Image Processing Reference

In-Depth Information

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

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