Ultrasound Computed Tomography
何祚明 陳彥甫 2002/06/12
Introduction Conventional X-ray image is the superposition of all the
planes normal to the direction of propagation. The tomography image is effectively an image of a slice
taken through a 3-D volume. An estimated 10-15% of breast cancers evade detection by
mammography. Poor differentiation of malignant tumors from highly
common cysts (while ultrasound can do so with accuracies of 90-100%).
UCT can provide not only structural/density information, but also tissue compressibility and speed of sound maps
Tomography Time of flight or intensity attenuation Array transducer D data (only tf) Rotation : 2-D data (tf on range and angle ) Scan : 3-D data (y position, tf and angle )
AB
Tomography
R
Areconstruct
Reconstruction Method
Attenuation method Iterative method
– Algebraic Reconstruction Technique (ART) Direct reconstruction
– Fourier transform Alternative direct reconstruction
– Back projection– Filtered back projection
Central Section Theorem
f (x,y)
g90(y’)
dxyxf
dxdyyyyxfyg
),(
)'(),()'(
x
y y’
Ambiguity Angle
f(x,y)
x
y
dxdyRyxyxfRg )sincos(),()(
R
xcos+ ysin= RRy
Rx
sin,cos
Equations
dxdyvyuxiyxf
dxdyyxiyxf
dxdyyxiyxf
dxdydRRiRyxyxf
RgTFDG
))(2exp(),(
))sincos(2exp(),(
))sincos(2exp(),(
)2exp()sincos(),(
)}(.{.)1()(
•1D FT of projection function
2D FT
Cont’d 2D Fourier transform
(u,v) in polar coordinates is (cos, sin)
2 D inverse Fourier Transform
),(|),()( sin,cos fvufG vu
dxdyvyuxjyxfvuf )](2exp[),(),(
0
2
0)]sincos(2exp[)(
)](2exp[)(),(
dyxjGd
dudvvyuxjGyxf
f(x,y)x
y
R
u
v
g(R)
G()
Figures
Algorithm
solve coordinate problems (polar to rectangular coordinates)
1. 1D FT each of the projections g(R) G()
2. Integrate G() in and -- avoid coordinate transformation problems
g(R) G() =F(u,v) f (x, y)
0
2
0)]sincos(2exp[)(),(
dyxjGdyxf
1D FT
Back Projection Reconstruction Back projection (at one angle)
Integration of all ambiguity angle
dRRyxRgyxb )sincos()(),(
0
)sincos()(),( ddRRyxRgyxfb
Cont’d Drawback of back projection method
So, we integrate this function after added 1/
ryxfFDyxfFFD
ddRyxiFyxfb
1),(12),(),(2
)sincos(2exp(),(),(
11
2
0 0
ddRyxiF
ddRdRiRyxFyxfb
0
0
)sincos(2exp(),(
)2exp()sincos(),(),(
Disadvantage
FT{1/1/r : blurring function f (x,y) convolves 1/r Additional filter is needed
x
y y y
x x
1/r object Backprojected object
Filter Design
Design filter at spatial frequency domain Cone filter
Spatial filter u
v
1/r
22),( vuvuC
( )(sin)2(sin2)(
2)(
02
02
0
000
RcRcRC
trirectC
Signal Processing Flow Backprojection all angles, g(R) Fb() Forward 2D FT to get: Apply filter on to get F (u,v) Inverse 2D FT to f (x,y)
),(F
),(F
Fb() F(u,v) f (x, y)
2D FT 2D IFT
),(Fg(R)
Filtering
Algebraic Reconstruction Technique (ART)
The iterative process starts with all values set to a constant, such as the mean or 0.
At each iteration, the difference between the measured projection data for a given projection and the sum of all reconstructed elements along the line defining the projection is computed.
This difference is then evenly divided among the N reconstruction elements.
Algebraic Reconstruction Technique (ART)
1. Projection Step2. Comparison Step3. Backprojection Step4. Update Step
N
fgff
N
i
qijj
qij
qij
11
Additive ART :q
ijN
i
qij
jqij f
f
gf
1
1
Multiplicative ART :
Algebraic Reconstruction Technique (ART)
N elements per line
fij (calculated element)
cross section
gj (measured projection)
Simulation Results I.•Using ART to reconstruct :
Simulated Data Reconstruction Result
10 20 30 40 50 60 70 80 90 100
102030405060708090
100 10 20 30 40 50 60 70 80 90 100
102030405060708090
100
Simulation Results I.
10 20 30 40 50 60 70 80 90 100
102030405060708090
100
Simulated Data
10 20 30 40 50 60 70 80
1020304050607080
Reconstruction Result
•Using direct Fourier transform to reconstruct :
Simulation Results II. •Using ART to reconstruct :
Reconstruction ResultSimulated Data
10 20 30 40 50 60 70 80 90 100
102030405060708090
100 10 20 30 40 50 60 70 80 90 100
102030405060708090
100
Simulation Results II.
Reconstruction ResultSimulated Data
10 20 30 40 50 60 70 80 90 100
102030405060708090
100 5 10 15 20
2468
101214161820
•Using direct Fourier transform to reconstruct :
Experiment Architecture
TransducerPhantomTransducer
y
x
Water Tank
Transmitter
Receiver
2 mm/step
2 degrees/rotate
Experiment Architecture
Breast Phantom
Eraser Phantom
Rubber Ball
Time-of-flightt1
t2
t1
t2
t3
t3
Reconstructed Image•Breast phantom
10 20 30 40 50 60 70 80
1020
3040
50
6070
80
ART Direct Fourier Transform
10 20 30 40 50 60 70 80
10
20
30
40
50
60
70
80
Reconstructed Image•Breast phantom, using ART
10 20 30 40 50 60 70 80
10
20
30
40
50
60
70
80
Reconstructed Image•Rubber ball
2 4 6 8 10 12 14 16 18 20
2468
101214161820
5 10 15 20
2468
101214161820
ART Direct Fourier Transform
Reconstructed Image•Eraser
5 10 15 20 25 30 35 40
5
10
15
20
25
30
35
405 10 15 20 25 30 35 40
510152025303540
Direct Fourier TransformART
Conclusions
ART reconstruction is much faster than other reconstruction method but must notice the velocity profile normalization.
Direct fourier reconstruction is effective but less efficiency.
Backprojection reconstruction needed additional filter to cancel integration errors.
Future Works
Correct velocity profile estimation Discuss backprojection method errors and
additional filter design method Discuss diffraction errors in ultrasound
References [1] Albert Macovski, “Medical Imaging Systems” [2] Matthew O’Donnell, “X-Ray Computed Tomography
(CT)”, University of Michigan [3] Douglas C. Noll, “Computed Tomography Notes”,
University of Michigan, 2001 [4] Brian Borchers, “Tomography Lecture Notes”, New
Mexico Tech., March 2000 [5] James F. Greenleaf et al., “Clinical Imaging with
Transmissive Ultrasonic Computerized Tomography”, IEEE trans.on Biomedical Engineering vol. 28 no. 2 1981
