click to edit master title style literature review single-slice rebinning method for helical...
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Click to edit Master title styleLiterature Review
Single-Slice Rebinning Method forHelical Cone-Beam CT
Frédéric Noo, Michel Defrise, Rolf ClackdoylePhysics in Medicine and Biology
Vol 44, 1999
Henry ChenMay 13, 2011
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Background
Computed Tomography
Tomography
Basis for CAT scan, MRI, PET, SPECT, etc.
Greek “tomos”: part or section
Cross-sectional imagingtechnique using transmissionor reflection data frommultiple angles
Computed Tomography (CT)
A form of tomographic reconstruction on computers
Usually refers to X-ray CT– Positron (PET)– Gamma rays (SPECT)
Cross-Sections by X-Ray Projections
Project X-ray through biological tissue;measure total absorption of ray by tissue
Projection Pθ(t) is the Radontransform of object functionf(x,y):
Total set of projections calledsinogram
, cos sinP t f x y x y t dxdy
X-Ray Projection Example
Phantom and Sinogram
Shepp-Logan Phantom
CT Reconstruction
Restore image from projection data
Inverse Radon transform
Most common algorithm is filtered backprojection– “Smear” each projection over image plane
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
Backprojection Result
Need filtering (high-pass), interpolation
FBP Algorithm
Input: sinogram sino(θ, n) Output: image img(x,y)
for each θfilter sino(θ,*)for each x
for each yn = x cos θ + y sin θimg(x,y) = sino(θ, n) + img(x,y)
O(n3) algorithm– But highly parallelizable, given sufficient memory
bandwidth; not computationally intensive
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Single-Slice Rebinning Methodfor Helical Cone-Beam CT
3-D Tomography
Construct 3-D image using sequence ofcross sectional images
Requires many passes of ionizing x-ray radiation
Helical Cone-Beam Scanner
Helical traversal reduces total number of passes; increases scanning speed and reduces x-ray exposure
However, need to convertcone-beam data into stackof fan-beam slice images
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)
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
Rebinning Geometry
Rebinning Equation
Each fan-beam value is just weighted version of cone-beam projection data
2 2
2 2 2, ,z
u Dp u g u v
u v D
2 2u Dv z
RD
fan sino
fan-beam projection length
cone-beam projection length
CB sino
Simulation Comparisons
A: Single detector row (no oblique rays), 5mm pitch
CSH-HS algorithm
B: Seven detector rows, 25mm pitchCB-SSRB algorithm
C: Seven detector rows, 100mm pitchFull 3-D backprojection algorithm
D: Seven detector rows, 100mm pitchCB-SSRB algorithm
Simulation Comparisons
Simulation Comparisons
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
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)
References
http://en.wikipedia.org/wiki/Tomography
http://en.wikipedia.org/wiki/Computed_tomography
Kak, A. C., Slaney, M., Principles of Computerized Tomographic Imaging, IEEE Press, 1988