1. GPU-based Image Processing Methods in Higher Dimensions and their Application to Tomography Reconstruction Szirmay-Kalos, László Budapest Uni of Tech Sapporo, 2010

2. Positron Emission Tomography e-e- e+e+ Line Of Response: y Intensity: x

3. Iterative Maximum Likelihood Reconstruction Measured detector response Source intensity as a 3D voxel array Source estimation Source correction Compute expected detector response Expected detector response

4. Ill-posed reconstruction error Iteration number Maximum likelihood estimate

5. Regularization Additional information –Penalty term added to the likelihood Prevents overfitting TV norm (L1 optimization) –No smoothness condition –Preserves edges

6. TV minimalization In steepest descent search the derivative of the TV term is needed: –Function |x| cannot be differentiated: Add a small term (blurring) Primal-dual methods –Only local values are needed: parallelization xVxV

7. Detector scattering compensation Path probability inside the detector can be pre-computed or measured photon crystals intercrystal scattering absorption Electronics number of hits

8. Pre-computation 

9. L w L = Quasi-Monte Carlo filtering

10. Random sampling undersampling oversampling

11. Delta-Sigma modulator Filter kernel pixels

12. Filter kernel Delta-Sigma modulator

13. Filter kernel Floyd-Steinberg halftoning!

14. Sampling with Sigma-Delta modulation

15. GPU Implementation Simulation step: GPU: Quasi-SIMD massively parallel machine –Gathering = threads to equations (outputs) –“No” conditional statements or variable length loops Reconstruction algorithm –Geometric LOR marching: threads to LORs (adjoint problem) –LOR filtering: threads to output LORs –TV regularization: threads to voxels high dim. integrals 10 8 voxels 10 8 LORs

16. TV regularization results =0.005 =0.05 =0.008 No TV

17. TV results =0.001 =0.0005 =0.0001 =0.005

18. Scattering in the detector 3D reconstruction, no detector scattering compensation Detector scattering compensation 2D reconstruction: SSRB + OSEM

19. F18 mouse

20. Conclusions Image processing algorithms can be and are worth being generalized to higher dimensions, but beware the curse of dimensions and use Monte Carlo methods. GPUs are good platforms for image processing, but adopt the gathering view and refrain from conditionals.