Born Approximation Observations - Introduction to Imaging from Scattered Fields - page 69

Sparse signal

k -space of sparse signal

(a) (b)

Equispaced undersampling
Random undersampling

Equispaced 8-fold u-sampling Random 8-fold u-sampling

(c) (d)

Figure 6.2 (a) Sparse signal to be sampled. (b) The k -space representation of original signal in (a) with
equispaced undersampling and random undersampling sample locations shown. (c) Resulting reconstruction of
original sampled signal using equispaced samples. (d) Resulting reconstruction of original sampled signal using
random samples.

have the receiver at any location between the chosen locations to be selected
in a Gaussian distribution centered on the original receiver location. This in
a sense creates, for lack of a better term, what we define here as a noise band-
width (NBW) around the chosen receiver location. For example, if 36 receivers
are chosen, then the maximum noise bandwidth would be 360/36 = 10. This
means that the receiver location could be any location within ±5° around the
original receiver location. Naturally, as fewer receivers are used, the NBW can
increase if desired.
As the consequences of doing this will be a function of frequency, this
approach was initially tested for three different frequencies to observe the
effects. The three incident frequencies used are 8, 5 and 2 GHz. The simplest case
was modeled first, that being a circle or cylinder with a radius of λ respective
of the incident frequency. In conjunction with testing this possible remedy for
the aliasing, the degree of freedom criteria was examined as well. For each case
in this first series of tests, the minimum number of degrees of freedom is calcu-
lated to be π( n ) and for these tests n = 1.225 so this results in N = π(1.225) = 3.85.
The results for these series of tests are shown in Table 6.7, and it is evident from
these images that the random spacing technique is effective.
With the aliasing issue now addressed to some extent, the degrees of free-
dom demonstration was then run for the targets comprising two circles, one
square, two squares, and two triangles as before for the tests with sources.
These images are shown in Tables 6.8 through 6.10. As already mentioned,
though the aliasing has disappeared, there is now a noise element added to
the images from the randomized detector spacing. This being the case, the
effect of the degrees of freedom on the reconstructed images is not as pro-
nounced as it was for the demonstrations involving the sources, but there