BMME 560 & BME 590I Medical Imaging: X-ray, CT, and Nuclear Methods Tomography Part 1
Today Exam 1 Tomography –Introduction –3D imaging methods –The Forward Problem –Radon Transform
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
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
Tomography 3D object 2D projection Obtaining 3D information about an object when we can only observe 2D projections
Tomography To locate a point object in 3D, we only need two views.
Tomography To reconstruct a distributed object in 3D, we need a lot of views.
Tomography What is wrong with 2D imaging? 1. 2.
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
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.
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
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.
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
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
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
Tomography The idea of longitudinal tomography has been reborn as “tomosynthesis” This is an example of “limited-angle” tomography
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”
Tomographic Reconstruction The problem f(x,y) p(t, ) t t y x s Given p(t, ) for 0< < Find f(x,y)
Tomographic Reconstruction Where do the projection data come from? –In X-ray, recall the imaging equation X-rays Detector I0I0 Subject x,y,z
Transmission Tomography X-rays Detector I0I0 Subject x,y,z The line integral of attenuation in this projection direction
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
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
Tomographic Reconstruction Mathematically, we do not care where the projections come from f(x,y) p(t, ) t t y x s
Tomographic Reconstruction Note the following relationships f(x,y) p(t, ) t t y x s
Tomographic Reconstruction Rewrite into a general equation using a delta function f(x,y) p(t, ) t t y x s
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
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