Transcript Document

Tomographic Image
Reconstruction
Miljenko Markovic
Overview
• Image creation
• Image reconstruction
– Brute force
– Iterative techniques
– Backprojection
– Filtered backprojection
Image Creation
• Tomogram
– image of a slice taken through a 3D volume
detector
x-ray source
collimator
object
• Projection
– Attenuation profile through the object
– The projection function represents the summation
of the attenuation coefficients along a given X-ray
path
Image Creation
• Sinogram
– 2D data set – result of
stacking all the
projections together
• Radon transform
– Transformation of a
function (image) into
the sinogram, p(r)
– Computes projections
of an image along
specified directions
Image Reconstruction
• Process of estimating an image from a
set of projections
• Several algorithms exist to accomplish
this task:
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–
Brute force
Iterative techniques
Backprojection
Filtered backprojection
Brute Force
• projection set defines a system of
simultaneous linear equations - can be
solved using algorithms from linear
algebra
• not practical for real systems (can have
hundreds of simultaneous equations for a
single slice)
Iterative Reconstruction
• Known as algebraic reconstruction
technique – ART, consists of three steps:
– Make an initial guess at the solution
– Compute projections based on the guess
– Refine the guess based on the weighted
difference between the actual projections and
the desired projections
– Original reconstruction method used in
medical imaging
– Works, but is slow and susceptible to noise
Backprojection
• Propagates sinogram back into the image
space along the projection paths (inverse
Radon transform)
• Backprojection image is a blurred version of
the original image
• The projection theorem (central slice
theorem) - provides an answer to inverse
Radon transform problem
– Set of 1D Fourier transform of the Radon
transform of a function is the 2D Fourier transform
of that function
Fourier Reconstruction
• Calculate the 1D Fourier transform of all
projections [p(r) = P(k)]
• Place P(k) on polar grid to get P(k,)
• Resample in Cartesian space to get
F(kx,ky)
• Calculate the 2D inverse Fourier
transform of F(kx,ky) to get f(x,y) – image
• Resultant image is noisy
Fourier Reconstruction
Filtered Backprojection
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Take projections - sinogram
Transform data to the frequency domain
Filter data
Inverse transform – smoothed sinogram
Backproject
Filtered Backprojection
Filtered Backprojection
1. ramp filter + nearest
neighbor algorithm
2. ramp & Hamming filter +
nearest neighbor algorithm
3. ramp filter + linear
interpolation
4. ramp & Hamming filter +
linear interpolation
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References
• Image Processing: The Core of Nuclear
Cardiology, Scott M. Leonard, MS, CNMT,
Northwestern University, ppt presentation
• Xiang Li , Jun Ni and Ge Wang, Parallel
iterative cone beam CT image reconstruction
on a PC cluster, Journal of X-Ray Science
and Technology 13 (2005) 63–72
• HARISH P. HlRlYANNAlAH, X-ray Computed
Tomography for Medical Imaging, IEEE
SIGNAL PROCESSING MAGAZINE
Thank you