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

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

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

Ill-posed reconstruction error Iteration number Maximum likelihood estimate

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

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

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

Pre-computation 

L w L = Quasi-Monte Carlo filtering

Random sampling undersampling oversampling

Delta-Sigma modulator Filter kernel pixels

Filter kernel Delta-Sigma modulator

Filter kernel Floyd-Steinberg halftoning!

Sampling with Sigma-Delta modulation

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

TV regularization results =0.005 =0.05 =0.008 No TV

TV results =0.001 = = =0.005

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

F18 mouse

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.