Resolution and Degrees of Freedom - Introduction to Imaging from Scattered Fields - page 213

Appendix E: Resolution and
Degrees of Freedom
We gain some additional understanding of the inverse scattering problem
and its challenges by considering the scattered field measurement process,
regarded as Fourier data, as a truncated sampling process. For a finite object
of width D the Whittaker-Shannon sampling theorem demands a sampling
rate of at least B min = 1/ D in the frequency spectrum. The representation of the
k -space spectrum is only complete, however, if an infinite number of samples
are available covering the entire k -space. Since our k -space volume is physi-
cally limited to include only propagating waves, it is necessarily a truncated
set of samples.
The image estimate based on a truncated set of k -space samples provides
only a low spatial frequency estimate of the signal. For a spectrum sampled
at the Nyquist rate determined by the object support, we can easily verify that
the P -matrix of the PDFTs is diagonal. This means that at this sampling rate
the PDFT will not improve the image estimate beyond the classical limit, and
the model of the signal is already represented optimally in a least square sense
by the available truncated sampling expansion, that is, by a discrete Fourier
series.
It is worth contemplating why Fourier data sampled at the Nyquist rate
precludes any hope of bandwidth extrapolation. The sampling expansion is
constructed to obtain orthogonal interpolation functions. In other words, the
zeros of the interpolating sinc function perfectly coincide with the location
of sampling points. Thus, the data are assumed to be independent and do
not contain any information related to other sampling points. Points on the
sampling grid of the spectrum outside the window of measured data do not
contribute to the points inside this window. In turn, the latter cannot be used
to estimate points on the sampling grid outside the data window. It is well
known that spectral data sampled at the Nyquist rate do not contain sufficient
information for bandwidth extrapolation but that a higher sampling rate is
required. This is precisely the context to which the PDFT algorithm is appli-
cable. For oversampled data, the samples are no longer independent, and the
coefficients of the PDFT reconstruction must be selected to ensure data con-
sistency as a result of convolution with the interpolating function. This inter-
dependency has two consequences. First, it provides the freedom to balance
the PDFT coefficients to obtain improved signal resolution (and bandwidth
extrapolation). Second, the image reconstruction from interdependent sam-
ples results in high susceptibility to noise in the measured data. In particular,
the improved image resolution is the result of a delicate interference between
different interpolating functions, all of which carry the main portion of their
energy outside the data window. Thus, even small errors inside the window
are amplified in the extrapolated region. It has been shown that this confines
213