Image Processing Reference
In-Depth Information
areas, but that the image of the target itself, no matter how close or far it is in
appearance from the original target, should basically remain the same as the
number of sources is increased.
Using the criteria described above, it seems evident that images obtained
when measurements are below the degree of freedom threshold do not seem to
be well formed and do changes from one image to the next, while the images
obtained above the degrees of freedom threshold do seem to have a consistent
reconstructed image that changes very little as the number of sources increases.
This would strongly suggest that the concept of the degrees of freedom pre-
sented above for this scenario is valid. This issue will be examined further in
relation to the number of receivers in later sections (Tables 6.2 through 6.6).
6.3 RequIReMentS FoR degReeS oF FReedoM
The requirement for the minimum number of degrees of freedom demon-
strated in the previous section for the number of sources is also similarly
applicable for the minimum number of receivers as well. This being the
case, it seems appropriate to demonstrate this concept as well on the num-
ber of receivers much like those done for the sources. This is not as simple
as it might seem. It has been observed that as one continues to reduce the
number of receivers and keep the receivers equally spaced and at the same
distance from the target, the natural geometrical consequence is that the
spacing between the receivers is ever increasing. Eventually, as is already
known in 1-D signal processing (Lustig et al., 2007), this will eventually lead
to aliasing issues in the resulting reconstructed image. This is exactly what
happens in this case. This is a very significant issue, especially dealing with
receiver numbers in the 5-20 range for the specific examples investigated
here. This issue prevents us from following the direct approach used in
the previous section for the sources. Before this aspect of the requirements
of minimum degrees of freedom can be sufficiently examined, the issue
of aliasing has to be addressed. One approach utilized in 1-D signal pro-
cessing to address aliasing in undersampled signals is to randomly space
the samples in lieu of using equally spaced samples (Lustig et  al., 2007).
This is a common technique that when employed eliminates or greatly
reduces the effects of aliasing, and as long as the original signal strength or
signal-to-noise ratio is high enough, the original signal will be evident in
the presence of the spatial noise introduced by reducing and randomizing
the receiver locations. This process is illustrated in Figure 6.2 taken from
Lustig et al. (2007). This same method is the technique that is implemented
in 2-D in the current algorithm code in an attempt to disperse the aliasing
as done in 1-D. This will of course introduce some element of noise to the
reconstructed images, but the signal-to-noise ratio should be sufficient to
produce a useful image for examining the effects of the degrees of freedom
on the number of receivers.
Since the total number of receivers used in this model is 360, any multiple
of 360 can be used for the number of receivers and still maintains the equal
spacing of the receivers. This being the case, the MATLAB ® code was modi-
fied such that for any undersampled quantity of receivers used, that is, less
than 360, the user has the ability to use the random generator in MATLAB to
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