Massively Parallel GPU-friendly Algorithms for PET. Szirmay-Kalos László, Budapest, University of Technology and Economics

Transcription

1 Massively Parallel GPU-friendly Algorithms for PET Szirmay-Kalos László, Budapest, University of Technology and Economics

2 (GP)GPU: CUDA (OpenCL) Multiprocessor N Multiprocessor 2 Multiprocessor 1 Scalar Processor 1 Shared memory Scalar Processor M Cache Device memory Instruction Execution Unit Massively parallel: #threads > 10 4 Independence: synchronization and write collisions should be avoided SIMD: conditional statements are not welcome Coalesced memory access

3 PET physics P e + P N e - Line Of Response (LOR)

4 Mediso nanoscan PET AnyScan PET

5 Maximum-likelihood reconstruction A 13 A 11 x 1 x 2 A 12 x 3 x 4 A 14 Expected number of hits: ~ y A x Maximize the probability of the measurement data y: ~ y 1 A 13 x 3 A 11 x 1 A 14 x A 4 12 x 2 x arg max Pr{ y ~ y ( x)}

6 Iterative solution Activity in voxels Physics Simulation Measured values Expected values Correction

7 Computational challenges Numbers of LORs and voxels: hundred millions! System matrix A: elements (PetaBytes) Probability that a positron of a voxel is detected by a LOR Patient dependent Not sparse if accurate simulation is needed Do not store, estimate on-the-fly Matrix elements are high dimensional integrals Monte Carlo quadrature Reuse of computation High performance (parallel) computational platform Minimize the effect of estimation error

8 Numerical integration f: integrand g: target density 0 1 samples

9 Quadrature error f/g g f importance sampling stratification

10 Effect of stratification undersampling oversampling

11 Direct physical simulation Input driven, scattering type algorithm Thread = photon Photons scatter different number of times The same detector is hit: write collision Random memory access Cannot mimic the detectors source detectors

12 GPU friendly approach Output driven, gathering type Thread = importon SIMD: grouping importons No write collision: LOR-driven Cannot mimic the source source detectors

13 Multiple Importance Sampling f(x) g b (x) g a (x) f(x i ) g a (x i ) f(x i ) g b (x i ) f(x i ) g a (x i )+g b (x i )

14 Direct gamma photon contribution Detector module 1 Detector module 2 LOR 5D Integration Accuracy for given sample number Cost of a sample l voxel x( v )ddv

15 Output- or LOR-driven sampling w Detector module 1 l l Detector module 2 LOR u Pros: Gathering Thread coherence Texture coherence Uniform on detectors Low-cost samples due to reuse Cons: Cannot mimick activity g Nray ( l, ) 2 D G( u, w) l 1

16 Input or voxel-driven sampling Detector module 1 Detector module 2 Pros: Can mimick activity l LOR Cons: Write collisions Less coherence u g 2 ( l, ) N voxel x( l ) l D cos( ) u xdv 2

17 Multiple Importance Sampling voxel voxel + l w u G D N l g ), ( ), ( 2 ray 1 ), ( ), ( ), ( 2 1 l g l g l g v x D u l l x N l g d ) cos( ) ( ), ( 2 voxel 2

18 Multiple Importance Sampling

19 Input-driven scattered photon transport Monte Carlo simulation: Free path Absorption? Scattering direction source voxel fetches absorption scattering

20 Free Path Sampling u s For special cross section functions (t), it can be solved analytically.

21 Ray marching u 1 exp( s) i i Complexity grows with the resolution Slow in high resolution low density media

22 Mix virtual particles to obtain a density that can be solved analytically photon Virtual collision Real collision Real material particle Virtual particle effective resolution 64 billion sample points

23 Output-driven single-scattered photon transport with reuse s 2 s 2 s 2 z 2 s 1 z 1 1. Scattering points 2. Ray marching between scattering points and detectors s 1 z 1 s 1 3. Combination

24 Multiple Importance Sampling 0 scattering 0 scattering 0, 1, 2, scattering 1 scattering LOR-driven unscattered contribution Voxel-driven unscattered contribution Photon transport LOR-driven single scatter MIS

25 Detector response photon absorption scattering Electronics hits Problem: The domain is 4 dimensional.

26 Quasi-Monte Carlo filtering L L w L X x) w( x) dx = ( L( X x( t)) dt 1 0

27 without with Detector Scattering Compensation

28 Back projection Measured values voxels Physics Simulation y Correction Expected values y~ Monte Carlo simulation or simplification ~ y ( n) A x ( n) Random or deterministic estimation x T ( n 1) ~ x ( n) y A ( y T A 1 n) Ratio of measured and computed values

29 2D study Measured sinogram

30 Backprojection with unbiased forward projection Contribution to x V outlier y / ~ y L L x T ( n 1) ~ x ( n) y / yˆ L L y A ( y T A 1 n) Ratio of measured and computed values ŷ L y~l Samples ŷ L

31 Reduce bias and outliers Averaging iteration: ~ ( n) n yl (1 n) ~ ( 1) yl Metropolis iteration: Ignore outliers randomly Acceptance with probability yˆ n L a min{ˆ y L L / ~ y n / (n) L,1} n

32 Recons res 1.3 mm voxels res 0.1 mm voxels

33 Conclusions GPU is an effective tool for computing tens of thousands of parallel threads having no conditionals and collisions. The problem must be interpreted and solved to keep this requirement in mind. Randomization (Monte Carlo) can help structure the problem in this way.