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)
ExampleCreate projection dataUse the radon functionThe radon function applies the radon
transform to an image