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Presented by Scott Lichtor Introduction to Tomography

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Overview Problem Statement Tomographic Applications The Mathematics Necessary Math Fourier Slice Theorem Filtered Backprojection Matlab Example

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Problem Can’t see inside of people to diagnose problems Can’t see inside of machinery to diagnose problems How do take a picture of a place where you can’t fit a camera?

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Solution Tomography Reconstructs a function using line integrals Goal: recover the interior structure of a body using exterior measurements Routine for medicine, earth sciences Image taken from

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Tomography Applications Single photon emission computed tomography (SPECT) is used for gamma imaging Gamma-emitting radio-isotope is injected into the body Gamma camera returns a 2-D image of the object Reconstruction then returns a 3-D image of the object Used for medical imaging (tumor imaging, functional brain imaging) Image taken from

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Tomography Applications Positron emission tomography (PET) acquires data from electron- positron annihilation Positron-emitting tracer is injected into the body System detects gamma rays produced by tracer Uses PET to reconstruct 3-D image Used for oncology, neurology, cardiology, etc. Image taken from

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Tomography Applications Computed tomography (CT) is used for X-ray imaging X-rays are produced and sent through the body Record the line integrals Calculate the shape of the imaged object Used extensively for medical imaging Also used for non-destructive materials testing Image taken from

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Tomography I’ll focus on X-ray tomography Get interior structure of body by X-raying the object from many different directions When an X-ray goes through an object, it is attenuated by the object Very dense objects will weaken the strength of the ray considerable Less dense objects will affect the strength of the ray less

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History of Computed Tomography Alessandro 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

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Necessary Mathematics Line integrals are integrals along a line Coordinate system: (x,y)->( Ѳ,t) Ѳ : Angle, t: distance along source Fourier Transform: F(w) = ∫f(t)e -j2 π wt dt F(u,v)=∫∫f(x,y)e -j2 π (ux+vy) dxdy Image taken from

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Tomography A 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)ds P Ѳ (t) = ∫∫f(x,y) δ (x cos ( Ѳ )+y sin ( Ѳ )-t)dxdy Radon transform

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Fourier Slice Theorem Object function is f Fourier transform of f is F Projection P Fourier transform of P is S F(u,v)=∫∫f(x,y)e -j2 π (ux+vy) dxdy S Ѳ (w) = ∫P Ѳ (t)e -j2 π wt dt To demonstrate the Fourier Slice Theorem, let Ѳ =0

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Fourier Slice Theorem Suppose v=0 F(u,0) = ∫∫f(x,y)e -j2 π ux dxdy = ∫(∫f(x,y)dy)e -j2 π ux dx P Ѳ =0 (x) = ∫ f(x,y)dy So F(u,0) = ∫ P Ѳ =0 (x) e -j2 π ux dx There’s a relationship between the projection data and the object image Specifically, each projection gives a slice of the Fourier transform of the overall image Image taken from

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Filtered Backprojection Filtered backprojection is the algorithm used to reconstruct the object image Idea: use the projection data to get slices of the Fourier transform of the object image. Then, calculate the object image Image taken from

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Filtered Backprojection Procedure: 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)

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Example Matlab illustration Typical image to reconstruct:

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Example Create projection data Use the radon function The radon function applies the radon transform to an image

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Example 18 projections

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Example 36 projections

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Example 90 projections

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Example The imradon function reconstructs images from projection data

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Example Reconstruction with 36 projections

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Example Reconstruction with 36 projections

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Example Reconstruction with 90 projections

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Sources An Introduction to X-ray tomography and Radon Transforms, by Eric Todd Quinto Principles of Computerized Tomographic Imaging, by Avinash C. Kak and Malcolm Slaney Wikipedia

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