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Optimization of Acquisition Geometry for Intra-operative Tomographic - - PowerPoint PPT Presentation

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Optimization of Acquisition Geometry for Intra-operative Tomographic Imaging J. Vogel T. Reichl J. Gardiazabal N. Navab T. Lasser Computer-Aided Medical Procedures, Technische Universit at M unchen Institute of Biomathematics and

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Optimization of Acquisition Geometry for Intra-operative Tomographic Imaging

  • J. Vogel
  • T. Reichl
  • J. Gardiazabal
  • N. Navab
  • T. Lasser

Computer-Aided Medical Procedures, Technische Universit¨ at M¨ unchen Institute of Biomathematics and Biometry, HelmholtzZentrum M¨ unchen

MICCAI October 4, 2012

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Motivation

◮ Enable flexible intra-operative functional imaging ◮ Identify cancer tissue during surgery using radioactive tracer

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SPECT

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SPECT

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SPECT

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SPECT

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SPECT

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SPECT

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SPECT

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Diagnostic SPECT

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Freehand SPECT

SurgicEye Press Picture

  • T. Wendler et al., Eur J Nucl Med Mol Imaging 37 (8), 2010

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Freehand SPECT

Courtesy of Aslı Okur & Thomas Wendler

  • T. Wendler et al., Eur J Nucl Med Mol Imaging 37 (8), 2010

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Intra-Op Tomographic Imaging

Robotic SPECT C-Arm CT

Siemens Press Picture

◮ Problem: Find optimal and reproducible trajectories!

  • Y. Murayama et al., Neurosurgery 68, 2011

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Sneak Peek

◮ Optimize sensor trajectory for tomographic reconstruction ◮ Directly use mathematical framework ◮ Control robotic arm accordingly

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Algebraic Reconstruction: Discretized Signal

f : Ω ⊂ R3 → R f ≈ f =

i xi · bi

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Algebraic Reconstruction: Measurement Model

Mj : (Ω → R) → R mj ≈ Mj (

i xi · bi)

mj = Mj(f ) mj =

  • Lj

f (x)dx

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Algebraic Reconstruction: Measurement Model

Mj : (Ω → R) → R mj ≈ Mj (

i xi · bi)

mj = Mj(f ) =

i xi · Mj(bi) =: aji

mj =

  • Lj

f (x)dx

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Algebraic Reconstruction: Linear System

mj = aT

j , x

m =      — a1 —      x

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Algebraic Reconstruction: Linear System

mj = aT

j , x

m =      — a1 — — a2 —      x

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Algebraic Reconstruction: Linear System

mj = aT

j , x

m =      — a1 — — a2 — — a3 — . . .      x = A · x

◮ A contains the geometry, m the measurements ◮ Kernel and rank of A are quality indicators

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Singular Value Spectrum of a System Matrix

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singular value index singular value intensity

A = U    σ1 σ2 ...   

  • =: S

V T η(A) =

  • i σi

◮ SVD is slow

  • T. Lasser et al., Medical Image Analysis 11 (4), 2007 P. C. Hansen et al., Johns Hopkins University Press 2012

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Pivoted QR Decomposition

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singular value index singular value intensity

A = Q    r11 r12 · · · r22 · · · ...   

  • =: R

PT η(A) = diag(R)ℓ1 =

  • i |rii|

◮ Pivoted QR is considerably faster

  • P. C. Hansen et al., Johns Hopkins University Press 2012 T. Lasser et al., Medical Image Analysis 11 (4), 2007

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current location current destination pi possible next destinations

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current location current destination pi possible next destinations currently optimal η

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current destination pi current location possible next destinations currently optimal η

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current destination pi current location possible next destinations currently optimal η

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current destination pi current location possible next destinations currently optimal η

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Robot Control: Overview

◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface

current location pi current dest. pi+1

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Robot Control: Exploring the Surface

basis functions bi bounding triangle-mesh

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Robot Control: Exploring the Surface

random pose, evaluate η

  • Ai

— a∗ —

  • ◮ Predicted matrix Ai after robot reaches current destination pi

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Robot Control: Exploring the Surface

high need for measurement low need highest energy next destination

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Real-Time Implementation

◮ Small voxel basis (10 × 10 × 7 = 700 basis functions) for

  • ptimization (finer for actual reconstruction)

◮ Mesh of 200–450 triangles, evaluated in parallel ◮ In-place decomposition using LAPACK’s SGEQP3

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Experiments: Simulation

500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction deviation of major hotspot robot−2 robot−1 random expert grid Alexander Hartl contributed. 16

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Experiments: Simulation

500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction deviation of major hotspot robot−2 robot−1 random expert grid Alexander Hartl contributed. 16

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Experiments: Simulation

500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction deviation of major hotspot robot−2 robot−1 random expert grid Alexander Hartl contributed. 16

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Experiments: Simulation

500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction deviation of major hotspot robot−2 robot−1 random expert grid Alexander Hartl contributed. 16

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Experiments: Real World

Setup Result

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Conclusion

◮ Sensor trajectory optimization for tomographic reconstruction ◮ General approach directly using the mathematical framework ◮ Real-time implementation available

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Acknowledgements

◮ DFG SFB 824 ◮ DFG Cluster of Excellence MAP ◮ European Union FP7 grant No 25698 ◮ Looking forward to meet you at our poster (Th-2-AG-01),

today 3:00 – 4:30 pm

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 522 measurements, case id 94

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 522 measurements, case id 171

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 853 measurements, case id 190

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 853 measurements, case id 249

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 939 measurements, case id 167

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Singular Value Spectrum: Sample Cases

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singular value index singular value intensity

700 basis functions, 939 measurements, case id 233

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SVD vs. QR Spectra: Sample Cases

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singular value index singular value intensity

700 basis functions, 522 measurements, case id 94

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SVD vs. QR Spectra: Sample Cases

10

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singular value index singular value intensity

700 basis functions, 522 measurements, case id 94

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SVD vs. QR Spectra: Sample Cases

10

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singular value index singular value intensity

700 basis functions, 853 measurements, case id 190

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SVD vs. QR Spectra: Sample Cases

10

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singular value index singular value intensity

700 basis functions, 853 measurements, case id 190

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SVD vs. QR Spectra: Sample Cases

10

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singular value index singular value intensity

700 basis functions, 939 measurements, case id 167

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SVD vs. QR Spectra: Sample Cases

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singular value index singular value intensity

700 basis functions, 939 measurements, case id 167

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SVD vs. QR Spectra: Random Paths

10 20 30 40 50 60 70 80 90 100 0.05 0.1 0.15 0.2 0.25 iterations energy QR SVD

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SVD vs. QR Spectra: Random Paths

10 20 30 40 50 60 70 80 90 100 0.05 0.1 0.15 0.2 0.25 iterations energy QR SVD

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SVD vs. QR Spectra: Random Paths

10 20 30 40 50 60 70 80 90 100 0.05 0.1 0.15 0.2 0.25 iterations energy QR SVD

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SVD vs. QR Spectra: Random Paths

10 20 30 40 50 60 70 80 90 100 0.05 0.1 0.15 0.2 0.25 iterations energy QR SVD

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Robot Control: Correlation Path – Energy

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Robot Control: Correlation Path – Energy

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