Transcript Single-Slice Rebinning Method for Helical Cone-Beam CT Literature Review

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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
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Background
Computed Tomography
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
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 Radon
transform of object function
f(x,y):
P  t   



 
f  x, y    x cos  y sin   t  dxdy
 Total set of projections called
sinogram
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 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
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Single-Slice Rebinning Method
for Helical Cone-Beam CT
3-D Tomography
 Construct 3-D image using sequence of
cross 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 convert
cone-beam data into stack
of 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 conebeam 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
fan-beam projection length
fan sino
pz  , u 
u 2  D2
u v  D
2
2
2
g  u, v  CB sino
cone-beam projection length
u 2  D2
v
z
RD
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
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