introduction Analytical methods Iterative methods

2. Image Reconstruction
in Nuclear Medicine
Seyedeh Shokoofeh Mousavi Gazafroudi
Seyedeh Shabnam Mousavi Gazafroudi
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3. Outline
 Introduction & Definition
 Image reconstruction’s method
– Analytical methods
– Iterative methods
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4. INTRODUCTION
 Images of the inside of the human body can be obtained non
invasively using tomographic acquisition and processing
technique
 these techniques are used to obtain images of gamma emitter
distribution after its administration
 reconstructed images are obtained given a set of their
projections using rotating gamma camera
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5. Definition
 Ray: a single transmission measurement through the patient made
by a single detector at a given moment in time
 Projection: a series of rays that pass through the patient at the
same orientation
 Sinogram : a type of display that shows the data acquired for one
slice before reconstruction
 Line of response: a line in space that determined by computer of
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7. Procedure
 the purpose is to acquire a large number of transmission
measurements through the patient at different position
 Prior to the reconstruction, the LOR data is often rebind into a
sinogram . This intermediate data representation corresponds to
projection data that can be used by conventional tomographic
reconstruction codes
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8. RECONSTRUCTION
METHODS
Analytical
Back projection
Filtered back
projection
Gridding
Iterative
Algebraic
ART, MART, SMART
Statistical
Weighted
CG, CD, ISRA
Likelihood
EM, OSEM, MLEM,
SAGE
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9. Reconstruction algorithm
G = A . F
Measures Model Image
Analytical reconstructions (e.g. FBP) Iterative reconstructions (e.g. MLEM or OSEM)
F(0) G’ = A . F(k) Compare G’ with G
F(k+1) = F(k) x (or +) G
A -1.G = F
Measures ImageModel -1
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10. Analytical methods
 Are based on a continuous modelling and the reconstruction process
consists of the inversion of measurement equations
 Image is “Radon transform” of radioactive distribution in the body
 The most frequently used is the filtered back-projection (FBP) algorithm
 Are efficient and elegant but they are unable to handle complicated factors
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11. Analytical algorithms properties
 very fast
 Direct inversion of the projection formula
 Corrections for physical factors are difficult
 It needs a lot of filtering trade-off between blurring and noise
 Quantitative imaging is difficult
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20. Iterative methods
 The principle is to find a solution to reconstruct an image of a
tomographic slice from projection by successive estimates
 Are more versatile but less efficient
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21. Iterative algorithms properties
 Amplification of noise
 long calculation time
 discreetness of data included in the model
 it is easy to model and handle projection noise & imaging physics
 quantitative imaging is possible
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22. Iterative Algorithm steps
(1) Make the first arbitrary estimate of the slice (homogeneous image F(0))
(2) Project the estimated slice into its projections (G’)
(3) Compare the projections of the estimate with measured projections
(4) Obtain the correction factors
(4) Stop or continue:
if the correction factors are approaching zero,
if they do not change in subsequent iterations,
if the maximum number of iterations was achieved, then finish; otherwise
(5) Apply corrections to the estimate thus make the new estimate of the slice
(6) Go to step (2)
F(0) G’ = A . F(k)
Compare G’ with G
F(k+1) = F(k) x (or +) G
G = A . F
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23. Iterative reconstruction steps
Current
estimate Measured
projection
Compare
(e.g. – or / )
Error
projection
projection
Estimated
projection
image space projection space
backprojection
Error
image
Update
Iteration 1
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24. Iterative steps
Current
estimate
Measured
projection
Compare
(e.g. - or / )
Error
projection
projection Estimated
projection
image space projection space
backprojectionError
image
Update
Estimated
projection
Estimated
projection
Iteration 2
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25. Iterative Algorithm
Current
estimate
Measured
projection
Compare
(e.g. - or / )
New
projection
projection
image space projection space
backprojection
new
image
Estimated
projection
Estimated
projection
Estimated
projection
Estimated
projection
Iteration N
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27. 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
MMMNMM
NN
NN
pfwfwfw
pfwfwfw
pfwfwfw







2211
22222121
11212111
Contribution factor of
the nth image element
to the mth ray sum.
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28. 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.
M>N noisy measurements oscillate in the neighborhood of
the intersections of the hyperplanes.
M<N under-determined.
Any a priori information about image is easily introduced into
the iterative procedure.
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29. ART
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36. MART
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37. Iterative methods
 Iterative statistical methods are based on iterative correction algorithms .
Iterative methods are relatively easy to model:
 the reconstruction starts using an initial estimate of the image (generally a
constant image),
 projection data is computed from this image,
 the estimated projections are compared with the measured projections,
 corrections are made to correct the estimated image, and
 the algorithm iterates until convergence of the estimated and measured
projection sets.
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38. Iterative Method
Statistical
 Statistical Nature of data
 Signal is due to emitted photons not amount of activity in organs
 For a uniform and constant amount of activity, there is not uniform
and constant number of emission
 Number of photons emitted from each voxel and in a constant time
follows Poisson distribution
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39. Maximum Likelihood reconstruction
 ML-EM and its derivative, OS-EM are gold standard reconstruction techniques in nuclear
medicine.
 They suffer from factors that limit the quantitative analysis of their results, hence limit their
exploitation during the treatment planning process and during treatment monitoring.
 ML-EM usually requires approximately 20-50 iterations to reach an acceptable solution.
Considering that ML-EM requires one forward projection and one back projection at each
iteration, this overall processing time is considerably more than the filtered back projection
approach, but leads to a potentially more accurate reconstruction.
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40. Ordered Subsets Expectation Maximization
 OSEM was introduced in 1994 [27] to reduce reconstruction time of
conventional ML-EM
 This slight modification of ML-EM uses subsets of the entire data set for
each image update
 There are many approaches for dividing the projection space into subsets.
Most approaches use non-overlapping subsets
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41. Conjugate Gradient
 In CG algorithms, the correction is additive and may generate negative
values in the reconstructed image.
 Each new estimate is found by adding a quantity to the current estimate
 the conjugate gradient algorithm applied to g=Af converges in a number of
iterations equal to the number m of equations of the system
 because of rounding errors in the determination of the conjugate
directions, the solution is not reached in m iterations.
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42. Filtered back-projection
 Very fast
 Direct inversion of projection
formula
 No Corrections for scatter, non-uniform attenuation and other
physical factors
 filtering - trade-off between blurring and noise
 Quantitative imaging difficult
Iterative reconstruction
 Discreteness of data included in the model
 Easy to model and handle projection noise
 Easy to model the imaging physics
 Quantitative imaging possible
 Long calculation time
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43. THANKS FOR YOUR ATTENTION
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