324
Biomedical Signal and Image Processing
As discussed in Chapter 13, Equation 16.31 constitutes a tomographic equation. If
several scans are taken in several directions, the process described earlier will reduce
to a standard nondiffracting tomography problem that was discussed in Chapter 13.
This shows how different ultrasound attenuation characteristics of tissues can be
used to form an image of the biological systems.
Next, we discuss a similar ultrasound tomographic system called TOF tomography whose setup is essentially the same as that of the attenuation tomography.
However, TOF tomography is based on a set of completely different characteristics
of the tissues, which make the resulting images very different from those of attenuation tomography.
16.6.2 ULTRASOUND TIME-OF-FLIGHT TOMOGRAPHY
Additional information is retrieved from the TOF of each transmission to obtain
details on the speed of sound in the region the sound traverses through. The speed
of sound gives details on either the density of the medium or the elastic modulus of
the medium, thus providing additional secondary anatomical details. Based on the
measurement of the TOF, the time “t d ” is defined as the total time it takes the signal
to travel from the transmitter to the receiver and is measured to perform TOF tomography as describe later.
In order to describe TOF tomography, we start by reconsidering Equation 16.26:
l
l
− j b ( ,
xy, f) dx − a ( ,
xy f d
) x
Y f
( ) = AH ( )
f H ( )
f e
− jb W ( )
f l W −a W ( )
f l
∫
∫
W e
0
e
0
1
X f
(16.3
2
e
( )
2)
From this relationship, the terms describing the delay from the output to the input
l
−
−a
j ∫ b ( ,
xy ,f) d x
are e
W ( )
f l W (cause by water on both sides) and e
0
(caused by the biological
tissues). This means that the total delay measured from the source to the receiver can
be related to these terms as follows:
⎡
l
⎤
− j ⎢ b W ( )
f l W + ∫ b (x, ,
yf)d x ⎥
⎢
⎥
e
⎣
0
⎦
= e
− j f
2 p t d
(16.33)
Taking the logarithm of both sides of Equation 16.32,
( )
∫
l
b W f l W + b(x, y, f ) dx = 2 pf t d
(16.34)
0
or
∫
l
b( ,
x y, f ) dx = 2 pft d − b W ( f ) l W
(16.35)
0
, ,
Biomedical Signal and Image Processing
As discussed in Chapter 13, Equation 16.31 constitutes a tomographic equation. If
several scans are taken in several directions, the process described earlier will reduce
to a standard nondiffracting tomography problem that was discussed in Chapter 13.
This shows how different ultrasound attenuation characteristics of tissues can be
used to form an image of the biological systems.
Next, we discuss a similar ultrasound tomographic system called TOF tomography whose setup is essentially the same as that of the attenuation tomography.
However, TOF tomography is based on a set of completely different characteristics
of the tissues, which make the resulting images very different from those of attenuation tomography.
16.6.2 ULTRASOUND TIME-OF-FLIGHT TOMOGRAPHY
Additional information is retrieved from the TOF of each transmission to obtain
details on the speed of sound in the region the sound traverses through. The speed
of sound gives details on either the density of the medium or the elastic modulus of
the medium, thus providing additional secondary anatomical details. Based on the
measurement of the TOF, the time “t d ” is defined as the total time it takes the signal
to travel from the transmitter to the receiver and is measured to perform TOF tomography as describe later.
In order to describe TOF tomography, we start by reconsidering Equation 16.26:
l
l
− j b ( ,
xy, f) dx − a ( ,
xy f d
) x
Y f
( ) = AH ( )
f H ( )
f e
− jb W ( )
f l W −a W ( )
f l
∫
∫
W e
0
e
0
1
X f
(16.3
2
e
( )
2)
From this relationship, the terms describing the delay from the output to the input
l
−
−a
j ∫ b ( ,
xy ,f) d x
are e
W ( )
f l W (cause by water on both sides) and e
0
(caused by the biological
tissues). This means that the total delay measured from the source to the receiver can
be related to these terms as follows:
⎡
l
⎤
− j ⎢ b W ( )
f l W + ∫ b (x, ,
yf)d x ⎥
⎢
⎥
e
⎣
0
⎦
= e
− j f
2 p t d
(16.33)
Taking the logarithm of both sides of Equation 16.32,
( )
∫
l
b W f l W + b(x, y, f ) dx = 2 pf t d
(16.34)
0
or
∫
l
b( ,
x y, f ) dx = 2 pft d − b W ( f ) l W
(16.35)
0
, ,