June 5, Nesterov s Method for Accelerated Penalized-Likelihood Statistical Reconstruction for C-arm Cone-Beam CT Adam S. Wang, J. Webster Stayman, Yoshito Otake, Gerhard Kleinszig, Sebastian Vogt, Jeffrey H. Siewerdsen Biomedical Engineering, Johns Hopkins University XP Division, Siemens Healthcare Johns Hopkins University Schools of Medicine and Engineering Acknowledgments The I-STAR Laboratory Imaging for Surgery, Therapy, and Radiology Hopkins Collaborators Surgery Z. Gokaslan (Neuro Spine) A. Khanna (Ortho Spine) G. Gallia (Neuro Skull Base) D. Reh (Otolaryngology) Funding Support NIH Fellowship F3-EB757 AAPM Research Seed Funding Siemens Healthcare (XP Division) Johns Hopkins University Schools of Medicine and Engineering Salt Lake City, UT (June )
June 5, Motivation Intraoperative C-arm cone-beam CT High-precision surgical guidance Detection of complications OR quality assurance Model-based reconstruction for C-arm CBCT Model physics and statistics in imaging chain Reduce dose, noise, and artifact Extend CBCT to soft-tissue imaging Enable novel source-detector trajectories Task-driven imaging Incorporate a wealth of prior information Planning CT, previous CBCT scans CT CBCT Motivation Iterative Reconstruction Tends to be Slow Typical PL reconstruction: ~ iterations, ~3 hrs Incomplete Data Slows Convergence Further a) Cone-beam artifacts (e.g., away from central slice) b) Longitudinal truncation (e.g., 5 cm coverage) c) Incomplete orbit (e.g., ~ orbit) d) Lateral truncation (e.g., 5 cm FOV) Proposed Solution Apply momentum-based methods to accelerate reconstruction time by an order of magnitude, Utilize fast, GPU-based projectors RMSE Time (Hours) z (a) (b) Kim, Ramani, and Fessler, Fully3D 3 (c) y x (d) Salt Lake City, UT (June )
June 5, Anthropomorphic Head Phantom Human skull encased in tissueequivalent plastic.7 mm diameter spheres for contrast Benchtop CBCT Untruncated projections Varian 33CB detector SAD = cm, SDD = cm kvp, mas orbit, 9 projections # subsets M {, 33,, } Mobile C-arm CBCT Truncated projections Varian 33+ detector Same geometry and technique Methods Bench C-arm Penalized Likelihood Model-based image reconstruction Basic Poisson statistical model of quantum noise Edge-preserving Huber penalty = ; Bench: =, C-arm: = Methods Objective Log-likelihood Regularization =argmax{φ ;" $ ;" % } % =' ' ( ) * ) ) *, - =,, > Penalty Recon Quadratic Linear Huber -δ δ Difference Quadratic Linear Reconstructions Separable footprints () projector in CUDA... mm 3 isotropic voxels Recon FOV encompasses entire object Head: 3 3 33 voxels Bench C-arm Salt Lake City, UT (June ) 3
June 5, Conventional SQS-M Initialize < For >,,3,,A iterations For B,,3,,C subsets ) DE F forward project ) $G C E H F IG likelihood gradient 3) JC E H F (K M(D++ likelihood curvature ) N% *,- G ) * reg. gradient P Q R ) * reg. curvature μ *,- 5) Δ T GUVWX YUVZ \Δ U 5 Δ PL Reconstruction compute update update image 5 μ * $G N% J μ * μ P 5 5.5 5-5 -.5-5 -5-5 -5 Accelerated Reconstruction Conventional SQS-M Initialize < For >,,3,,A For B,,3,,C Compute Δ \Δ U update image Nesterov Acceleration (Nes-M+ Initialize <,`,g For >,,3,,A iterations For B,,3,,C subsets Compute Δ image update ` `\gδ accumulated updates g (\ \g +/ momentum weight j \Δ U conventional update \ < \` j U momentum image Δ ` 5 \Δ (<+ \` 5.5 5 5 -.5-5 -5 - -5-5 Salt Lake City, UT (June )
June 5, 5 3 Accelerated Reconstruction Momentum weight t # Subiterations Nesterov Acceleration (Nes-M) Initialize = <,`=,g= For >=,,3,,A iterations For B=,,3,,C subsets Compute Δ image update ` `+gδ accumulated updates g (+ +g )/ momentum weight j +Δ U + j < +` U conventional update momentum image Δ ` 5 +Δ (<) +` 5.5 5 5 -.5-5 -5 - -5-5 Acceleration Factor 7 5 3 SQS- Equivalent SQS- Acceleration Factor Use objective function Φ to compare progress of algorithms A and B: min> k s.t. Φ k m n ;" Φ p m q ;" Errstsuvwxyz {vrwyu: AF > p => p /> k Baseline algorithm: SQS- (no subsets) Limit-cycle / unstable when C too large Nesterov AF increases with more iterations Faster convergence rate than SQS SQS-M Acceleration SQS- Acceleration Factor 3 Nes-M Acceleration Nes- Nes-9 Nes- Equivalent SQS- Salt Lake City, UT (June ) 5
