Image Processing Reference

In-Depth Information

(a)

×10
-4

(b)

×10
-3

0

40

5

30

4

3.5

3

2.5

2

1.5

1

0.5

4.5

50

20

4

10

100

3.5

0

3

150

-10

2.5

-20

2

200

-30

1.5

-40

250

-40

-30

-20

-10

0

10

20

30

40

0

50

100

150

200

250

(d)

(c)

×10
-5

40

0

4

0.04

30

3.5

50

0.035

20

3

10

0.03

100

2.5

0

-10

2

0.025

150

1.5

1

0.5

0.02

-20

200

0.015

-30

-40

0.01

250

-40

-20

0

20

40

0

50

100

150

200

250

Figure 11.5
(a) Ewald circle using 12 sources and 10 receivers randomly spaced. (b) Resulting reconstructed

image from (a). (c) Ewald circle using 36 source and 360 receivers equally spaced for comparison with (a) and (b).

(d) Resulting image from (c).

sources and 360 receivers. It is entirely possible that the minimum has been

met in this case. There simply is not enough evidence at this point to say con-

clusively if this describes the overall degrees of freedom relationship.

Having now observed the evidence that there does seem to be an overall

degrees of freedom criterion, it seems appropriate not to test this idea a bit

more thoroughly to see if this concept holds true and if so, whether there

appears to be a consistent threshold that we can identify and possibly even

quantify as an overall degrees of freedom criterion. To do this, we will exam-

ine two different targets over a range of permittivites for the scenarios of the

number of randomly but equally spaced data points that correspond to
N
2
/2,

N
2
, 2
N
2
and 4
N
2
. These images are then compared to the maximum degrees

of freedom case for this experiment which is
N
= 12,960 for comparison. The

results from testing circular targets are shown in Table 11.2. It is clear from

these images that, for the cases of
N
2
/2 and
N
2
total data points, the degrees of

freedom are clearly not satisfied for any of the permittivity cases. The images

are starting to take shape for the 2
N
2
case for all permittivity values, but still

appear as if they have not quite satisfied the overall number of degrees of free-

dom. Once the number of data points reaches the 4
N
2
case, it is evident that

the image is fairly well formed for all permittivity ranges. This strongly sug-

gests that the overall degrees of freedom criterion have been satisfied at this

point for all permittivities for this class of target.

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