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

▶
tomography workshop
SMART_READER_LITE
LIVE PREVIEW

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

Jul 13, 2023 331 likes •942 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

slide-1
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

slide-2
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

slide-3
SLIDE 3

Outline

X-ray attenuation as line integral Construction of the sinogram

slide-4
SLIDE 4
slide-5
SLIDE 5
slide-6
SLIDE 6

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

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

slide-7
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.

slide-8
SLIDE 8

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

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

slide-9
SLIDE 9

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

X-ray source

1000

photon count

1000

  • ✲

Detector

slide-10
SLIDE 10

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

1000 1000 1000

photon count

1000 500 250

  • ✲
  • ✲
  • ✲
slide-11
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

  • ✲
  • ✲
  • ✲
slide-12
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

slide-13
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.

slide-14
SLIDE 14

FIPS Computational Blog

slide-15
SLIDE 15

Outline

X-ray attenuation as line integral Construction of the sinogram

slide-16
SLIDE 16

Construction of the sinogram

Angle of X-rays: 3.0 degrees

slide-17
SLIDE 17

Construction of the sinogram

Angle of X-rays: 12.2 degrees

slide-18
SLIDE 18

Construction of the sinogram

Angle of X-rays: 21.5 degrees

slide-19
SLIDE 19

Construction of the sinogram

Angle of X-rays: 30.7 degrees

slide-20
SLIDE 20

Construction of the sinogram

Angle of X-rays: 39.9 degrees

slide-21
SLIDE 21

Construction of the sinogram

Angle of X-rays: 49.2 degrees

slide-22
SLIDE 22

Construction of the sinogram

Angle of X-rays: 58.4 degrees

slide-23
SLIDE 23

Construction of the sinogram

Angle of X-rays: 67.6 degrees

slide-24
SLIDE 24

Construction of the sinogram

Angle of X-rays: 76.8 degrees

slide-25
SLIDE 25

Construction of the sinogram

Angle of X-rays: 86.1 degrees

slide-26
SLIDE 26

Construction of the sinogram

Angle of X-rays: 95.3 degrees

slide-27
SLIDE 27

Construction of the sinogram

Angle of X-rays: 104.5 degrees

slide-28
SLIDE 28

Construction of the sinogram

Angle of X-rays: 113.8 degrees

slide-29
SLIDE 29

Construction of the sinogram

Angle of X-rays: 123.0 degrees

slide-30
SLIDE 30

Construction of the sinogram

Angle of X-rays: 132.2 degrees

slide-31
SLIDE 31

Construction of the sinogram

Angle of X-rays: 141.5 degrees

slide-32
SLIDE 32

Construction of the sinogram

Angle of X-rays: 150.7 degrees

slide-33
SLIDE 33

Construction of the sinogram

Angle of X-rays: 159.9 degrees

slide-34
SLIDE 34

Construction of the sinogram

Angle of X-rays: 169.2 degrees

slide-35
SLIDE 35

Construction of the sinogram

Angle of X-rays: 178.4 degrees

slide-36
SLIDE 36

Construction of the sinogram

Angle of X-rays: 187.6 degrees

slide-37
SLIDE 37

Construction of the sinogram

Angle of X-rays: 196.8 degrees

slide-38
SLIDE 38

Construction of the sinogram

Angle of X-rays: 206.1 degrees

slide-39
SLIDE 39

Construction of the sinogram

Angle of X-rays: 215.3 degrees

slide-40
SLIDE 40

Construction of the sinogram

Angle of X-rays: 224.5 degrees

slide-41
SLIDE 41

Construction of the sinogram

Angle of X-rays: 233.8 degrees

slide-42
SLIDE 42

Construction of the sinogram

Angle of X-rays: 243.0 degrees

slide-43
SLIDE 43

Construction of the sinogram

Angle of X-rays: 252.2 degrees

slide-44
SLIDE 44

Construction of the sinogram

Angle of X-rays: 261.5 degrees

slide-45
SLIDE 45

Construction of the sinogram

Angle of X-rays: 270.7 degrees

slide-46
SLIDE 46

Construction of the sinogram

Angle of X-rays: 279.9 degrees

slide-47
SLIDE 47

Construction of the sinogram

Angle of X-rays: 289.2 degrees

slide-48
SLIDE 48

Construction of the sinogram

Angle of X-rays: 298.4 degrees

slide-49
SLIDE 49

Construction of the sinogram

Angle of X-rays: 307.6 degrees

slide-50
SLIDE 50

Construction of the sinogram

Angle of X-rays: 316.8 degrees

slide-51
SLIDE 51

Construction of the sinogram

Angle of X-rays: 326.1 degrees

slide-52
SLIDE 52

Construction of the sinogram

Angle of X-rays: 335.3 degrees

slide-53
SLIDE 53

Construction of the sinogram

Angle of X-rays: 344.5 degrees

slide-54
SLIDE 54

Construction of the sinogram

Angle of X-rays: 353.8 degrees

slide-55
SLIDE 55

We have object and data for the inverse problem

❅

  • A

f ∈ R32×32 Af ∈ R49×39

slide-56
SLIDE 56

Illustration of the ill-posedness of tomography

Difference 0.02672

❅

  • ❅
  • A

A

slide-57
SLIDE 57

Illustration of the ill-posedness of tomography

Difference 0.00899

❅

  • ❅
  • A

A

slide-58
SLIDE 58

Illustration of the ill-posedness of tomography

Difference 0.00254

❅

  • ❅
  • A

A

slide-59
SLIDE 59

Illustration of the ill-posedness of tomography

Difference 0.00124

❅

  • ❅
  • A

A

slide-60
SLIDE 60

Illustration of the ill-posedness of tomography

Difference 0.00004

❅

  • ❅
  • A

A

Core Hubs:
All Converters PDF Hub Video Hub Audio Hub Image Hub File Compressor