Tomographic Image Reconstruction

1
Tomographic Image Reconstruction

• Miljenko Markovic

2
Overview

• Image creation
• Image reconstruction
• Brute force
• Iterative techniques
• Backprojection
• Filtered backprojection

3
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

4
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

5
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

6
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)

7
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

8
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

9
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

10
Fourier Reconstruction
11
Filtered Backprojection

• Take projections - sinogram
• Transform data to the frequency domain
• Filter data
• Inverse transform smoothed sinogram
• Backproject

12
Filtered Backprojection
13
Filtered Backprojection

• ramp filter nearest neighbor algorithm
• ramp Hamming filter nearest neighbor
  algorithm
• ramp filter linear interpolation
• ramp Hamming filter linear interpolation

1
2
4
3
14
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) 6372
• HARISH P. HlRlYANNAlAH, X-ray Computed
  Tomography for Medical Imaging, IEEE SIGNAL
  PROCESSING MAGAZINE

15
Thank you