Thermoacoustic Tomography – Inherently 3D Reconstruction
Thermoacoustic Tomography
An Inherently 3D Generalized Radon Inversion Problem
G Ambartsoumian Texas A&M D Finch Oregon State UniversitySK Patch* GE Healthcare Rakesh U. Delaware
Thermoacoustic Tomography – Inherently 3D Reconstruction
Outline
NIR diffuses. . .
Xrays propagate straight through, image recovery stable.
Sound waves also propagate, permitting stable inversion.
supp f• WHY TCT – images, ... • Ties to wave equation• Physics - forward problem • Inversion formulae for
complete data• Inverting incomplete data
Backup• Wave Fronts • Recon Background
– Xray CT– Spherical Transforms
Thermoacoustic Tomography – Inherently 3D Reconstruction
TransScan R&D
EIT
US
MRI
slice
UW-MadisonTransScan R&D
Xray
proje
ction
Images across modalities
TCT
slice
Kruger/OptoSonics, Inc.
ART opticalPrototype TCT
already competes w/conventional
scans!
Thermoacoustic Tomography – Inherently 3D Reconstruction
Ductal Carcinoma in situ (DCIS)
Images courtesy R. Kruger, OptoSonics, Inc.
Thermoacoustic Tomography – Inherently 3D Reconstruction
TCT Changes During Chemo (TCT V2.4)
Longitudinal changes during primary chemotherapy. Tumor mass (arrows) appears to have decreased markedly.
Baseline 7 weeks Pre-Surgery
Images courtesy R. Kruger, OptoSonics, Inc.
Thermoacoustic Tomography – Inherently 3D Reconstruction
Fibrocystic Breast(Extra Dense Breast)
cyst
cysts
Images courtesy R. Kruger, OptoSonics, Inc.
Thermoacoustic Tomography – Inherently 3D Reconstruction
Thermoacoustics (Kruger, Wang, . . . )
RF/NIR heating thermal expansion
pressure waves US signal
C t
C t
???
breast
waveguides
Kruger, Stantz, Kiser. Proc. SPIE 2002.
Thermoacoustic Tomography – Inherently 3D Reconstruction
Measured Data
• Integrate f over spheres
• Centers of spheres on sphere
• Partial data only for mammography
S+ upper hemisphere
S- lower hemisphere
inadmissable transducer
θppθ
drfrrfRTCT
1
2,
Thermoacoustic Tomography – Inherently 3D Reconstruction
imaging object
filtered inversion (complete data)
• Backproject data (thanks to V. Palamodov!)
• Switch order of integration
• Use manifold identity (4x!)
• f Riesz potential • f after high-pass filter zdzzzhzdzh
zzMn
R
n
Mz n
))(()()(
0)(|1
Use co-area formula
Thermoacoustic Tomography – Inherently 3D Reconstruction
filtered inversion (complete data)
pypypxpyypxp y
ddfR
1
222
1
3
ppxppxp
dfRTCT
,1
1
x
p
pθpxpxppxp θ
ddf
1
2
1
1
yppxpyypy
ddfR
1
222
3
yppxpypypy
ddfRR
33
22212
Thermoacoustic Tomography – Inherently 3D Reconstruction
filtered inversion (-identity)
22
221 1
21
11
hpp
dshxy
y
xy
yppxp
pxyp
df
dfRR
TCT
3
2,1
1
x
y
h21 h
pypxp
ppxy
ppxpyp 2232221
112
3
ddR
xy
2
Thermoacoustic Tomography – Inherently 3D Reconstruction
Inversion Formulae (complete data)
)(8
1)( 2,2
xRfxf sz
x
ppxppxp
x dfRTCT
,1
8
1
12
-filtered
ppxp
pxp
dfRTCT
,1
8
1
12
FBP
Thermoacoustic Tomography – Inherently 3D Reconstruction
Numerical Results (G Ambartsoumian)
256x256 images from (N,N,Nr) = (400,200,200)
FBP with 1/weighting FBP w/experimentalweighting
Thermoacoustic Tomography – Inherently 3D Reconstruction
FBP filtered
Simulated data sans noise (N,N,Nr) = (800,400,512)
machine precision,
as it should be
Thermoacoustic Tomography – Inherently 3D Reconstruction
full scan data w/o noise
(N,N,Nr) = (800,400,512)
Low Contrast Detectability
abs = 0.002 abs = 0.004 abs = 0.006
Thermoacoustic Tomography – Inherently 3D Reconstruction
Low Contrast Detectability
Thermoacoustic Tomography – Inherently 3D Reconstruction
Partial Scan Reconstructions (N,N,Nr) = (400,200,200)
FBP with ½ data in filtered with ½ data in
Thermoacoustic Tomography – Inherently 3D Reconstruction
Consistency Conditions – Necessary, but maybe not sufficient
rfRrfR TCTTCT ,, pp even wrt r
xxpxpp dfdrrfRrMomnR
k
R
TCTk
k
,
ppp kSk QMom n 12
xxppxxp dfMomnR
k
k 22
2 2
Thermoacoustic Tomography – Inherently 3D Reconstruction
Consistency Conditions – Implications
1
1
2 ,~
ˆr
TCTkk drrfRrPc pp
0
2
~,
nkkTCT rPcrfR pp
1
1
2
0
,r
TCTl
k
l
kl drrfRrc p
pp l
k
l
klk Qcc
0
poly of degree k in p!!
measure ck on S- ;
evaluate ck on S+
Thermoacoustic Tomography – Inherently 3D Reconstruction
Polynomial Expansion Accuracy
High-order expansions required!!!
