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
projec
tion
Images across modalities
TCT
slice
Kruger/OptoSonics, Inc.
ART optica
lPrototype 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
inadmissabletransducer
( )( ) ( ) ?pp?
drfrrfRTCT ∫=
+=1
2, θ
Thermoacoustic Tomography – Inherently 3D Reconstruction
imaging object
ρ−filtered inversion (complete data)
• Backproject data (thanks to V. Palamodov!)
• Switch order of integration
• Use δ−manifold identity (4x!) →
• f ∗Riesz potential →• f after high-pass filter
{ }( ) zdzzzhzdzh
zzMn
R
n
Mz n
))(()()(
0)(|1 φδφ
φ
∫∫ ∇=
==−
∈
Use co-area formula
Thermoacoustic Tomography – Inherently 3D Reconstruction
ρ−filtered inversion (complete data)
( ) ( ) ( ) pypypxpyypxp y
ddfR
∫ ∫= ∈
−−−−
−=
1
22 21
3
δ
( )( ) ppxppxp
dfRTCT −−∫
=
,1
1
x
p
( ) p?pxpxppxp ?
ddf∫ ∫= =
−−+
−=
1
2
1
1θ
( ) ( ) yppxpyypy
ddfR
−−−= ∫∫
=∈ 1
2223
δ
( ) ( ) ( ) yppxpypypy
ddfRR
−−−−= ∫∫
∈∈ 33
222 12 δδ
Thermoacoustic Tomography – Inherently 3D Reconstruction
ρ−filtered inversion (δ-identity)
∫−=+ −−
=22
221 1
21
11
hpp
dshxy
( )( ) ( )y
xyy
ppxppx
yp
df
dfRR
TCT ∫∫∈= −
=−− 3
2,1
1
π
x
y
h21 h−
( ) ( ) ( )∫∫−=−∈
−−
=−−−−pypxp
ppxy
ppxpyp 223222 11
123
ddR
δδδ
xy −=
π2
Thermoacoustic Tomography – Inherently 3D Reconstruction
Inversion Formulae (complete data)
[ ] )(8
1)( 2,2 xRfxf sz∗∆
−= xπ
( )( ) ppxppxp
x dfRTCT −−
∆−
= ∫=
,1
81
12π
ρ-filtered
:( ) ( ) ppxp
pxp
dfRTCT −″−
−= ∫
=
,1
81
12π
:FBP
Thermoacoustic Tomography – Inherently 3D Reconstruction
Numerical Results (G Ambartsoumian)256x256 images from (Nφ,Nθ,Nr) = (400,200,200)
FBP with 1/ρ weighting FBP w/experimental weighting
Thermoacoustic Tomography – Inherently 3D Reconstruction
FBP ρ−filtered
Simulated data sans noise (Nφ,Nθ,Nr) = (800,400,512)
machine precision,
as it should be
Thermoacoustic Tomography – Inherently 3D Reconstruction
full scan data w/o noise
(Nφ,Nθ,Nr) = (800,400,512)
Low Contrast Detectability
σabs = 0.002σabs = 0.004σabs = 0.006
Thermoacoustic Tomography – Inherently 3D Reconstruction Low Contrast Detectability
Thermoacoustic Tomography – Inherently 3D Reconstruction
Partial Scan Reconstructions (Nφ,Nθ,Nr) = (400,200,200)
FBP with ½ data in θ ρ−filtered with ½ data in φ
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 ∫ +•−= 222 2
Thermoacoustic Tomography – Inherently 3D Reconstruction
Consistency Conditions –Implications
( ) ( ) ( )∫−=
≡1
12 ,~ˆ
rTCTkk drrfRrPc pp
( ) ( ) ( )∑∞
=
=0
2~
,n
kkTCT 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 ExpansionAccuracy
High-order expansions required!!!
( ) ( )
( ) ( )∑∞
=
=
=
02
3321
~
,,,,
nkk
TCTTCT
rPc
rpfRrpppfR
p
deg 8
deg 16deg 24
R TCTf
(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 noiseσ = 0.01
½ scan only ½ scan+deg-10 extrap
Thermoacoustic Tomography – Inherently 3D Reconstruction
full scan data w/o noise
(Nφ,Nθ,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 ∆− τα
DISCLAIMER - WIP
•
•
•
• soundspeed c=1500m/s
tce τ1.0− PNT Wells, Biomedical Ultrasonics
where
• τ ∼ t are dual Fourier variables
• b ~ 1
• α ~ 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 θ transducer response – Finch
Thermoacoustic Tomography – Inherently 3D Reconstruction
Back-Up slides
Thermoacoustic Tomography – Inherently 3D Reconstruction
Simulated data sans noise (Nφ,Nθ,Nr) = (800,400,512)
FBP ρ−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 ,)( σσ ∧∧ = 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
– Katsevichline ∫ → 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