From filtered back-projection to statistical tomography reconstruction

idtools.com.au
From Filtered Back-Projection to
statistical tomography reconstruction
Daniel Pelliccia
Instruments
& Data Tools

© 2018 Instruments & Data Tools idtools.com.au
Overview
Filtered-Back Projection
● Analytical method based on the Radon
Transform
Algebraic methods
● Solving a linear system of equations
iteratively
Statistical methods
● Incorporating the knowledge of the
particle statistics in the reconstruction
This tutorial is an overview of the philosophy
behind different types of reconstruction
algorithms for tomography, their strengths and
pitfalls when it comes to. neutron tomography

Tomography reconstruction methods
Reconstruction methods
Analytical Iterative
Algebraic StatisticalFBP
…
ART
SART
SIRT
MART
…
Max Likelihood
Least Squares

Filtered Back-Projection (FBP)
Parallel Beam
Projection
1. From intensity to attenuation
2. Radon Transfom (Projection)
3. To reconstruct the slice we run
an Inverse Radon Transform

Parallel Beam
Projection
1. Radon Transfom (Projection: the
phyisical system does that for us)
2. To reconstruct the slice we run
an Inverse Radon Transform

sinogramslice
Radon Transform
Inverse Radon Transform
projection
back-projection

Pros and Cons of FBP
• Computationally very efficient
• Implemented in countless
software packages
• It requires a ‘large’ number of projections
(compared to the number of pixels)
• It requires equally spaced projections
• Doesn’t handle the noise well
• It doesn’t handle other deviations from
the ideal case

1. The slice to be reconstructed is modelled
as a discrete array of pixels (with
unknown value) from the outset
2. The set of projections through the object
is written as a linear system of equations.
Algebraic Reconstruction Technique (ART)
“ray sums”

Algebraic Reconstruction Technique (ART)
… is solving the Sudoku!
ray sum = 45

Recall the Filtered Back-Projection
It approximates the projection as
being taken on a line

ART accounts for the width of the beam (detector pixel)
1. The beam width is determined by the
detector pixel size, or resolution element
2. The beam does not traverse all pixels
equally, so each pixel is weighted by the
corresponding cross section
3. The linear system of equations (the ray
sum) is generalised to
1 2
N

Why do we need an iterative method?
Tomography reconstruction is modelled as a linear system of equations
or in matrix form

The number of equations scales quadratically
with the linear number of pixels

Why do we need an iterative method?
Tomography reconstruction is modelled as a linear system of equations
or in matrix form
It is generally impossible, or unreasonable, to look for a solution in closed form since:
• The number of equation is prohibitively large ( megapixel images)
• Randomness introduced by noise  The linear system may not have a solution, or the
solution may not be unique
For these reasons iterative solvers are the best approaches.

Iterative reconstruction process
adapted from
M. Beister et al, Physica
Medica 28, 94 (2012)
forward
projection
back
projection
initial
estimate
Calculated
projections
Measured
projections
compare
stop?
Difference
new
estimate
Final
image
Input Iterative loop Output
Y
N

Iterative reconstruction process (more technical)
The basic projection operation is formally the inner product between
any row i of the system matrix W and the (unknown) attenuation vector
The attenuation vector values are iteratively refined using the
so called Kaczmarz method
Normalisation
factor
Next
iterate
Previous
iterate
Relaxation
factor Difference
between
calculated and
measured
projections

Pros and Cons of algebraic reconstruction techniques
• The reconstruction problem is
discretised from the outset
• ‘Reasonable’ results are
obtained with a lower number
of projections compared with
FBP.
• It handles noise better
• Computationally very
demanding
• It may require some trial and
error to set the parameters to
ensure convergence

An example: comparison between FBP and SART
on neutron tomography data taken at DINGO
FBP: 720 projections ART: 720 projections
2 iterations

An example: comparison between FBP and SART
on neutron tomography data taken at DINGO
FBP: 72 projections ART: 72 projections
10 iterations

Statistical methods
The key idea of statistical methods is to incorporate
counting statistics of the detected photons into the
reconstruction process.
The detected intensity becomes a random variable . All
projections measurement are independent random variables
whose expectation value is
Expectation
value Mean number
of neutrons
withoutthe
sample
Detection
noise

What do we mean by ‘incorporate the counting statistics’?
1. Particle statistics of the incoming neutron beam
2. Statistics of detection noise
3. Weighing different ‘rays’ differently according to the
corresponding attenuation

Advantages of statistical reconstruction methods
for neutron tomography
1. Use the raw measurements rather than the logarithms, to avoid biases due to the
nonlinearity of the logarithm applied to the transmission data (valid in general)
2. Increase sample throughput by making better use of low counting statistics
3. Limit activation problems
4. Speed-up dynamic studies

Convex Algorithm Statistical Image Reconstruction
Difference between measured and
estimated projection
Normalisationfactor that weighs less low
countingvoxels
Next
iterate
Previous
iterate

Statistical reconstruction applied to DINGO data
• Maxwell-Boltzmann thermal spectrum peaked
at about 1.5 A
• Flux: 1.1  107 n/cm2/s
• Detector: Scintillation screen, mirror and
2048x2048 pixel CCD camera (Andor IKON-L)
• Detector: 53.25x53.25mm (26x26 μm)
• 720 projections: 9 hr tomographic scan (in
2014)
Image source: ansto.gov.au
U. Garbe et al., Physics Procedia 88, 13-18 (2017)
J. Brown, U. Garbe and D. Pelliccia, arXiv:1806.02741 (2018)

Statistical reconstruction applied to DINGO data /2
• Cylindrical sample Ti:S.S:Al

Statistical reconstruction applied to DINGO data /2

Take-home message on statistical image reconstruction
• SIR comparable performance to FBP with 12.5% projections
• Equivalent to reducing the scan times at DINGO down to the order of a ∼1-2 hrs

Acknowledgements
Dr Jeremy M. C. Brown
Department of Radiation Science and Technology,
Delft University of Technology, The Netherlands
Dr Ulf Garbe
Australian Centre for Neutron Scattering,
ANSTO

References
• T. M. Buzug, Computed Tomography, Springer-Verlag (2012)
• M. Beister, D. Kolditz, W. A. Kalender, Iterative reconstruction methods in X-ray CT, Physica
Medica 28, 94-108 (2012)
• J. A. Fessler, Statistical Image Reconstruction Methods for Transmission Tomography, book
chapter in SPIE Handbook of Medical Imaging 1, 1-70 (2000)
• J. Brown, U. Garbe and D. Pelliccia, Statistical Image Reconstruction for High-Throughput
Thermal Neutron Computed Tomography, arXiv:1806.02741 (2018)

From filtered back-projection to statistical tomography reconstruction

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