1. 2 Unknown 3 4 5 6 7 8 9 10 Backprojection usually produce a blurred version of the image.

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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

𝑝𝜃 (𝑟 )=− ln ¿

Unknown𝑝𝜃 (𝑟 )

[𝑟𝑠]=[ cos𝜃 sin𝜃−sin 𝜃 cos𝜃 ][ 𝑥𝑦 ]

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What’s Backprojection ?

Example : 2 projections

(projecting)

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What’s Backprojection ?

Example : 2 projections

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What’s Backprojection ?

Example : 2 projections

(backprojecting)

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What’s Backprojection ?

Example : 2 projections

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What’s Backprojection ?

Example : 2 projections

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What’s 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:

𝑏(𝑥 𝑖 , 𝑦 𝑗)=𝐵 ¿

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Reminder : Central Slice Theorem

¿ } 1D-FT{}

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Remainder : Central Slice Theorem

2D-FT(I) 1D-FT(Radon(I))

0°

10°

90°

120°

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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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Let’s do some calculus

𝑓 (𝑥 , 𝑦)=∬𝐹 (𝑘𝑥 ,𝑘𝑦)𝑒+2 𝜋 𝑗𝑘𝑥 𝑥𝑒+ 2𝜋 𝑗 𝑘𝑦 𝑦𝑑𝑘𝑥𝑑𝑘𝑦

2D Inverse Fourier Transform

Function we want to reconstruct

Let’s change F from cartesian coordinates to polar coordinates

¿ ¿

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From Cartesian to Polar

{𝑘𝑥=𝑘cos𝜃𝑘𝑦=𝑘 sin𝜃

¿{ 𝑘=√𝑘𝑥

2+𝑘𝑦2

𝜃=tan− 1(𝑘𝑦

𝑘𝑥)

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With

Form half lines to full lines :

𝑓 (𝑥 , 𝑦)=∫0

𝜋

∫−∞

∞

𝐹 (𝑘 ,𝜃)∙|𝑘|∙𝑒𝑖2𝜋 𝑘𝑟𝑑𝑘𝑑𝜃

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Now the Central Slice Theorem become simply :

=P

And therefore:

𝑓 (𝑥 , 𝑦)=∫0

𝜋

∫∞

∞

𝑃 (𝑘 ,𝜃) ∙|𝑘|∙𝑒𝑖2 𝜋𝑘𝑟 𝑑𝑘𝑑𝜃

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Note that is a filter in the K-space. Let’s define the filtered projection in K-space :

𝑝∗ (𝑟 ,𝜃 )≜∫−∞

∞

𝑃∗ (𝑘 , 𝜃 )𝑒𝑖 2𝜋𝑘𝑟 𝑑𝑘

)

And its 1D inverse Fourier transform from k to r.

In the Radon domain it’s a convolution over r :

)

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𝑓 (𝑥 , 𝑦)=∫0

𝜋

∫∞

∞

𝑃 (𝑘 ,𝜃) ∙|𝑘|∙𝑒𝑖2 𝜋𝑘𝑟 𝑑𝑘𝑑𝜃

𝑓 (𝑥 , 𝑦)=∫0

𝜋

∫∞

∞

𝑃∗(𝑘 ,𝜃)𝑒𝑖 2𝜋𝑘𝑟 𝑑𝑘𝑑𝜃

𝑓 (𝑥 , 𝑦)=∫0

𝜋

𝑝∗ (𝑟 , 𝜃 )𝑑𝜃

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Filtered Backprojection

𝑓 (𝑥 , 𝑦)=∫0

𝜋

𝑝∗ (𝑟 , 𝜃 )𝑑𝜃

Note that it’s a backprojection ! 𝑓 (𝑥 , 𝑦 )=B {𝑝∗ (𝑟 ,𝜃 ) }=B {𝑝 (𝑟 , 𝜃 )∗𝑞 (𝑟 )}

This is called Filtered Backprojection

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FBP : Ramp Filter (Ram-Lak)

In Frequency domain

|𝑘|

Fundamentals of Medical ImagingPaul Suentes

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FBP : Ramp Filter (Ram-Lak) In space domain :

𝑞 (𝑟 )=𝑘𝑚𝑎𝑥

2

4𝜋 2 (𝑠𝑖𝑛𝑐 (𝑘𝑚𝑎𝑥 ∙𝑟 )− 12𝑠𝑖𝑛𝑐2(𝑘𝑚𝑎𝑥 ∙𝑟

2 )) A sample at discrete value of gives this simple filter :

𝑞 (𝑛)={14𝑛=0

−1𝑛2𝜋 2 𝑛𝑖𝑠𝑜𝑑𝑑

0𝑛𝑖𝑠𝑒𝑣𝑒𝑛

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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)

𝑞 (𝑛)={14𝑛=0

−1𝑛2𝜋 2 𝑛𝑖𝑠𝑜𝑑𝑑

0𝑛𝑖𝑠𝑒𝑣𝑒𝑛

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, Hann…)(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

Σ

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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 … 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

Next week in Introduction to Medical imaging :

Magnetic Resonance Image Reconstruction