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Tutorial 3CT Image Reconstruction
Part II
Alexandre Kassel
Introduction to Medical Imaging
046831
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Tutorial Overview
Backprojection Filtered Backprojection Other Methods
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Recall : Projection
ππ (π )=β ln ΒΏ
Unknownππ (π )
[ππ ]=[ cosπ sinπβsin π cosπ ][ π₯π¦ ]
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Whatβs Backprojection ?
Example : 2 projections
(projecting)
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Whatβs Backprojection ?
Example : 2 projections
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Whatβs Backprojection ?
Example : 2 projections
(backprojecting)
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Whatβs Backprojection ?
Example : 2 projections
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Whatβs Backprojection ?
Example : 2 projections
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Whatβs Backprojection ?
Example : 2 projections
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Backprojection From 2 Projections
From 10 Projections
From 90 Projections :
From 4 Projections
Backprojection usually produce a blurred version of the image.
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BP: Numerical Example
3
1
3
1
3
0 5 3 3 0
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BP : Numerical Example
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
3
1
3
1
3
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BP: Numerical Example
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 5 3 3 0
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0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0 1 0.6 0.6 0
0.6 1.6 1.2 1.2 0.6
0.2 1.2 0.8 0.8 0.2
0.6 1.6 1.2 1.2 0.6
0.2 1.2 0.8 0.8 0.2
0.6 1.6 1.2 1.2 0.6
BP: Numerical Example
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BP : Mathematical DefinitionThe Backprojection is given by :
And the discrete version:
π(π₯ π , π¦ π)=π΅ ΒΏ
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Reminder : Central Slice Theorem
ΒΏ } 1D-FT{}
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Remainder : Central Slice Theorem
2D-FT(I) 1D-FT(Radon(I))
0Β°
10Β°
90Β°
120Β°
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Remainder : Direct Fourier Reconstruction
We discussed the problematic of interpolating into the Fourier Domain. Can we find a way to avoid doing this ?
Fundamentals of Medical ImagingPaul Suentes
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Letβs do some calculus
π (π₯ , π¦)=β¬πΉ (ππ₯ ,ππ¦)π+2 π πππ₯ π₯π+ 2π π ππ¦ π¦πππ₯πππ¦
2D Inverse Fourier Transform
Function we want to reconstruct
Letβs change F from cartesian coordinates to polar coordinates
ΒΏ ΒΏ
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From Cartesian to Polar
{ππ₯=πcosπππ¦=π sinπ
ΒΏ{ π=βππ₯
2+ππ¦2
π=tanβ 1(ππ¦
ππ₯)
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With
Form half lines to full lines :
π (π₯ , π¦)=β«0
π
β«ββ
β
πΉ (π ,π)β|π|βππ2π ππππππ
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Now the Central Slice Theorem become simply :
=P
And therefore:
π (π₯ , π¦)=β«0
π
β«β
β
π (π ,π) β|π|βππ2 πππ ππππ
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Note that is a filter in the K-space. Letβs define the filtered projection in K-space :
πβ (π ,π )ββ«ββ
β
πβ (π , π )ππ 2πππ ππ
)
And its 1D inverse Fourier transform from k to r.
In the Radon domain itβs a convolution over r :
)
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π (π₯ , π¦)=β«0
π
β«β
β
π (π ,π) β|π|βππ2 πππ ππππ
π (π₯ , π¦)=β«0
π
β«β
β
πβ(π ,π)ππ 2πππ ππππ
π (π₯ , π¦)=β«0
π
πβ (π , π )ππ
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Filtered Backprojection
π (π₯ , π¦)=β«0
π
πβ (π , π )ππ
Note that itβs a backprojection ! π (π₯ , π¦ )=B {πβ (π ,π ) }=B {π (π , π )βπ (π )}
This is called Filtered Backprojection
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FBP : Ramp Filter (Ram-Lak)
In Frequency domain
|π|
Fundamentals of Medical ImagingPaul Suentes
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FBP : Ramp Filter (Ram-Lak) In space domain :
π (π )=ππππ₯
2
4π 2 (π πππ (ππππ₯ βπ )β 12π πππ2(ππππ₯ βπ
2 )) A sample at discrete value of gives this simple filter :
π (π)={14π=0
β1π2π 2 πππ πππ
0πππ ππ£ππ
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FBP : Ramp Filter (Ram-Lak)
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25Ram-Lak filter in space domain
n
q(n)
π (π)={14π=0
β1π2π 2 πππ πππ
0πππ ππ£ππ
Discrete filter in space domain :
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FBP : Ramp Filter (Ram-Lak)
-5 -4 -3 -2 -1 0 1 2 3 4 5-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25Ram-Lak filter in space domain
n
q(n)
The Ramp Filter is also called the Ram-Lak filter after Ramachandran and Lakshiminarayanan
Problem : High frequencies are unreliable because of noise and aliasing. And Ram-Lak filter enhances them.
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FBP : Smoothed window (Hamming, Hannβ¦)(in frequency domain)
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FBP: A two steps algorithm
Ram-Lak Filter
(or smoothed version of
it)
Projections Backproject
Reconstructed image
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Filtered backprojection : Results Examples(from 360 projections)
No filtered Ram-Lak
Ram-Lak Hamming
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A.R.TAlgebraic Reconstruction Technique
3
1
3
1
3
0 5 3 3 0
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A.R.T(Rectification by difference)
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
3
1
3
1
3
0 5 3 3 0
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A.R.T(Rectification By difference)
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
3
1
3
1
3
0 5 3 3 0
2.2 2.2 2.2 2.2 2.2
Ξ£
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A.R.T(Rectification By difference)
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
0.2 0.2 0.2 0.2 0.2
0.6 0.6 0.6 0.6 0.6
3
1
3
1
3
0 5 3 3 0
-2.2 2.8 0.8 0.8 -2.2 Rectification
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A.R.T(Rectification By difference)
0.16 1.16 0.76 0.76 0.16
-0.22
0.76 0.36 0.36-
0.22
0.16 1.16 0.76 0.76 0.16
-0.22
0.76 0.36 0.36-
0.22
0.16 1.16 0.76 0.76 0.16
3
1
3
1
3
0 5 3 3 0
-2.2 2.8 0.8 0.8 -2.2 Rectification
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And we continue until convergence β¦ We can prove A.R.T is converging. For an
unique solution we need N projections for a NxN matrix.
A.R.T is accurate but very slow. Some elaborate techniques were developed with improved efficiency.
Current CT devices are using FBP anyway.
A.R.TAlgebraic Reconstruction Technique
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Next week in Introduction to Medical imaging :
Magnetic Resonance Image Reconstruction