It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.

No abstract

No abstract

For pt.II see J. Phys. C: Solid State Phys., vol.21, p.4437 (1988). A formalism for an effective-Hamiltonian description of electron dynamics in inhomogeneously distorted crystal lattices which was recently developed by Brown in 1983 and Brown and Oldfield in 1988 is extended to include the effects of an externally applied magnetic field with slow spatial variation. Spatially averaged distortion-modified operators for velocity and acceleration are derived for wavepackets constructed from suitably defined Wannier functions and simplified expressions appropriate to the small-wavevector approximation are presented for these operators. The formalism is well suited to applications concerning the effects of lattice defects on magnetic orbits and magnetotransport properties.

No abstract