Truly 3D Tomography Since 1983
tomos - Greek for slice
Xray CT
measures line
integrals
k
HighSpeed mode in Warp3: k = 1.2°
Lightspeed Recon assumes k = 0
8-slice Warp3
recon X
is 2D
Cone-beam backprojector required!!
Xray CT: HW vs. Cone-Beam
8 row; 9:1 pitch; 2.50mm slice width
Warp3
Feldkamp
shading
artifacts
(w,l) = (300,0)
Thermoacoustics (Kruger, Wang, . .
.)
breast
Ct
Ct
waveguides
Kruger, Stantz, Kiser. Proc. SPIE 2002.
 RF/NIR heating 
 thermal expansion 
 pressure waves 
 US signal
???
Measured Data - Spherical Integrals
RTCT f p, r  = r  f p  r  dθ
2
θ =1
inadmissable
transducer
S+ upper
hemisphere
S- lower
hemisphere
• Integrate f over
spheres
• Centers of spheres
on sphere
• Partial data only for
mammography
Xray CT Reconstruction Primer
Math fundamentals
a. Projection-Slice on blackboard
b. Fourier inversion
c. Xray inversion formula
d. FBP (Filtered BackProjection), aka “Radon”
VCT – FDK & Grangeat
Research
a. Public domain
b. GE - primarily CRD for GEAE
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
n-Dim Xray Inversion
Recover a function f(x) from line
integrals in 2 steps: backproject,
then high-pass filter.
Xf ( x, o) =
 f ( x  to) dt
data
 Xf ( x, o) do
BP
R1
X * Xf ( x) =

S n1

f ( x) =  X Xf   
*


filter
( x)
n-Dim FBP
Recover function f (x) from
(n-1) dim planar integrals in
2 steps: high-pass filter,
then backproject.
data
filter
BP
measure
2-Dim FBP
smooth(coarsen(smooth f ))) = f
so
Rf ( s, o) =
filter

f ( x ) d 1x
x o = s


Rf (s, o) =  Rf   , o (s, o)


backproject
f ( x) =  Rf ( x  o, o) do
S1
FDK - perturbation of 2D FBP
Pb,x = plane defined by source
position b and a horizontal line
on detector containing x
b
x
k
fix reconstruction point x,
for each source position b
update f(x) as if reconstructing plane Pb,x
end
Grangeat’s technique
line integrals  plane integrals
“fan” of line integrals in
g
s
b
want plane integral
t
Radon Inversion
Pitch Constraint
Pitch < 2(#rows-1)
R
R
Triangulate
Radon planes
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).
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
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
Rat Recon @ CRD
high res & contrast
5° cone angle,
270m resolution,
circular trajectory,
FDK recon
AX
SAG
COR