Truly 3D Tomography Since 1983 tomos - Greek for slice
Xray CT measures line integrals
HighSpeed mode in Warp3: = 1.2°
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
Xray CT: HW vs. Cone-Beam8 row; 9:1 pitch; 2.50mm slice width
Warp3 Feldkamp
shading artifacts
(w,l) = (300,0)
Thermoacoustics (Kruger, Wang, . . . )
RF/NIR heating thermal expansion pressure waves US signal
C t
C t
???
breast
waveguides
Kruger, Stantz, Kiser. Proc. SPIE 2002.
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,
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
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 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
n-Dim FBPRecover function f (x) from (n-1) dim planar integrals in 2 steps: high-pass filter, then backproject.
data
filter
BP
2-Dim FBPso
sx
xdxfsRfo
o 1)(),(
smooth(coarsen(smooth f ))) = f
measure
backproject ),(,),( ooo sRfsRf
filter
ooo 1
),()(S
dxRfxf
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
Grangeat’s technique line integrals plane integrals
“fan” of line integrals in
want plane integral
ts
Radon Inversion Pitch Constraint
R
R
Triangulate Radon planes
Pitch < 2(#rows-1)
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 @ CRDhigh res & contrast
AX
SAG
COR
5° cone angle, 270m resolution, circular trajectory, FDK recon