Conditions are obtained for the cepstrum of anl1 sequence having arbitrary support on theM-dimensional lattice to exist also as anl1 sequence with arbitrary support. These conditions are used to show that the cepstrum of a sequence with support on a half-space will, when it exists, also have support on the same half-space. This result is used, in turn, to describe the support of the cepstrum of a sequence with bounded support on a half-space. The relationship between the existence of the cepstrum and bounded-input, bounded-output (BIBO) stability for all three of these cases is considered. Finally, the derivatives of the inversion operator, the homomorphic transform operator, and its inverse are calculated. These results are also useful in the one-dimensional case.

Some properties of the multidimensional complex cepstrum and their relationship to the stability of multidimensional systems

A new type of differential cepstrum (DC) named as partially DC (PDC) is introduced. While compared with the previously defined DC, the PDC saves half the computational complexity since only one dimension of the partial derivative is considered. Nevertheless, for the calculation of the complex cepstrum (CC), using the PDC avoids the phase-unwrapping routine as well. One application is to construct the minimum-phase sequence from a mixed-phase one through a windowing technique, which can be further applied to filter stabilization. Another application is performing phase unwrapping on 2-D phase data to which we propose a much simpler technique using the PDC. These applications can be implemented in a more efficient and convenient way with the PDC than with the traditional DC.

Two-Dimensional Partially Differential Cepstrum and Its Applications on Filter Stabilization and Phase Unwrapping

: Conventional algorithms for media characterization in diffraction tomography rely on weakly scattering approximations such as the Born or Rytov methods. In this report, we investigate the utility of a nonlinear filtering procedure applied to single-view backpropagated images of strongly scattering two-dimensional objects. With a single illumination direction and constant wave number k0, exact scattered far fields are backpropagated to form an image of the scattering potential times the total field. The multiplicative field, in this case, precludes one from ascertaining quantitative information on the scattering potential. We apply a differential cepstral filter to single-view backpropagated images in order to remove the perturbing field. Phase wrapping problems associated with cepstral filtering procedures are avoided in the differential cepstrum, since phase ambiguities introduced by the complex algorithm are avoided in its definition. Exact scattering solutions are used from several strongly scattering cylinders ranging from k0alpha(square root of (epsilon tau) - 1) = 2 to k0alpha(square root of (epsilon tau) - 1) = 41 where alpha is the maximum dimension of the object and epsilon tau is the relative permittivity. For the lower range, the filtered single-view backpropagated images are superior to the unfiltered. As the range increases, object features become less resolved due to the limited Fourier data available. Results and limitations of the proposed methods are discussed.

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA345619&Location=U2&doc=GetTRDoc.pdf

Image Reconstruction From Strongly Scattering Penetrable Objects Using a Differential Cepstral Filter

The aim of this paper is to stimulate further research in multidimensional (N-D) homomorphic deconvolution systems to deal with multivariable signals. In this paper an attempt is made to present a comprehensive treatment of all the four deconvolution systems, viz. the logarithmic cepstrum (LC), the differential cepstrum (DC), the spectral root cepstrum (SRC) and the generalized cepstrum (GC) as a natural extension of the one-dimensional cases. The properties and computation formulae of these N-D systems are discussed highlighting the main differences and advantages.

D. Raghuramireddy

Papers

Multidimensional Homomorphic deconvolution systems