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1
Reconstruction Technique
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Parallel Projection
prxII
eII
x
x
x
)/ln(
sincos yxl
Projection in Cartesian and polar system:
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Projection in Cartesian and polar system:
dssfpr
y
x
s
dsyxfprline
),()(
cossin
sincos
:coordinate s)(t, in the
),()(),(
5
Algebraic (Iterative) Reconstruction Technique (ART)
A B
C D
13
16
10
1214
5
6
Algebraic (Iterative) Reconstruction Technique (ART)
1.5=4/6
We have originally output from detector:
We start with the first prediction:
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Algebraic (Iterative) Reconstruction Technique (ART)
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ART Algorithm
)(),(),(
)(),(),(
)(N/)(Pr)(Pre
1
1
ppp
ppp
p
eyxfyxf
eyxfyxf
For each actual projection prθ(l), predict pixel value and obtain predicted prθp(l) value
)(N/)(Pr)(Pre p
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ART Algorithm
pryxwN
pr
yxw
yxf
yxfyxwprp
pp
each along ),( of sum theis )(
line projection
each of width e within thpixel of
area toingcorrespond weight a is ),(
valuepixel predicted is ),(
),(),()(
:Where
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Fourier Slice Theorem:1D Fourier transform of a parallel projection is equal to a slice of 2D FT of the original object
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Fourier Slice Theorem:
2)()( jeprS
FT of
2D FT of object:
dxdyeyxfvuF vyuxj )(2),(),(
)(pr
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Fourier Slice Theorem:
1D FT of object along line v=0 or =0
)()()0,( 02
0 uSdxexpruF uxj
dydxeyxfuF uxj
lpr
2
)(0
),()0,(
y
x
s
l
cos
sin
sin
cos
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FT of in (l , s) coordinate system:
)sin,cos(
),(
),()(
)sincos(2
2
wwF
dxdyeyxf
dledsslfwS
yxwj
wlj
)(pr
y
x
s
l
cos
sin
sin
cos
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Summing 1D FT of projections of object at a number of angles gives an estimate of 2D FT of the object
(projections are inserted along radial lines)
y
x
s
l
cos
sin
sin
cos
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2π|ω|/K (K=180)ω = √(u2=v2)
Deconvolution filter
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Algorithm for FT reconstruction:
1) For each angle θ between 0 to 180 for all (l)
2) Measure projection prθ
3) Fourier transform it to find Sθ
4) Multiply it by weighting function 2π|w|/K (K=180)(ie. filtering (weighting) each FT data lines to estimate a pie-shaped wedge from line)
5- Inserting data from all projections into 2D FT place
6- Doing interpolation in frequency domain to fill the gap in high frequency regions.
5) Inverse FT 6) Interpolation data if necessary
y
x
s
l
cos
sin
sin
cos
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Continuous and discrete version of the Shepp and Logan filter to reduce the emphasis given by HF components
