Presented by Scott Lichtor

Introduction to Tomography

OverviewProblem StatementTomographic ApplicationsThe Mathematics

Necessary MathFourier Slice TheoremFiltered Backprojection

Matlab Example

ProblemCan’t see inside of people to diagnose

problemsCan’t see inside of machinery to diagnose

problemsHow do take a picture of a place where you

can’t fit a camera?

SolutionTomography

Reconstructs a function using line integralsGoal: recover the interior structure of a body

using exterior measurementsRoutine for medicine, earth sciences

Image taken from http://media-2.web.britannica.com

Tomography ApplicationsSingle photon emission computed tomography

(SPECT) is used for gamma imaging Gamma-emitting radio-isotope is injected into

the bodyGamma camera returns a 2-D image of the

objectReconstruction then returns a 3-D image of the

objectUsed for medical imaging (tumor imaging,

functional brain imaging)

Image taken from http://www.biocompresearch.org

Tomography ApplicationsPositron emission tomography (PET) acquires

data from electron-positron annihilationPositron-emitting tracer is injected into the

bodySystem detects gamma rays produced by

tracerUses PET to reconstruct 3-D imageUsed for oncology, neurology, cardiology, etc.

Image taken from http://www.ibfm.cnr.it

Tomography ApplicationsComputed tomography (CT) is used for

X-ray imagingX-rays are produced and sent through

the bodyRecord the line integralsCalculate the shape of the imaged

objectUsed extensively for medical imagingAlso used for non-destructive materials

testing

Image taken from http://www.csmc.edu

TomographyI’ll focus on X-ray tomographyGet interior structure of body by X-raying

the object from many different directionsWhen an X-ray goes through an object, it is

attenuated by the objectVery dense objects will weaken the

strength of the ray considerableLess dense objects will affect the strength

of the ray less

History of Computed TomographyAlessandro Vallebona proposed

representing a slice of the body on radiographic film in the early 1900s

First commercially viable CT scanner invented by Sir Godfrey Hounsfield at EMI Laboratories in 1972

Originally, water tanks were needed for imaging on humans

Necessary MathematicsLine integrals are integrals along a lineCoordinate system: (x,y)->(Ѳ,t)Ѳ: Angle, t: distance along sourceFourier Transform: F(w) = ∫f(t)e-j2πwtdtF(u,v)=∫∫f(x,y)e-j2π(ux+vy)dxdy

Image taken from http://www.mindef.gov.sg

TomographyA projection is composed of a bunch of line

integrals Easiest example: line integrals with the

same Ѳ but different t’s (parallel line integrals).

The value of a line integral: P Ѳ(t) = ∫(Ѳ,t)line f(x,y)dsP Ѳ(t) = ∫∫f(x,y)δ(x cos (Ѳ)+y sin (Ѳ)-t)dxdyRadon transform

Fourier Slice TheoremObject function is fFourier transform of f is FProjection PFourier transform of P is SF(u,v)=∫∫f(x,y)e-j2π(ux+vy)dxdyS Ѳ (w) = ∫P Ѳ (t)e-j2πwtdtTo demonstrate the Fourier Slice Theorem,

let Ѳ=0

Fourier Slice TheoremSuppose v=0F(u,0) = ∫∫f(x,y)e-j2πuxdxdy = ∫(∫f(x,y)dy)e-j2πuxdx• P Ѳ=0(x) = ∫ f(x,y)dy

So F(u,0) = ∫ P Ѳ=0(x) e-j2πuxdxThere’s a relationship between the projection

data and the object imageSpecifically, each projection gives a slice of the

Fourier transform of the overall image

Image taken from http://www.eng.warwick.ac.uk

Filtered BackprojectionFiltered backprojection is the algorithm used to

reconstruct the object imageIdea: use the projection data to get slices of

the Fourier transform of the object image. Then, calculate the object image

Image taken from http://www.eng.warwick.ac.uk

Filtered BackprojectionProcedure:For all angles K

1. Get projections P Ѳ

2. Apply Fourier transform and get S

Ѳ (w)

3. Place the inverse Fourier transforms of the projections on the approximation of the original image

In this way an approximation of the original image can be obtained (this is only the algorithm for parallel projections)

ExampleMatlab illustrationTypical image to reconstruct:

ExampleCreate projection dataUse the radon functionThe radon function applies the radon

transform to an image

Example18 projections

Example36 projections

Example90 projections

ExampleThe imradon function reconstructs images

from projection data

ExampleReconstruction with 36 projections

ExampleReconstruction with 36 projections

ExampleReconstruction with 90 projections

SourcesAn Introduction to X-ray tomography and

Radon Transforms, by Eric Todd QuintoPrinciples of Computerized Tomographic

Imaging, by Avinash C. Kak and Malcolm Slaney

Wikipedia