1. Single-Slice Rebinning Method for Helical Cone-Beam CT
Literature Review
Single-Slice Rebinning Method for
Helical Cone-Beam CT
Frédéric Noo, Michel Defrise, Rolf Clackdoyle
Physics in Medicine and Biology
Vol 44, 1999
Henry Chen
May 13, 2011

2. Background
Computed Tomography

3. Tomography Basis for CAT scan, MRI, PET, SPECT, etc.
Greek “tomos”: part or section
Cross-sectional imaging technique using transmission or reflection data from multiple angles

4. Computed Tomography (CT)
A form of tomographic reconstruction on computers
Usually refers to X-ray CT
Positron (PET)
Gamma rays (SPECT)

5. Cross-Sections by X-Ray Projections
Project X-ray through biological tissue; measure total absorption of ray by tissue
Projection Pθ(t) is the Radon transform of object function f(x,y):
Total set of projections called sinogram

6. X-Ray Projection Example

7. Phantom and Sinogram
Shepp-Logan Phantom

8. CT Reconstruction Restore image from projection data
Inverse Radon transform
Most common algorithm is filtered backprojection
“Smear” each projection over image plane

9. Fourier Slice Theorem
Fourier transform of a 2-D object projected onto a line is equal to a slice of a 2-D Fourier transform of the object
Allows image reconstruction from projection data
Overlay FT of projections in 2-D Fourier domain
If carried out in space domain, becomes backprojection procedure

19. Backprojection Result
Need filtering (high-pass), interpolation

20. FBP Algorithm Input: sinogram sino(θ, n) Output: image img(x,y)
for each θ
filter sino(θ,*)
for each x
for each y
n = x cos θ + y sin θ
img(x,y) = sino(θ, n) + img(x,y)
O(n3) algorithm
But highly parallelizable, given sufficient memory bandwidth; not computationally intensive

21. Single-Slice Rebinning Method for Helical Cone-Beam CT

22. 3-D Tomography
Construct 3-D image using sequence of cross sectional images
Requires many passes of ionizing x-ray radiation

23. Helical Cone-Beam Scanner
Helical traversal reduces total number of passes; increases scanning speed and reduces x-ray exposure
However, need to convert cone-beam data into stack of fan-beam slice images

24. Rebinning Maps projection data into fan-beam slices
Virtual fan source mapped to surface of cone source
Each slice requires CB projections from 1 revolution centered on that slice (+/- 0.5P)

25. Rebinning
Fan-beam “projection” estimated from value of cone-beam projection in same vertical plane
Use closest cone-beam source directly above or below virtual fan-beam source
Use oblique ray passing through M

26. Rebinning Geometry

27. Rebinning Equation
Each fan-beam value is just weighted version of cone-beam projection data
fan-beam projection length
fan sino
CB sino
cone-beam projection length

28. Simulation Comparisons
A: Single detector row (no oblique rays), 5mm pitch CSH-HS algorithm
B: Seven detector rows, 25mm pitch CB-SSRB algorithm
C: Seven detector rows, 100mm pitch Full 3-D backprojection algorithm
D: Seven detector rows, 100mm pitch CB-SSRB algorithm

29. Simulation Comparisons

30. Simulation Comparisons

31. Performance Comparison
Runtime for (C): 276s CPU, using 27 ray-sums
Runtime for (D): 1310s CPU, using 57 ray-sum
Runtimes for (A) and (B) about equal

32. Conclusions
Like all CT, CB-SSRB is not exact; reconstruction artifacts can affect image quality
However, good performance for large-pitch helixes
5x larger pitch = 5x fewer projection measurements
Selection of oblique rays integral to performance
Need multiple detector rows (7 vs. 1)

33. References http://en.wikipedia.org/wiki/Tomography
Kak, A. C., Slaney, M., Principles of Computerized Tomographic Imaging, IEEE Press, 1988