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

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

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

Outline X-ray attenuation as line integral Construction of the sinogram

X-ray intensity attenuates inside matter, here shown with a homogeneous block https://www.youtube.com/watch?v=IfXo2S1xXCQ

Formula for X-ray attenuation along a line inside homogeneous matter An X-ray with intensity I 0 enters a homogeneous physical body. I 0 I 1 ✲ • � �� � s The intensity I 1 of the X-ray when it exits the material is I 1 = I 0 e − µ s , where s is the length of the path of the X-ray inside the body and µ > 0 is X-ray attenuation coefficient.

X-ray intensity attenuates inside matter, here shown with two homogeneous blocks https://www.youtube.com/watch?v=Z_IBFQcn0l8

A digital X-ray detector counts how many photons arrive at each pixel photon count 1000 ✲ • 1000 X-ray source Detector

Adding material between the source and detector reveals the exponential X-ray attenuation law photon count 1000 ✲ • 1000 1000 ✲ 500 • 1000 ✲ 250 •

We take logarithm of the photon counts to compensate for the exponential attenuation law photon count log 1000 ✲ • 1000 6.9 1000 ✲ 500 6.2 • 1000 ✲ 250 5.5 •

Final calibration step is to subtract the logarithms from the empty space value (here 6.9) photon line count log integral 1000 ✲ • 1000 6.9 0.0 1000 ✲ 500 6.2 • 0.7 1000 ✲ 250 5.5 • 1.4

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: � b a f ( x ) dx . I 1 = I 0 e − We can also write it in the form � b − log( I 1 / I 0 ) = f ( x ) dx , a where I 0 is known from calibration and I 1 from measurement.

FIPS Computational Blog

Outline X-ray attenuation as line integral Construction of the sinogram

Construction of the sinogram Angle of X-rays: 3.0 degrees

Construction of the sinogram Angle of X-rays: 12.2 degrees

Construction of the sinogram Angle of X-rays: 21.5 degrees

Construction of the sinogram Angle of X-rays: 30.7 degrees

Construction of the sinogram Angle of X-rays: 39.9 degrees

Construction of the sinogram Angle of X-rays: 49.2 degrees

Construction of the sinogram Angle of X-rays: 58.4 degrees

Construction of the sinogram Angle of X-rays: 67.6 degrees

Construction of the sinogram Angle of X-rays: 76.8 degrees

Construction of the sinogram Angle of X-rays: 86.1 degrees

Construction of the sinogram Angle of X-rays: 95.3 degrees

Construction of the sinogram Angle of X-rays: 104.5 degrees

Construction of the sinogram Angle of X-rays: 113.8 degrees

Construction of the sinogram Angle of X-rays: 123.0 degrees

Construction of the sinogram Angle of X-rays: 132.2 degrees

Construction of the sinogram Angle of X-rays: 141.5 degrees

Construction of the sinogram Angle of X-rays: 150.7 degrees

Construction of the sinogram Angle of X-rays: 159.9 degrees

Construction of the sinogram Angle of X-rays: 169.2 degrees

Construction of the sinogram Angle of X-rays: 178.4 degrees

Construction of the sinogram Angle of X-rays: 187.6 degrees

Construction of the sinogram Angle of X-rays: 196.8 degrees

Construction of the sinogram Angle of X-rays: 206.1 degrees

Construction of the sinogram Angle of X-rays: 215.3 degrees

Construction of the sinogram Angle of X-rays: 224.5 degrees

Construction of the sinogram Angle of X-rays: 233.8 degrees

Construction of the sinogram Angle of X-rays: 243.0 degrees

Construction of the sinogram Angle of X-rays: 252.2 degrees

Construction of the sinogram Angle of X-rays: 261.5 degrees

Construction of the sinogram Angle of X-rays: 270.7 degrees

Construction of the sinogram Angle of X-rays: 279.9 degrees

Construction of the sinogram Angle of X-rays: 289.2 degrees

Construction of the sinogram Angle of X-rays: 298.4 degrees

Construction of the sinogram Angle of X-rays: 307.6 degrees

Construction of the sinogram Angle of X-rays: 316.8 degrees

Construction of the sinogram Angle of X-rays: 326.1 degrees

Construction of the sinogram Angle of X-rays: 335.3 degrees

Construction of the sinogram Angle of X-rays: 344.5 degrees

Construction of the sinogram Angle of X-rays: 353.8 degrees

We have object and data for the inverse problem A ❅ � f ∈ R 32 × 32 Af ∈ R 49 × 39

Illustration of the ill-posedness of tomography A ❅ � Difference 0.02672 A ❅ �

Illustration of the ill-posedness of tomography A ❅ � Difference 0.00899 A ❅ �

Illustration of the ill-posedness of tomography A ❅ � Difference 0.00254 A ❅ �

Illustration of the ill-posedness of tomography A ❅ � Difference 0.00124 A ❅ �

Illustration of the ill-posedness of tomography A ❅ � Difference 0.00004 A ❅ �