7/28/2019 Analytic Reconstruction Algorithm

1/26

7/28/2019 Analytic Reconstruction Algorithm

2/26

Iterative ReconstructionAlgorithm

7/28/2019 Analytic Reconstruction Algorithm

3/26

Analytic ReconstructionMethods

FBP Quick

Inaccuracy in emission tomography

7/28/2019 Analytic Reconstruction Algorithm

4/26

Iterative ReconstructionMethods

Allow for a rich description of theblurring and attenuation mechanisms inthe imaging process

Iterative, meaning that the estimated

image is progressively refine in arepetitive calculation

7/28/2019 Analytic Reconstruction Algorithm

5/26

FBP vs. Iterative Techniques

Efficiency vs. accuracy

Noise texture and image detail can looksignificantly different

7/28/2019 Analytic Reconstruction Algorithm

6/26

Projection

7/28/2019 Analytic Reconstruction Algorithm

7/26

Sinogram

7/28/2019 Analytic Reconstruction Algorithm

8/26

Central Slice Theorem

7/28/2019 Analytic Reconstruction Algorithm

9/26

Backprojection

7/28/2019 Analytic Reconstruction Algorithm

10/26

Inverse Distance Weighting ofDirectBackprojection

7/28/2019 Analytic Reconstruction Algorithm

11/26

Filtered Backprojection

7/28/2019 Analytic Reconstruction Algorithm

12/26

Iterative ReconstructionMethods

Two main components of any iterativemethods:

The criterion for selecting the best imagesolution

The algorithmfor finding that solution

7/28/2019 Analytic Reconstruction Algorithm

13/26

Linear Model of theImaging Process

7/28/2019 Analytic Reconstruction Algorithm

14/26

Projection Process

7/28/2019 Analytic Reconstruction Algorithm

15/26

Iterative ReconstructionAlgorithm

7/28/2019 Analytic Reconstruction Algorithm

16/26

Image Reconstruction Criteria

Maximum-Likelihood Criterion

The probability law p(g;f) for the observation g is

determined by some unknown deterministicparameter vector f

Choose the reconstructed image f to be theobject function f for which the measured datawould have had the greatest likelihood p(g;f).

7/28/2019 Analytic Reconstruction Algorithm

17/26

The Maximum-Likelihood

Expectation-MaximizationAlgorithm

7/28/2019 Analytic Reconstruction Algorithm

18/26

Properties of ML-EM

Asymptotically unbiased: as the numberof observation becomes large, theestimates become unbiased.

Asymptotically efficient: for large data

records, they yield the minimumvariance

7/28/2019 Analytic Reconstruction Algorithm

19/26

Properties of ML-EM

To reduced variance, by introducingspatial smoothing in the images

Low pass filtering

Prematurely stopping an ML algorithmbefore it actually reaches the ML solution

7/28/2019 Analytic Reconstruction Algorithm

20/26

Shortcoming of ML methods

The convergence of the algorithm isslow. A usable solution may require 30-50 iterations.

ML criterion yields very noisy

reconstructed images

7/28/2019 Analytic Reconstruction Algorithm

21/26

Images Reconstructed byML-EM

7/28/2019 Analytic Reconstruction Algorithm

22/26

Ordered-Subsets EM

7/28/2019 Analytic Reconstruction Algorithm

23/26

Properties of OS-EM OS-EM at n iteration reaches rough the same

point of convergence as ML-EM at (numberof subsets) x n iterations

Generally requires fewer than 7 iterations

As ML-EM, low spatial frequencies convergefirst, with higher spatial frequencies improvingwith father iterations

7/28/2019 Analytic Reconstruction Algorithm

24/26

Properties of OS-EM The principle cost of using the subset method

is an increase in image noise for the samelevel of bias as compared to ML-EM.

Users should be wary of using a largenumber of subsets; modest acceleration of8-

10 times is possible with very little increase innoise.

7/28/2019 Analytic Reconstruction Algorithm

25/26

Images Reconstructed byOS-EM

7/28/2019 Analytic Reconstruction Algorithm

26/26

Thanks for your attention.