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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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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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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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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Again, one wants to solve g Af, where g is the vector
of values in the sinogram, A is a given matrix, and f is the
unknown vector of pixel values in the image to be reconstructed.