June 5, Head Phantom (Benchtop Study) Minimal truncation Can use high number of subsets RMSD =. HU Nesterov uses 9.3 fewer iters Artifact from incomplete orbit slow to converge } Bench RMSD 5 ( ƒ) } ~~ SQS- SQS- RMSD RMSD =. HU ( ) } s Nes- Nes-9 Nes- Difference 5 RMSD } -5-5 - - Almost fully truncated Slower convergence, especially outside C- arm FOV Use fewer subsets for stability Choose higher RMSD Head Phantom (C-arm Study) RMSD } ( Š ) } ~~ SQS- SQS- RMSD RMSD =. HU 9. fewer iterations (Œ ) } s Nes- Nes-9 Nes- 5 5-5 C-arm -5 Salt Lake City, UT (June )
June 5, Projector Selection High-Fidelity Projector Separable Footprints TT Voxel-driven Cast shadow onto detector, trapezoid basis functions Accurate (but slow) Source Faster Projectors Forward-Project (Siddon) Ray-driven Voxel weighted by intersection length with ray Voxel Siddon Detector Back-Project (Peters) 3 Voxel-driven Match voxel size (. mm ~. mag) with pixel size (.3 mm) Convolve detector with 3 3 window prior to backprojection Peters Long et al, IEEE TMI Siddon, Med Phys () 95 3 Peters et al, IEEE TNS 9 Faster Projectors -SP able to achieve same RMSD Mismatched projectors assisted by smoothing and regularization RMSD SQS- SQS- RMSD Nes- Nes-9 Nes- -SP } ( Š ) } ~~ 9. fewer iterations RMSD =. HU (Œ ) } s.5 more iterations ( Œ) } s ~ Difference 5 5 - -5 - -5 Salt Lake City, UT (June ) 7
June 5, Reconstruction Time CUDA implementation on x GeForce GTX Titan Black (NVIDIA) 7 7 9 data 3 3 33 volume ( Š ) } ~~ (Œ ) } s ( Œ) } s ~ 5 5 = Separable Footprints, trapezoidal basis SP = Siddon forward-, Peters backprojector -5-5 Time/Iter (sec) # SP +7.9% +.3% -7.3% Forward Project Backproject (likelihood gradient) Backproject (likelihood curvature) Regularization Update Volume / Variables SP 9 97 3 Reconstruction Time CUDA implementation on x GeForce GTX Titan Black (NVIDIA) 7 7 9 data 3 3 33 volume ( Š ) } ~~ (Œ ) } s ( Œ) } s ~ 5 5 = Separable Footprints, trapezoidal basis SP = Siddon forward-, Peters backprojector -5-5 Total Time (sec),, SP 37 3. 5.3 Salt Lake City, UT (June )
June 5, Cadaver Abdomen Realistic Soft Tissue Imaging Fresh cadaver (non-fixed) Larger recon volume (5 35 33 voxels) to cover entire object kvp, mas (3. mgy) PL parameters: β =, δ = e-3 } MDCT CBCT Incomplete data slows convergence Choose higher RMSD Larger M unstable Excellent agreement in soft-tissue Some differences at high-contrast edges Cadaver Reconstructions RMSD SQS- SQS- RMSD Nes- Nes-9 Nes- -SP } (Œƒ ) } ~~ C-arm RMSD =. HU (ŒŒ) } s ( ) } s ~ 95 sec 959 sec 97 sec 5 5 Difference -5 - -5 - Salt Lake City, UT (June ) 9
June 5, Discussion Accelerated PL Reconstructions SQS: ~ iterations Nesterov: ~ iterations SP projectors: ~3- iterations much faster per iteration Further Acceleration Is Possible Use pre-computed curvature (save BP) Spatially non-uniform updates Relaxed momentum (increased stability) Optimized momentum method 3 Multi-GPU Speedup observed for Siddon/Peters fast enough that overhead costs may limit benefit Multi-resolution reconstruction (especially outside imaging FOV) (e.g., Abdomen: ~% of voxels outside FOV) SP Total Time (sec),, 95 959 97 Cadaver C-arm 9.9.9 Recon Kim et al, IEEE TMI 3 Kim and Fessler, IEEE MIC 3 3 Kim and Fessler, CT Meeting Conclusions Nesterov-Accelerated PL Reconstruction Helps overcome slower convergence of incomplete data Nesterov s method ~ speedup Fast projectors same image quality with ~5 speedup ~-3 minute iterative reconstructions possible Increased compatibility with clinical workflow Increased adoption of MBIR for C-arm CBCT sec 97 sec Future Work Further acceleration Automatic selection of M (# subsets) Incorporation of convergence criteria Salt Lake City, UT (June )