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Biomedical Imaging I
Class 9 – Ultrasound Imaging
Doppler Ultrasonography;
Image Reconstruction
11/09/05
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 1
Doppler Effect
Change in ultrasound frequency caused by motion of source (which can be
a scatterer) and/or receiver relative to the background medium
fR 
θ
fR  f
fR 
c
c v
R
f
c
c v
f
More generally,
fR 
c
f
c  v cos 
Effect of receiver motion is different from that of source motion (Why?).
Combining both effects gives:
observer velocity
 c  v  cos 

fd  fR  f  
 1  f , Doppler shift
 c  v cos 

source velocity
If v = v’,
 2v cos  
fd  
f

 c  v cos  
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 2
Clinical Application of Doppler Effect
2v cos 
If v = v’, f d  
 c  v cos 

f

see: C. Holcombe et al., “Blood flow in breast cancer
and fibroadenoma estimated by colour Doppler
ultrasonography,” British J. Surgery 82, 787-788
(1995).
But why would it ever be the case that source and detector both are
moving in the same direction with the same speed?
How about if the medium is moving past a stationary source and
detector?
Ultrasound
transmitter
Artery
Limb
Ultrasound
receiver
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 3
Net Doppler Shift
Source
Receiver
c
λ
f = wave frequency [s-1]
λ = wavelength [m]
c = wave (or phase, or
propagation) velocity [m-s-1]
c = λf
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 4
Net Doppler Shift
Case 1: source in motion relative
to medium and receiver
v
v = source speed [cm-s-1]
T = 1/f = wave period [s]
vT = v/f = distance source travels
λ´ = λ – v/f
between emission of successive
wavefronts (crests) [m]
= c/f – v/f = (c – v)/f
f’ = c/λ´ = [c/(c – v)]f
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 5
Net Doppler Shift
Case 2: receiver in motion relative
to medium and source
v
v = receiver speed [m-s-1]
c’ = c + v = wave propagation
speed in receiver’s frame of
reference [m-s-1]
λ´ = λ
c´ = c + v
f’ = c´/λ = [(c + v)/c]f
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 6
Net Doppler Shift
Case 3: source and detector in
motion relative to each other and
medium
vs
vr
f’net = f’sf’r/f
= [c/(c – vs)][(c + vr)/c]f
= [(c + vr)/(c – vs)]f
fd = f’net – f = [(c + vr)/(c – vs) - 1]f
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 7
Nonlinear Features of Ultrasound Wave Propagation
Pressure p can be expressed as a function of density η:
p  ps
condensation
0 ,0
   0
 A
 0
¶p
A  0
¶
 1
 B
 2
2
   0 

 
 0 
, B  0
s 0 ,0
In combination with the fact that c 2 
¶2 p
¶ 2
s 0 ,0
¶p
, we get
¶ s ,
0
B
¶c
 20c 0
A
¶p
2
0
s 0 ,0
Note that if B  0, then c is a function of p. Wave crests (regions of
compression) propagate faster than wave troughs (regions of rarefaction)!
Observable significance of this dependence is...?
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 8
Nonlinear Features of Ultrasound: Shock Waves
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 9
Ultrasound Computed Tomography
Recommended supplemental reading:
A. J. Devaney, “A filtered backpropagation algorithm for
diffraction tomography,” Ultrasonic Imaging 4, 336-350 (1982).
J. F. Greenleaf, “Computerized tomography with ultrasound,”
Proceedings of the IEEE 71, 330-337 (1983).
H. Schomberg, W. Beil, G. C. McKinnon, R. Proksa, and O.
Tschendel, “Ultrasound computerized tomography,” Acta
Electronica 26, 121-128 (1984).
J. Ylitalo, J. Koivukangas, and J. Oksman, “Ultrasonic reflection
mode computed tomography through a skullbone,” IEEE
Transactions on Biomedical Engineering 37, 1059-1066 (1990).
Kak and Slaney, Chapters 6 (Tomographic Imaging with
Diffracting Sources) and 8 (Reflection Tomography)
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 10
Ultrasound Computed Tomography
Elementary Forms of Ultrasound CT Types
Ultrasonic Refractive Index Tomography
B
• Projection: A 1  n  x , y   ds  VwT d , T d  T  Tw
Ultrasonic Attenuation Tomography
B
• Projection: A   x , y  ds
• Three methods for estimating attenuation line integral:
– Energy-Ratio Method
– Division of Transforms Followed by Averaging Method
– Frequency-Shift Method
All of the foregoing are predicated on an assumption of
negligible refraction/diffraction/scattering of ultrasound
beams in the medium
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 11
Ultrasound Computed Tomography
Kak and Slaney,
pp. 153, 154
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 12
Ultrasound Computed Tomography
Photograph
Ultrasound Refractive
Index CT Image
Kak and Slaney, p. 154
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 13
Ultrasound Computed Tomography
Ultrasonic Attenuation CT Images
E-ratio Method
Division of
Transforms Method
f-shift Method
Kak and Slaney, pp. 156-158
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 14
Energy-Ratio Method
x(t) = incident ultrasound pulse, y(t) = detected transmitted
ultrasound pulse, yw(t) = detected pulse for transmission through
water
FT  X(f), Y(f), Yw(f)
Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|.
E(fk) = energy (or power), at frequency fk, in H(f).
Consider any two specific frequencies, f1 and f2, for which E(f1) and
E(f2) can be reliably and accurately determined. Then in principle:
B
A
  x , y  ds 
E
1
ln 1
2 f 2  f 1  E 2
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 15
Division of Transforms Followed by Averaging Method
x(t) = incident ultrasound pulse, y(t) = detected transmitted
ultrasound pulse, yw(t) = detected pulse for transmission through
water
FT  X(f), Y(f), Yw(f), Yw(f)
Transfer function: H(f) = Y(f)/X(f), |H(f)| = |Y(f)/Yw(f)|.
HA(f) = -ln|H(f)| = -ln|Y(f)/Yw(f)|.
In principle:
B
A
1
  x , y  ds 
2f 2
f 2 f 2
f
2 f 2
1
H A f  df 
2f 1
f1 f1
f
1 f 1
H A f  df
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 16
Frequency-Shift Method
x(t) = incident ultrasound pulse, y(t) = detected transmitted
ultrasound pulse
FT  X(f), Y(f), Yw(f)
f0 = frequency at which Yw(f) is maximal
fr = frequency at which Y(f) is maximal
σ2 = width of |Yw(f)|
In principle:
B
A
  x , y  ds 
f0  fr
2 2
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 17
Ultrasound Computed Tomography
Ultrasound CT
mammography...
...compared with x-ray CT
mammograms of the same patient.
Kak and Slaney, pp. 159-160
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 18
Diffraction Tomography with Ultrasound
What we can (attempt to) do when the “negligible
refraction/diffraction/scattering” criterion mentioned earlier is violated
Based upon treating ultrasound propagation through medium as a wave
phenomenon, not as a particle (i.e., ray) phenomenon
For homogeneous media, the Fourier Diffraction Theorem is analogous to
the central-slice theorem of x-ray CT
Heterogeneous media are treated as (we hope) small perturbations of a
homogeneous medium, to which an assumption such as the Born
approximation or Rytov approximation can be applied
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 19
Fourier Diffraction Theorem I
Arc radius is 
ultrasound frequency, or
wavenumber
Kak and Slaney, pp. 219
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 20
Fourier Diffraction Theorem II
Arc radius dependence
on wavenumber
Tomographic measurements
fill up Fourier space
Kak and Slaney, pp. 228, 229
BMI I FS05 – Class 9 “Ultrasound Imaging” Slide 21