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BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Tomography Part 1
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Today Exam 1 Tomography –Introduction –3D imaging methods –The Forward Problem –Radon Transform
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Tomography From Greek –Tomos = slice –Graph = picture Tomography is a technique for finding a 3D distribution from 2D projections –Computed tomography (CT) –Computer-assisted tomography (CAT) –Optical coherence tomography (OCT) – not really tomography
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Tomography Applications of tomography –X-ray CT –Nuclear medicine (SPECT and PET) –Diffuse optical tomography (DOT) –MRI (in the old days) –Ultrasound tomography –Impedance tomography
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Tomography 3D object 2D projection Obtaining 3D information about an object when we can only observe 2D projections
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Tomography To locate a point object in 3D, we only need two views.
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Tomography To reconstruct a distributed object in 3D, we need a lot of views.
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Tomography What is wrong with 2D imaging? 1. 2.
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The Earlier Example The intensity detected at a pixel is related to the sum of attenuation along the path. X-rays Detector 10 cm 3 cm =.01 cm -1 =.1 cm -1
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Tomography Overlap of anatomy can mask important features in certain views –Consider a cylinder with two spheres – one more dense than the background; one less dense.
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Tomography To solve the contrast pileup problem: –Subtract the background –Digital subtraction angiography (DSA) Image the subject Inject contrast agent Image again Subtract the two images to obtain a view of the contrast agent only Link: DSA Helps contrast pileup; still no 3D information
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Tomography 3D imaging –In projection radiography, 3D locations are inferred by the observer with knowledge of anatomical structures –Some 3D can be inferred by observer with use of stereo-view, multi-view, or rotational planar imaging.
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Tomography (Classical, Longitudinal, Motion) Tomography –Use an X-ray plate –Move X-ray system and plate longitudinally –One plane ends up in focus; everything else is out of focus
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Tomography (Classical, Longitudinal, Motion) Tomography –Use an X-ray plate –Move X-ray system and plate longitudinally –One plane ends up in focus; everything else is out of focus
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Tomography (Classical, Longitudinal, Motion) Tomography –Use an X-ray plate –Move X-ray system and plate longitudinally –One plane ends up in focus; everything else is out of focus
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Tomography The idea of longitudinal tomography has been reborn as “tomosynthesis” This is an example of “limited-angle” tomography
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Tomographic Reconstruction Define terms f(x,y) p(t, ) t t y x s The object The projection “Object space” or “image space” “Projection space” The “rotated frame”
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Tomographic Reconstruction The problem f(x,y) p(t, ) t t y x s Given p(t, ) for 0< < Find f(x,y)
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Tomographic Reconstruction Where do the projection data come from? –In X-ray, recall the imaging equation X-rays Detector I0I0 Subject x,y,z
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Transmission Tomography X-rays Detector I0I0 Subject x,y,z The line integral of attenuation in this projection direction
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Emission Tomography Detector Subject emits gamma photons f x,y,z The line integral of photon emissions in this projection direction Detector admits only photons traveling in parallel directions
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Other sources of projections Transmission –Ultrasound – transmit acoustic waves through subject –Optical – shining near-IR sources through subject Emission –MRI – setting up gradient field along one direction gives a sum of signals (like a projection) along that axis
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Tomographic Reconstruction Mathematically, we do not care where the projections come from f(x,y) p(t, ) t t y x s
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Tomographic Reconstruction Note the following relationships f(x,y) p(t, ) t t y x s
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Tomographic Reconstruction Rewrite into a general equation using a delta function f(x,y) p(t, ) t t y x s
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The Radon Transform Specifies the 1D projection of the 2D function f(x,y) at any angle Nonzero only along the line of projection at t
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Radon Transform Example Both delta functions have to be nonzero for the integral to be one. This occurs when the following conditions are met: Therefore, the transform is nonzero only when
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