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Truly 3D Tomography Since 1983 tomos - Greek for slice
Xray CT measures line integrals
HighSpeed mode in Warp3: = 1.2°
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Lightspeed Recon assumes
8-slice Warp3 recon is 2D
Cone-beam backprojector required!!
New CT systems are
64 slice & have
cone-beam BP and
~2.4
X
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Xray CT: HW vs. Cone-Beam8 row; 9:1 pitch; 2.50mm slice width
Warp3 Feldkamp
shading artifacts
(w,l) = (300,0)
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Thermoacoustics (Kruger, Wang, . . . )
RF/NIR heating thermal expansion pressure waves US signal
C t
C t
???
breast
waveguides
Kruger, Stantz, Kiser. Proc. SPIE 2002.
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Measured Data - Spherical Integrals
• 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,
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Xray CT Reconstruction Primer
Math fundamentals a. Projection-Slice on blackboardb. Fourier inversionc. Xray inversion formula d. FBP (Filtered BackProjection), aka “Radon”
VCT – FDK & GrangeatResearch
a. Public domain
b. GE - primarily CRD for GEAE
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n-Dim Fourier inversion of Radon data
Recover function f (x) from (n-1) dim planar integrals in 3 steps: many 1D FFTs, regrid, n-Dim IFFT.
data
proj-slice (1D FFT) regrid nD IFFT
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n-Dim Xray InversionRecover a function f(x) from line integrals in 2 steps: backproject, then high-pass filter.
1
1
),()(
)(),(
*
nS
R
dxXfxXfX
dttxfxXf
oo
oo
)()( * xXfXxf
data
BP
filter
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n-Dim FBPRecover function f (x) from (n-1) dim planar integrals in 2 steps: high-pass filter, then backproject.
data
filter
BP
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2-Dim FBPso
sx
xdxfsRfo
o 1)(),(
smooth(coarsen(smooth f ))) = f
measure
backproject ),(,),( ooo sRfsRf
filter
ooo 1
),()(S
dxRfxf
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FDK - perturbation of 2D FBP
x
Px = plane defined by source position and a horizontal line
on detector containing x
fix reconstruction point x,
for each source position update f(x) as if reconstructing plane Px end
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Grangeat’s technique line integrals plane integrals
“fan” of line integrals in
want plane integral
ts
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Radon Inversion Pitch Constraint
R
R
Triangulate Radon planes
Pitch < 2(#rows-1)
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Major Published Results
• HK Tuy, “An Inversion Formula for Cone-Beam Reconstructions,” SIAM J. Appl. Math, 43, pp. 546-552, (1983).
• LA Feldkamp, LC Davis, JW Kress, "Practical Cone-Beam Algorithm," JOSA A, 1 #6, pp. 612-619, (1984).
• KT Smith, "Inversion of the X-ray Transform," SIAM-AMS Proc., 14, pp. 41-52, (1984).
• D. Finch, “Cone Beam Reconstruction with Sources on a Curve,” SIAM J. Appl. Math, 45 #4, pp. 665-673, (1985).
• P. Grangeat, "Analyse d'un Systeme D'Imagerie 3D par reconstruction a partir de radiographies X en geometrie. conique," doctoral thesis, Ecole Nationale Superieure des Telecommunications, (1987).
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VCT Research at GE
• Kennan T. Smith - CRD summer visitor from Oregon State University; filtered backprojection algorithms
• Kwok Tam - CRD employee; implemented Grangeat's algorithm; long object problem
• Per-Erik Danielsson - CRD summer visitor ~90 from Linkoping University; Fourier implementation of Grangeat's algorithm
• Hui Hu - GEMS-ASL; compared FDK vs. Grangeat• SK Patch - range conditions on VCT data
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VCT in Action at GE
• MBPL(CRD) - Tam recons for GEAE projects - plagued by detector problems
• GEAE - circular FDK on high-res VCT data w/very small cone angle, high-contrast
• IEL(CRD) - circular FDK on Apollo data, high-contrast
• GEMS - helical FDK on Lightspeed data
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Rat Recon @ CRDhigh res & contrast
AX
SAG
COR
5° cone angle, 270m resolution, circular trajectory, FDK recon