02
3321
~
,,,,
nkk
TCTTCT
rPc
rpfRrpppfR
p
deg 8
deg 16deg 24
RT
CTf(
3/8,
r)r
f=1
f=0
Thermoacoustic Tomography – Inherently 3D Reconstruction
Polynomial Extrapolation – Stability
deg 24
Measure over p3[-1,0)
• rescale so q3[-1,1)
• fit measurements to another set of Leg. polys
P2
f=1
f=0
deg 8
deg 16
Evaluate for p3 [0, 1) i.e., q3[1,3)
P2
Thermoacoustic Tomography – Inherently 3D Reconstruction
measured data
extrapolated data
Thermoacoustic Tomography – Inherently 3D Reconstruction
additive white noise
½ scan only ½ scan+deg-10 extrap
Thermoacoustic Tomography – Inherently 3D Reconstruction
full scan data w/o noise
(N,N,Nr) = (800,400,512)
½ scan reconstruction –
zero-filling vs.
data extension
with respect to z-only
window width = 0.6
window width = 0.3window width = 0.3window width = 0.2
Thermoacoustic Tomography – Inherently 3D Reconstruction
½ scan FBP reconstruction –
0.2% “absolute” additive white noise
zero-filling vs.data extension
window width = 1.3deg 4
window width = 1.2
window width = 0.6deg 8 window width = 0.6deg 12 window width = 0.6deg 16
Thermoacoustic Tomography – Inherently 3D Reconstruction
Attenuation Blurs
xb
e
DISCLAIMER -
WIP
•
•
•
• soundspeed c=1500m/s
tce 1.0 PNT Wells, Biomedical Ultrasonics
where
• t are dual Fourier variables
• b ~ 1
• ~ 0.1 MHz-1 cm-1
x= 1
x= 2
x= 4 x= 6
Thermoacoustic Tomography – Inherently 3D Reconstruction
Heuristic Image Quality Impact
Ideal Object/Full Scan Attenuation-Full ScanAttenuation-Partial Scan
use 2D xray transform & exploit projection-sliceDISCLAIMER -
WIP
Thermoacoustic Tomography – Inherently 3D Reconstruction
non-Math Conclusions
• Positives– cheap ??– non-ionizing – high-res (exploits hyperbolic physics)– 2x depth penetration of ultrasound, sans
speckle– detect masses
• Issues– will not detect microcalcifications– contrast mechanism not understood– fundamental physics (attenuation, etc)
and HW constraints will impact IQ
GOAL : biannual
screening
TCT for small
low-contrast
masses xrays
miss.
Xrays for
precursors
(microcalcs)
Thermoacoustic Tomography – Inherently 3D Reconstruction
Math Conclusions
• FBP type inversion formulae• Partial scan - unstable outside of
“audible zone”– Palamodov– Davison & Grunbaum
• Attenuation – expect blurring
GOAL #1
Incorporate
fundamental
physics
GOAL #2
incorporate
hardware
constraints
– Anastasio et al , Xu et al – OK inside
• cos transducer response – Finch
Thermoacoustic Tomography – Inherently 3D Reconstruction
Back-Up slides
Thermoacoustic Tomography – Inherently 3D Reconstruction
Simulated data sans noise (N,N,Nr) = (800,400,512)
FBP filtered
Thermoacoustic Tomography – Inherently 3D Reconstruction
½ scan reconstruction –
0.2% “absolute” noise
zero-filling vs.data extension
window width = 0.8window width = 0.8
Thermoacoustic Tomography – Inherently 3D Reconstruction
Wave Fronts in 2-D Standard Radon
so
sx
xdxfsRfo
o 1)(),(
measure
w/resolution comparable to that of surface parameter s
Recover image edges tangent to measurement surface edges
oo ,)( RfRfprojection-slice theorem
Thermoacoustic Tomography – Inherently 3D Reconstruction
Wave Fronts in TCT
f=1
“indirect” information about vertical edges.
“direct” information about horizontal edges;
Thermoacoustic Tomography – Inherently 3D Reconstruction
Recon Background
Xray CT – line integrals
• 2D • 3D
– Grangeat• line plane • plane recon’d image
– Katsevich line recon’d
image
supp f
supp f
Spherical Transforms• 2D
– Circles centered on lines– Circles through a point
• 3D – Spheres centered on plane– Spheres through a point
Thermoacoustic Tomography – Inherently 3D Reconstruction
Recon Background
• 2D– Circles centered on lines– Circles through a point
• 3D – Spheres centered on plane– Spheres through a point
• 2DCircles centered on circles
(Norton)
• 3D Spheres centered on
sphere(Norton & Linzer, approximate inversion for complete data)
Spherical Transforms