Digital Signal Processing Reference
In-Depth Information
Fig. 3.3
Original and reconstructed K-space for getting MRI
T
2image
f
(
x
,
−
y
)
(refer Fig.
3.3
). The image in the spatial domain
f
(
x
,
−
y
)
is further properly arranged
to obtain
f
.
Thus the reconstructed
T
2
image matrix is as follows.
(
x
,
y
)
⎡
⎤
2
.
55 2
.
00 6
.
99 1
.
73 3
.
92 3
.
04 3
.
76 1
.
62 6
.
54 1
.
55 1
.
02
5
.
26 4
.
70 1
.
08 1
.
47 5
.
04 1
.
00 2
.
22 0
.
42 4
.
63 2
.
53 5
.
02
⎣
⎦
3
.
89 1
.
17 5
.
57 1
.
60 0
.
88 1
.
01 4
.
02 3
.
85 2
.
48 0
.
74 4
.
39
2
.
44 5
.
30 2
.
00 6
.
65 5
.
32 3
.
64 2
.
41 4
.
64 5
.
49 2
.
30 4
.
54
4
.
00 2
.
67 2
.
91 5
.
19 0
.
88 1
.
97 3
.
18 3
.
78 2
.
74 3
.
65 2
.
18
=
2
.
00 5
.
19 3
.
01 6
.
21 3
.
47 1
.
61 2
.
52 4
.
92 4
.
08 3
.
92 2
.
43
RECONSTRUCTEDIMAGE
1
96
2
.
40 2
.
85 2
.
01 2
.
00 1
.
00 1
.
96 5
.
30 1
.
91 2
.
90 2
.
67 2
.
66
3
.
87 4
.
12 7
.
80 4
.
25 6
.
18 4
.
52 3
.
84 4
.
84 3
.
05 2
.
89 1
.
13
1
.
60 2
.
71 1
.
00 1
.
93 5
.
29 2
.
00 5
.
75 7
.
01 1
.
91 2
.
13 4
.
02
2
.
96 2
.
41 2
.
73 0
.
42 3
.
23 2
.
07 4
.
69 4
.
96 3
.
90 2
.
22 4
.
76
.
49 4
.
83 2
.
00 3
.
33 3
.
30 3
.
66 3
.
13 1
.
00 4
.
52 1
.
59 1
.
The resultant magnitude of the magnetic moments at every pixel during dephasing
(refer Figs.
3.4
,
3.5
). Note that at
t
0, the value is the corresponding proton-density.
As time goes, the resultant magnitude gradually decreases. At
t
=
3, (note that
we allowed the magnetic moment to dephase for 0.3) the resultant magnitude (refer
Fig.
3.4
) is measured as follows.
=
0
.
⎡
⎤
2
.
55 2
.
00 6
.
99 1
.
73 3
.
92 3
.
04 3
.
76 1
.
62 6
.
54 1
.
55 1
.
02
5
.
.
.
.
.
.
.
.
.
.
.
⎣
26 4
70 1
08 1
47 5
04 1
00 2
22 0
42 4
63 2
53 5
02
⎦
3
.
89 1
.
17 5
.
57 1
.
60 0
.
88 1
.
01 4
.
02 3
.
85 2
.
48 0
.
74 4
.
39
2
.
44 5
.
30 2
.
00 6
.
65 5
.
32 3
.
64 2
.
41 4
.
64 5
.
49 2
.
30 4
.
54
4
.
00 2
.
67 2
.
91 5
.
19 0
.
88 1
.
97 3
.
18 3
.
78 2
.
74 3
.
65 2
.
18
2
.
00 5
.
19 3
.
01 6
.
21 3
.
47 1
.
61 2
.
52 4
.
92 4
.
08 3
.
92 2
.
43
RESULTANTMAGNITUDE
at t
=
0
.
3
1
.
49 4
.
83 2
.
00 3
.
33 3
.
30 3
.
66 3
.
13 1
.
00 4
.
52 1
.
59 1
.
96
2
.
40 2
.
85 2
.
01 2
.
00 1
.
00 1
.
96 5
.
30 1
.
91 2
.
90 2
.
67 2
.
66
3
13
1
.
60 2
.
71 1
.
00 1
.
93 5
.
29 2
.
00 5
.
75 7
.
01 1
.
91 2
.
13 4
.
02
2
.
96 2
.
41 2
.
73 0
.
42 3
.
23 2
.
07 4
.
69 4
.
96 3
.
90 2
.
22 4
.
76
.
87 4
.
12 7
.
80 4
.
25 6
.
18 4
.
52 3
.
84 4
.
84 3
.
05 2
.
89 1
.
(which is same as that of RECONSTRUCTEDIMAGE).
Thus
T
2
image gives the resultant magnitude of the magnetic moment at every
pixel after dephasing for some finite time duration (0.3 in our illustration). We also
know that the initial values of the magnetic moments (
t
=
0) at every pixel is
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