GPU-based Image Processing Methods in Higher Dimensions and their Application

to Tomography Reconstruction  

Szirmay-Kalos, LászlóBudapest Uni of Tech

Sapporo, 2010

Positron Emission Tomography

e-

e+

Line Of Response: y

Intensity: x

Iterative Maximum Likelihood Reconstruction

Measureddetectorresponse

Source intensity as a 3D voxel array

Source estimation

Source correction

Compute expecteddetector response

Expecteddetector response

Ill-posed reconstructionerror

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

x

dttf )('

V

dvvx )(

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

V

dvvx )(

xV

Detector scattering compensation

Path probability inside the detector can be pre-computed or measured

photon

crystals

intercrystalscattering

absorption

Electronicsnumber of hits

Pre-computation

q

dxxwxXL )()( 1

0

))(( dttxXL

L

w

L

=

Quasi-Monte Carlo filtering

Random sampling

Random sampling

undersampling

oversampling

Delta-Sigma modulator

Filter kernel

pixels

Filter kernel

Delta-Sigma modulator

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

xAy high dim. integrals

108 voxels108 LORs

TV regularization results

=0.005

=0.05=0.008

No TV

TV results

=0.001 =0.0005 =0.0001=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.