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