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Truly 3D Tomography Since 1983 tomos - Greek for slice

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

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Truly 3D Tomography Since 1983 tomos - Greek for slice

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  1. Truly 3D Tomography Since 1983tomos - 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-Beam8 row; 9:1 pitch; 2.50mm slice width Warp3 Feldkamp shading artifacts (w,l) = (300,0)

  4. C t C t Thermoacoustics (Kruger, Wang, . . . ) breast 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 fover 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. 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 Recover a function f(x) from line integrals in 2 steps: backproject, then high-pass filter. data BP filter

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

  10. measure 2-Dim FBP smooth(coarsen(smooth f ))) = f so 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. 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).

  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. AX Rat Recon @ CRDhigh res & contrast 5° cone angle, 270m resolution, circular trajectory, FDK recon SAG COR

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