Tomography workshop Samuli Siltanen Department of Mathematics and - - PowerPoint PPT Presentation

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Tomography workshop Samuli Siltanen Department of Mathematics and - - PowerPoint PPT Presentation

Jul 13, 2023 331 likes •941 views

Tomography workshop Samuli Siltanen Department of Mathematics and Statistics University of Helsinki, Finland samuli.siltanen@helsinki.fi www.siltanen-research.net Summer school University of Helsinki Kumpula Campus, June 1012, 2019 Lotus

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SLIDE 1

Tomography workshop

Samuli Siltanen

Department of Mathematics and Statistics University of Helsinki, Finland samuli.siltanen@helsinki.fi www.siltanen-research.net

Summer school University of Helsinki Kumpula Campus, June 10–12, 2019

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SLIDE 2

Lotus root tomography

YouTube search: “lotus tomography” www.youtube.com/watch?v=eWwD_EZuzBI&t=7s Video: thanks to Tatiana Bubba, Andreas Hauptmann and Juho Rimpeläinen

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SLIDE 3

Outline

X-ray attenuation as line integral Construction of the sinogram

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SLIDE 4
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SLIDE 5
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SLIDE 6

X-ray intensity attenuates inside matter, here shown with a homogeneous block

https://www.youtube.com/watch?v=IfXo2S1xXCQ

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SLIDE 7

Formula for X-ray attenuation along a line inside homogeneous matter

An X-ray with intensity I0 enters a homogeneous physical body.

I0 I1

  • ✲
  • s

The intensity I1 of the X-ray when it exits the material is

I1 = I0e−µs,

where s is the length of the path of the X-ray inside the body and µ > 0 is X-ray attenuation coefficient.

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SLIDE 8

X-ray intensity attenuates inside matter, here shown with two homogeneous blocks

https://www.youtube.com/watch?v=Z_IBFQcn0l8

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SLIDE 9

A digital X-ray detector counts how many photons arrive at each pixel

X-ray source

1000

photon count

1000

  • ✲

Detector

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SLIDE 10

Adding material between the source and detector reveals the exponential X-ray attenuation law

1000 1000 1000

photon count

1000 500 250

  • ✲
  • ✲
  • ✲
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SLIDE 11

We take logarithm of the photon counts to compensate for the exponential attenuation law

log

6.9 6.2 5.5

1000 1000 1000

photon count

1000 500 250

  • ✲
  • ✲
  • ✲
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SLIDE 12

Final calibration step is to subtract the logarithms from the empty space value (here 6.9)

log

6.9 6.2 5.5

1000 1000 1000

photon count

1000 500 250

  • ✲
  • ✲
  • ✲

line integral

0.0 0.7 1.4

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SLIDE 13

Formula for X-ray attenuation along a line: Beer-Lambert law

Let f : [a, b] → R be a nonnegative function modelling X-ray attenuation along a line inside a physical body. Beer-Lambert law connects the initial and final intensities: I1 = I0e−

b

a f (x)dx.

We can also write it in the form

− log(I1/I0) = b

a

f (x)dx,

where I0 is known from calibration and I1 from measurement.

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SLIDE 14

FIPS Computational Blog

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SLIDE 15

Outline

X-ray attenuation as line integral Construction of the sinogram

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SLIDE 16

Construction of the sinogram

Angle of X-rays: 3.0 degrees

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SLIDE 17

Construction of the sinogram

Angle of X-rays: 12.2 degrees

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SLIDE 18

Construction of the sinogram

Angle of X-rays: 21.5 degrees

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SLIDE 19

Construction of the sinogram

Angle of X-rays: 30.7 degrees

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SLIDE 20

Construction of the sinogram

Angle of X-rays: 39.9 degrees

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SLIDE 21

Construction of the sinogram

Angle of X-rays: 49.2 degrees

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SLIDE 22

Construction of the sinogram

Angle of X-rays: 58.4 degrees

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SLIDE 23

Construction of the sinogram

Angle of X-rays: 67.6 degrees

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SLIDE 24

Construction of the sinogram

Angle of X-rays: 76.8 degrees

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SLIDE 25

Construction of the sinogram

Angle of X-rays: 86.1 degrees

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SLIDE 26

Construction of the sinogram

Angle of X-rays: 95.3 degrees

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SLIDE 27

Construction of the sinogram

Angle of X-rays: 104.5 degrees

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SLIDE 28

Construction of the sinogram

Angle of X-rays: 113.8 degrees

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SLIDE 29

Construction of the sinogram

Angle of X-rays: 123.0 degrees

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SLIDE 30

Construction of the sinogram

Angle of X-rays: 132.2 degrees

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SLIDE 31

Construction of the sinogram

Angle of X-rays: 141.5 degrees

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SLIDE 32

Construction of the sinogram

Angle of X-rays: 150.7 degrees

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SLIDE 33

Construction of the sinogram

Angle of X-rays: 159.9 degrees

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SLIDE 34

Construction of the sinogram

Angle of X-rays: 169.2 degrees

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SLIDE 35

Construction of the sinogram

Angle of X-rays: 178.4 degrees

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SLIDE 36

Construction of the sinogram

Angle of X-rays: 187.6 degrees

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SLIDE 37

Construction of the sinogram

Angle of X-rays: 196.8 degrees

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SLIDE 38

Construction of the sinogram

Angle of X-rays: 206.1 degrees

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SLIDE 39

Construction of the sinogram

Angle of X-rays: 215.3 degrees

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SLIDE 40

Construction of the sinogram

Angle of X-rays: 224.5 degrees

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SLIDE 41

Construction of the sinogram

Angle of X-rays: 233.8 degrees

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SLIDE 42

Construction of the sinogram

Angle of X-rays: 243.0 degrees

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SLIDE 43

Construction of the sinogram

Angle of X-rays: 252.2 degrees

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SLIDE 44

Construction of the sinogram

Angle of X-rays: 261.5 degrees

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SLIDE 45

Construction of the sinogram

Angle of X-rays: 270.7 degrees

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SLIDE 46

Construction of the sinogram

Angle of X-rays: 279.9 degrees

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SLIDE 47

Construction of the sinogram

Angle of X-rays: 289.2 degrees

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SLIDE 48

Construction of the sinogram

Angle of X-rays: 298.4 degrees

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SLIDE 49

Construction of the sinogram

Angle of X-rays: 307.6 degrees

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SLIDE 50

Construction of the sinogram

Angle of X-rays: 316.8 degrees

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SLIDE 51

Construction of the sinogram

Angle of X-rays: 326.1 degrees

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SLIDE 52

Construction of the sinogram

Angle of X-rays: 335.3 degrees

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SLIDE 53

Construction of the sinogram

Angle of X-rays: 344.5 degrees

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SLIDE 54

Construction of the sinogram

Angle of X-rays: 353.8 degrees

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SLIDE 55

We have object and data for the inverse problem

❅

  • A

f ∈ R32×32 Af ∈ R49×39

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SLIDE 56

Illustration of the ill-posedness of tomography

Difference 0.02672

❅

  • ❅
  • A

A

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SLIDE 57

Illustration of the ill-posedness of tomography

Difference 0.00899

❅

  • ❅
  • A

A

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SLIDE 58

Illustration of the ill-posedness of tomography

Difference 0.00254

❅

  • ❅
  • A

A

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SLIDE 59

Illustration of the ill-posedness of tomography

Difference 0.00124

❅

  • ❅
  • A

A

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SLIDE 60

Illustration of the ill-posedness of tomography

Difference 0.00004

❅

  • ❅
  • A

A