bioengineering 280a lecture 3
TRANSCRIPT
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TT Liu, BE280A, UCSD, Fall 2006
Bioengineering 280APrinciples of Biomedical Imaging
Fall Quarter 2006Ultrasound Lecture 2
TT Liu, BE280A, UCSD, Fall 2006
Depth of Penetration
!
Assume system can handle L dB of loss, then
L = 20log10
Az
A0
"
# $
%
& '
We also have the definition
( = -1
z20log10
Az
A0
"
# $
%
& '
and the approximation
( =(0 f
Total range a wave can travel before attenuation L is
z = L
(0 f
Depth of penetration is
dp =L
2(0 f
2
TT Liu, BE280A, UCSD, Fall 2006
Depth of Penetration
22041085133202401
Depth ofPenetration(cm)
Frequency(MHz)
Assume L = 80 dB; α0= 1dB/cm/MHz
TT Liu, BE280A, UCSD, Fall 2006
Pulse Repetition and Frame Rate
!
Need to wait for echoes to return before transmitting new pulse
Pulse repetition interval is
TR "2dp
c
Pulse repetition rate is
fR =1
TR
If N pulses are required to form an image, then the
frame rate is
F =1
NTR
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TT Liu, BE280A, UCSD, Fall 2006
Example
!
N = 256, L= 80dB, c =1540m /s, "0 =1dB /cm /MHz
What frequency should be used to achieve a frame rate of 15 frame/sec?
TR =1
FN= 0.26ms
TR #2dp
c=
L
"0 fc
f #L
acTR=1.99MHz
TT Liu, BE280A, UCSD, Fall 2006
Huygen’s Principle
http://www.fink.com/thesis/chapter2.html http://www.cbem.imperial.ac.uk/ardan/diff/hfw.html
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TT Liu, BE280A, UCSD, Fall 2006
Huygen’s Principle
Anderson and Trahey 2000
!
Wavenumber
k =2"
#
!
Oliquity Factor
r01
TT Liu, BE280A, UCSD, Fall 2006
Small-Angle (paraxial) Approximation
Anderson and Trahey 2000
r01
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TT Liu, BE280A, UCSD, Fall 2006
Fresnel Approximation
Anderson and Trahey 2000
Approximates spherical wavefront with a parabolic phaseprofile
TT Liu, BE280A, UCSD, Fall 2006
Fresnel Approximation
!
U(x0,y0) =exp( jkz)
j"z## exp
jk
2zx1 $ x0( )
2+ y1 $ y0( )
2( )%
& '
(
) * s x1,y1( )dx1dy1
=exp( jkz)
j"zs(x0,y0)++exp
jk
2zx02 + y0
2( )%
& '
(
) *
%
& '
(
) *
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TT Liu, BE280A, UCSD, Fall 2006
Fraunhofer Approximation
!
kr01" kz 1+
1
2
x1# x
0
z
$
% &
'
( )
2
+1
2
y1# y
0
z
$
% &
'
( )
2$
% & &
'
( ) )
= kz 1+1
2z2x1
2 # 2x1x0
+ x0
2( ) +1
2z2y1
2 # 2y1y0
+ y0
2( )$
% &
'
( )
= kz +k
2zx1
2 + y1
2( ) +k
2zx0
2 + y0
2( ) #k
zx1x0
+ y1y0( )
" kz + +k
2zx0
2 + y0
2( ) #k
zx1x0
+ y1y0( )
Assume this termis negligible.
TT Liu, BE280A, UCSD, Fall 2006
Fraunhofer Condition
!
k
2zx1
2 + y1
2( )Phase term due to position on transducer is
Far-field condition is
!
k
2zx1
2 + y1
2( ) <<1
z >>k
2x1
2 + y1
2( ) ="
#x1
2 + y1
2( )
For a square DxD transducer, x1
2 + y1
2 = D2 /4
z >>"D2
4#$D
2
#
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TT Liu, BE280A, UCSD, Fall 2006
Fraunhofer Approximation
Anderson and Trahey 2000
Quadratic phase term
Fourier transform of thesource with
!
kx =x
0
"z ky =
y0
"z
TT Liu, BE280A, UCSD, Fall 2006
Plane Wave (Fraunhofer) Approximationz
x0
x1
θ
θd
r01
!
d = "x1sin#
sin# =x0
r01
$x0
z
d $ "x0x1
z
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TT Liu, BE280A, UCSD, Fall 2006
Plane Wave Approximation
!
