Image Reconstruction from Projections

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Image Reconstruction from Projections

• Antti Tuomas Jalava
• Jaime Garrido Ceca

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Overview

• Reconstruction methods
• Fourier slice theorem Fourier method
• Backprojection
• Filtered backprojection
• Algebraic reconstruction
• Diffractive tomography
• Display of CT images
• Tissue characterization with CT

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Projection Geometry

• Problem Reconstructing 2D Image.
• Given parallel-ray projections.
• 1D projection (Radon transform).
• Density distribution
• Ray AB
• Integral evaluated for different values of the
  ray offset t1.
• 1D projection or Radon transform.

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The Fourier Slice Theorem

• 1D Fourier Transform of 1D projection of 2D image
• is equal to the radial section (slice or
  profile) of the 2D Fourier Transform of the 2D
  image at the angle of the projection.

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The Fourier Slice Theorem

• How to obtain f(x,y) applying Fourier Slice
  Theorem
• Assumption we have projections available at all
  angles from 0º to 180º.
• From projections, we take their 1D Fourier
  transform.
• Fill the 2D Fourier Space with the corresponding
  radial sections.
• Take an inverse 2D Fourier transform to obtain
  
• Problem finite number of projections available
• Solution Interpolation is needed in 2D Fourier
  space.

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Backprojection

• Simplest reconstruction procedure
• Assumptions
• Rays Ideal straight lines.
• Image dimensionless points.
• Procedure
• Estimate of the density at a point by simply
  summing (integrating ) all the rays that pass
  through it at various angles.

• Problem
• Finite number of rays per projection
• Finite number of projections
• Interpolation is required.

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Backprojection

• BP produces a spoke-line pattern blurring
  details.
• Finite number of projections produces streaking
  artifacts.
• Reconstructed image modeled by convolution
  between PSF (impulse response) and the original
  image.
• Solution
• Applying deconvolution filters to the
  reconstructed image.
• Filtered BP technique.

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Point density function
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Filtered Backprojection

• After some manipulations, we get
• where
• In practice, smoothing window should be applied
  to reduce the amplification of high-frequency
  noise.

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Discrete Filtered Backprojection

• Projection in frequency domain is
  manipulated

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Discrete Filtered Backprojection

• The filtered projection may then be
  obtained as

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Discrete Filtered Backprojection

• Finally, we get this expression
• Algorithmic
• Measure projection.
• Compute filtered projection.
• Backproject the filtered.
• Repeat 1-3 all projection angles

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Filtered back projection 10
Original
Filtered back projection 1
Back projection 1
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Algebraic Reconstruction Techniques

• Projections seen as set of simultaneous
  equations.
• Kaczmarz method
• Iterative method.
• Implemented easily.
• Assumptions
• Discrete pixels.
• Image density is constant within each cell.
• Equations

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Algebraic Reconstruction Techniques

• Karzmarz method take the approach of successively
  and iteratively projecting an initial guess and
  its successors from one hyperplane to the next.
• In general, the mth estimate is obtained from the
  (m-1)th estimate as
• Because the image is updated by altering the
  pixels along each individual ray sum, the index
  of the updated estimate or of the iteration is
  equal to the index of the latest ray sum used.

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Algebraic Reconstruction Techniques

• Characteristics worth
• ART proceed ray by ray and it is iterative
• Small angles between hyperplanes
• Large number or iterations
• It should be reduced by using optimized
  ray-access schemes.
• MgtN noisy measurements oscillate in the
  neighborhood of the intersections of the
  hyperplanes.
• MltN under-determined.
• Any a priori information about image is easily
  introduced into the iterative procedure.

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Approximations to the Kaczmarz method

• We could rewrite reconstruction step at the nth
  pixel level as
• Corrections could also be multiplicative

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Approximations to the Kaczmarz method

• Generic ART procedure
• Prepare an initial estimate
• Compute ray sum
• Obtain difference between true ray sum and the
  computed ray sum and apply the correction.
• Perform Steps 2 and 3 for all rays available.
• Repeat Steps 2-4 as many times as required.

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Original
1. 178 angles dt 1 voxel width
3.
2.
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4.
5.

• Original again

6.
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Imaging with Diffraction Sources

• Non ionizing radiation
• Ultrasonic
• Electromagnetic (optical or thermal)
• Refraction and diffraction
• Fourier diffraction theorem

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Imaging with Diffraction Sources

• When an object, f(x,y), is illuminated with a
  plane wave the Fourier transform of the forward
  scattered fields measured on line TT gives the
  values of the 2-D transform, F(w1,w2), of the
  object along a circular arc in the frequency
  domain, as shown in the right half of the figure.

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Display of CT Images

• measured attenuation coefficient.
• attenuation coefficient of water
• When K 1000 units are called Hounsfield Units
• Air -1000 HU
• Water 0 HU
• Bone 1000 HU
• Study
• 86 healthy infants aged 0-5 years
• White matter 15 HU to 22 HU
• Gray matter 23 HU to 30 HU
• Difference between grey and white matter exactly
  8 HU (In all measurements)
• Boris P, Bundgaard F, Olsen A. Childs Nerv Syst.
  19873(3)175-7

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Microtomography

• µ-scale CT
• Volume few
• Nanotomography already introduced.
• Biomedical use
• Both dead and alive (in-vivo) rat and mouse
  scanning.
• Human skin samples, small tumors, mice bone for
  osteoporosis research.

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Estimation of Tissue Components with CT

• Manual segmentation of tumor by radiologist
• Parametric model for the tissue composition
• Gaussian mixture model
• Method to estimate the parameters of the model
• EM algorithm

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Gaussian Mixture Model (i)

• Fit M gaussian kernels to intensity histogram

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Gaussian Mixture Model (ii)

• Intensity value for voxel is a Gaussian random
  variable.
• Parameters for ith tissue
• Probability that voxel belonging to that tissue
  gets value x
• M number of different tissues in tumor
• the fraction of belonging to ith tissue
  (probability).
• Tumor as whole PDF is a mixture of M Gaussians

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Gaussian Mixture Model (iii)

• Tumor as whole PDF is a mixture of M Gaussians
• Probability of parameter set
• If nothing is known about
• Find that maximizes likelihood

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Gaussian Mixture Model (iv)

• Probability that jth voxel with value
  belongs to the ith tissue type
• EM algorithm (iterative, chapter 8) -gt

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Ending Remarks

• Some image manipulation tasks can be performed in
  1D in radon domain (edge detection etc.).
• Reconstruction heavily dependent on
  reconstruction algorithm (method).
• MRI images are usually reconstructed with Fourier
  method (according to book).
• CT allows fast 3D imaging
• So does MRI. MRI has better sensitivity
  especially with soft tissues.