evolution Equations on Gabor Transforms. In: Florack, L., Duits, R., Jongbloed, G., van Lieshout, MC., Davies, L. (eds) Mathematical Methods for Signal and Image Analysis and Representation. Computational Imaging and Vision, vol 41. Springer, London. https://doi.org/10.1007/978-1-4471-2353-8_8
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