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