1. Truly 3D Tomography Since 1983 tomos - Greek for slice
Xray CT measures line integrals k
HighSpeed mode in Warp3:  = 1.2°

2. Lightspeed Recon assumes k = 0
8-slice Warp3 recon is 2D X
New CT systems are 64 slice & have cone-beam BP and ~2.4
Cone-beam backprojector required!!

3. Xray CT: HW vs. Cone-Beam 8 row; 9:1 pitch; 2.50mm slice width
Warp3
Feldkamp
shading artifacts
(w,l) = (300,0)

4. Thermoacoustics (Kruger, Wang, . . . )
breast
C t
waveguides
Kruger, Stantz, Kiser. Proc. SPIE 2002.
 RF/NIR heating 
 thermal expansion 
 pressure waves 
 US signal
???

5. Measured Data - Spherical Integrals
S+ upper hemisphere
inadmissable transducer
Integrate f over spheres
Centers of spheres on sphere
Partial data only for mammography
S- lower hemisphere

6. 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

7. data 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

8. n-Dim Xray Inversion data BP
Recover a function f(x) from line integrals in 2 steps: backproject, then high-pass filter.
data BP
filter

9. n-Dim FBP data filter BP
Recover function f (x) from (n-1) dim planar integrals in 2 steps: high-pass filter, then backproject.
data
filter BP

10. 2-Dim FBP filter measure smooth(coarsen(smooth f ))) = f backprojectso
filter
backproject

11. FDK - perturbation of 2D FBP
Pb,x = plane defined by source position b and a horizontal line on detector containing x
fix reconstruction point x,
for each source position b update f(x) as if reconstructing plane Pb,x end x k b

12. Grangeat’s technique line integrals  plane integrals
“fan” of line integrals in g t s b
want plane integral

13. Radon Inversion Pitch Constraint
Pitch < 2(#rows-1)
Triangulate Radon planes R R

14. Major Published Results
HK Tuy, “An Inversion Formula for Cone-Beam Reconstructions,” SIAM J. Appl. Math, 43, pp , (1983).
LA Feldkamp, LC Davis, JW Kress, "Practical Cone-Beam Algorithm," JOSA A, 1 #6, pp , (1984).
KT Smith, "Inversion of the X-ray Transform," SIAM-AMS Proc., 14, pp , (1984).
D. Finch, “Cone Beam Reconstruction with Sources on a Curve,” SIAM J. Appl. Math, 45 #4, pp , (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).

15. 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

16. 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

17. Rat Recon @ CRD high res & contrastAX
Rat CRD high res & contrast
5° cone angle, 270m resolution, circular trajectory, FDK recon
SAG
COR