1
rexp( jkr) "
1
zexp jk z + d( )( ) =
1
zexp j
2#
$z %
x0x1
z
&
' (
)
* +
&
' (
)
* +
U(x0) = s(x1)%,
,
-1
rexp( jkr)dx1
" s(x1)%,
,
-1
zexp j
2#
$z %
x0x1
z
&
' (
)
* +
&
' (
)
* + dx1
=1
zexp j
2#z
$
&
' (
)
* + s(x1)%,
,
- exp %j2#x0x1
$z
&
' (
)
* + dx1
=1
zexp j
2#z
$
&
' (
)
* + s(x1)%,
,
- exp % j2#kxx1( )dx1
=1
zexp j
2#z
$
&
' (
)
* + F s(x)[ ]
k x=x0
$z
TT Liu, BE280A, UCSD, Fall 2006
Plane Wave Approximation
!
In general
U(x0,y0) =1
zexp j
2"z
#
$
% &
'
( ) F s(x,y)[ ]
k x=x0
#z,ky =
y0
#z,
Example
s(x,y) = rect(x /D)rect(y /D)
U(x0,y0) =1
zexp( jkz)D2 sinc Dkx( )sinc Dky( )
=1
zexp( jkz)D2 sinc D
x0
#z
$
% &
'
( ) sinc Dky
y0
#z
$
% &
'
( )
Zeros occur at x0 =n#z
D and y0 =
n#z
D
Beamwidth of the sinc function is #z
D
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TT Liu, BE280A, UCSD, Fall 2006
!
"z
D1
!
D1
!
"z
D2
!
D2
2
"
!
D1
2
"
!
D2
TT Liu, BE280A, UCSD, Fall 2006
Example
Anderson and Trahey 2000
!
rectx
D
"
# $
%
& ' rect
x
d
"
# $ %
& ' (1
dcomb
x
d
"
# $ %
& '
)
* +
,
- . /Dsinc(Dk
x)( d sinc(dk
x)comb(dk
x)[ ]
Sidelobes
Question: What should we do to reduce the sidelobes?
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TT Liu, BE280A, UCSD, Fall 2006
Transducer Dimension
Anderson and Trahey 2000!
Goal : Operate in the Fresnel Zone
z < D2 /"
Dopt # "zmax
Example
zmax = 20 cm
" = 0.5 mm
Dopt =1 cm
TT Liu, BE280A, UCSD, Fall 2006
Focusing in Fresnel Zone
!
U(x0,y0) =exp( jkz)
j"z## exp
jk
2zx1 $ x0( )
2+ y1 $ y0( )
2( )%
& '
(
) * s x1,y1( )dx1dy1
=exp( jkz)
j"z## exp
jk
2zx12 + y1
2( ) + x02 + y0
2( ) $ 2 x1x0 + y1y0( )( )%
& '
(
) * s x1,y1( )dx1dy1
=exp( jkz)
j"zexp
jk
2zx02 + y0
2( )%
& '
(
) *
exp( jkz)
j"z## exp
jk
2zx12 + y1
2( )%
& '
(
) * exp $
jk
zx1x0 + y1y0( )
%
& '
(
) * s x1,y1( )dx1dy1
!
Use time delays to compensate for this phase term
!
U(x0,y0) =exp( jkz)
j"zexp
jk
2zx02 + y0
2( )#
$ %
&
' ( F exp
jk
2zx12 + y1
2( )#
$ %
&
' ( s x1,y1( )
)
* +
,
- .
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TT Liu, BE280A, UCSD, Fall 2006
Focusing in Fresnel Zone
!
U(x0,y0) =exp( jkz)
j"zexp
jk
2zx02 + y0
2( )#
$ %
&
' ( F exp
jk
2zx12 + y1
2( )#
$ %
&
' ( s x1,y1( )
)
* +
,
- .
Make
!
s x1,y1( ) = s0 x1,y1( )exp "jk
2z0x12 + y1
2( )#
$ %
&
' (
!
U(x0,y0) =exp( jkz0)
j"z0exp
jk
2z0x02 + y0
2( )#
$ %
&
' ( F s x1,y1( )[ ]
At the focal depth z =z0
Beamwidth at the focal depth is:
!
"z0
D
TT Liu, BE280A, UCSD, Fall 2006
Acoustic Lens
!
"z0
D= "F
D
c > c0
z0
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TT Liu, BE280A, UCSD, Fall 2006
Depth of Focus
!
When z " z0, the phase term is #$ = exp %jk
2z0
x1
2 + y1
2( )&
' (
)
* + exp %
jk
2zx1
2 + y1
2( )&
' (
)
* +
and the lens is not perfectly focused.
Consider variation in the x - direction.
#$ =kx
2
2
1
z%
1
z0
&
' (
)
* +
For transducer of size D, x
2
2=D
2
4
If we want #$ =,D2
2-
1
z%
1
z0
&
' (
)
* + <1 radian then
1
z%
1
z0
<2-
,D2
The larger the D, the smaller the depth of focus.