Image Reconstruction

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Image Reconstruction

• T-61.182, Biomedical Image Analysis
• Seminar Presentation 7.4.2005
• Seppo Mattila Mika Pollari

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Overview

• Reconstruction from projections (general)
• projection geometry and radon transform
• Reconstruction methods
• Backprojection, (Fourier slice theorem), Filtered
  Backprojection, and Algebraic Reconstruction
  Technique
• Reconstruction examples
• MRI reconstruction
• examples

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Introduction

• Only photography (reflection) and planar x-ray
  (attenuation) measure spatial properties of the
  imaged object directly.
• Otherwise, measured parameters are some how
  related to spatial properties of imaged object.
• US (amplitude and time), CT, SPECT and PET
  (integral projections of parallel rays), MRI
  (amplitude, frequency and phase of the NMR
  signal) etc...
• Construct the object (image) which creates the
  measured parameters.

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Reconstruction From the Projections

• Projection is a line integral along the path
• CT measure the projections of passed photons,
  with different angles.
• MRI measure projection of NMR signal with
  different magnetic gradients (projection based
  MRI not used anymore).
• Assumption no notable diffraction.

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Projection Geometry and Radon Transformation

• Co-ordinate transformation
• Radon transformation

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Backprojection (BP)

• Simplest reconstruction method Integrate all
  possible rays that pass through the same point.
• Cause smearing and blurring. Method has nowadays
  only historical importance.

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Backprojection Graphically
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FBP ramp vs. smooth filter
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FST and FBP

• Starting from FST we end up to FBP without any
  approximations or assumptions

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Remarks of FBP

• From the previous equations its clear that the
  image is backprojection of filtered signal (
  ) and (w) is the ramp filter.
• FBP advantages
• Each projection may be filtered and backprojected
  while further projections are collected (on-line
  processing).
• No need for 2D inverse Fourier transformation.

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Algebraic Reconstruction Techniques (ART)

• Each object entity (image pixel/voxel) has
  physical property (grey-level value) such as
  attenuation coefficient
• All pixel in the rays path contribute to sum an
  amount which equals pixels area along the path
  (weight) times pixels physical property
  (grey-level value)
• We end up set of simultaneous equations

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ART Model

• Each ray sum
• Set of simultaneous equations

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Kaczmarz Method Solution to ART

• Each equation spans a hyperplane in n-dimensional
  space. If unique solution exist it is in
  intersection of the hyperplanes
• Solution is found iteratively by solving each ray
  equation at the time

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Reconstruction Examples Number of Projection
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Reconstruction Examples Projection Angle
BP
FBP (Butter)
ART
PSF
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Reconstruction Examples BP, FBP, and ART
Phantom
BP
FBP (Butter)
ART
Reconstruction of phantom with different
reconstruction methods using 90 projections from
interval 0-180 degree
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Display of CT Images Using Hounsfield Units

• Attenuation coefficients are normalized with
  respect of water
• Now mean and SD of of different tissues are known
  advanced (measured in HU)

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MRI Reconstruction
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MRI Signal
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Reconstruction Examples
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Effects of Sampling the K-space
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Summary I

• Reconstruct 2D CT image from 1D projections
• Backprojection (BP)
• - Only historical importance
• Filtered backprojection (FBP)
• - Most widely used technique
• - Large number of projections over 0-180
  required
• Algebraic Reconstruction Techniques (ART)
• - Better handling of sparse and non-uniform
  projections
• - Slow compared to FBP

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Summary II

• Reconstruct 2D MR image from measured current
  (1D)
• Spatial (x,y) info encoded in frequency and
  phase
• Collect data to spatial frequency domain
  (k-space)
• Reconstruction by inverse 2D FT
• non-invasive
• imaging sequences (T1, T2, fMRi etc.) lots
  of possibilities
• - image artifacts (distortions, ghosts, etc.)
• - more expensive