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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. Projection

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Presentation Transcript
  • Image creation
  • Image reconstruction
    • Brute force
    • Iterative techniques
    • Backprojection
    • Filtered backprojection
image creation
Image Creation
  • Tomogram
    • image of a slice taken through a 3D volume
  • Projection
    • Attenuation profile through the object
    • The projection function represents the summation of the attenuation coefficients along a given X-ray path
image creation4
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
Image Reconstruction
  • Process of estimating an image from a set of projections
  • Several algorithms exist to accomplish this task:
    • Brute force
    • Iterative techniques
    • Backprojection
    • Filtered backprojection
brute force
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
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
  • 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
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
filtered backprojection
Filtered Backprojection
  • Take projections - sinogram
  • Transform data to the frequency domain
  • Filter data
  • Inverse transform – smoothed sinogram
  • Backproject
filtered backprojection13
Filtered Backprojection
  • ramp filter + nearest neighbor algorithm
  • ramp & Hamming filter + nearest neighbor algorithm
  • ramp filter + linear interpolation
  • ramp & Hamming filter + linear interpolation





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