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

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 1

Motivation ◮ Enable flexible intra-operative functional imaging ◮ Identify cancer tissue during surgery using radioactive tracer 2

SPECT 3

SPECT 3

SPECT 3

SPECT 3

SPECT 3

SPECT 3

SPECT 3

Diagnostic SPECT 4

Freehand SPECT SurgicEye Press Picture T. Wendler et al., Eur J Nucl Med Mol Imaging 37 (8), 2010 5

Freehand SPECT Courtesy of Aslı Okur & Thomas Wendler T. Wendler et al., Eur J Nucl Med Mol Imaging 37 (8), 2010 5

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 6

Sneak Peek ◮ Optimize sensor trajectory for tomographic reconstruction ◮ Directly use mathematical framework ◮ Control robotic arm accordingly 7

Algebraic Reconstruction: Discretized Signal f = � f : Ω ⊂ R 3 → R f ≈ � i x i · b i 8

Algebraic Reconstruction: Measurement Model m j ≈ M j ( � M j : (Ω → R ) → R i x i · b i ) m j = M j ( f ) � m j = f ( x ) d x L j 9

Algebraic Reconstruction: Measurement Model m j ≈ M j ( � M j : (Ω → R ) → R i x i · b i ) = � m j = M j ( f ) i x i · M j ( b i ) � �� � � =: a ji m j = f ( x ) d x L j 9

Algebraic Reconstruction: Linear System � a T m j = j , x �  — a 1 —      = m    x  10

Algebraic Reconstruction: Linear System � a T m j = j , x �   — a 1 —   — a 2 —   = m   x   10

Algebraic Reconstruction: Linear System � a T m j = j , x �   — a 1 —   — a 2 —   = m   x  — a 3 —  . . . = A · x ◮ A contains the geometry, m the measurements ◮ Kernel and rank of A are quality indicators 10

Singular Value Spectrum of a System Matrix 1 10   0 10 σ 1   V T A = U  σ 2  singular value intensity −1 10 ... � �� � −2 10 =: S � −3 10 η ( A ) = i σ i −4 10 singular value index ◮ SVD is slow T. Lasser et al., Medical Image Analysis 11 (4), 2007 P. C. Hansen et al., Johns Hopkins University Press 2012 11

Pivoted QR Decomposition 1 10   0 10 r 11 r 12 · · ·   r 22 · · · P T A = Q   singular value intensity −1 10 ... � �� � −2 10 =: R � −3 10 η ( A ) = � diag ( R ) � ℓ 1 = i | r ii | −4 10 singular value index ◮ 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 12

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface possible next destinations current destination p i current location 13

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface possible next destinations currently optimal η current destination p i current location 13

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface possible next destinations currently optimal η current destination p i current location 13

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface possible next destinations currently optimal η current destination p i current location 13

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface possible next destinations currently optimal η current destination p i current location 13

Robot Control: Overview ◮ Find trajectory maximizing cost function η ◮ Constrain motion to bounding surface current dest. p i +1 current location p i 13

Robot Control: Exploring the Surface bounding triangle-mesh basis functions b i 14

Robot Control: Exploring the Surface � � A i random pose, evaluate η — a ∗ — ◮ Predicted matrix A i after robot reaches current destination p i 14

Robot Control: Exploring the Surface high need for measurement highest energy next destination low need 14

Real-Time Implementation ◮ Small voxel basis (10 × 10 × 7 = 700 basis functions) for optimization (finer for actual reconstruction) ◮ Mesh of 200–450 triangles, evaluated in parallel ◮ In-place decomposition using LAPACK’s SGEQP3 15

Experiments: Simulation grid expert random robot−1 robot−2 deviation of major hotspot 0 500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction Alexander Hartl contributed. 16

Experiments: Simulation grid expert random robot−1 robot−2 deviation of major hotspot 0 500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction Alexander Hartl contributed. 16

Experiments: Simulation grid expert random robot−1 robot−2 deviation of major hotspot 0 500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction Alexander Hartl contributed. 16

Experiments: Simulation grid expert random robot−1 robot−2 deviation of major hotspot 0 500 1000 1500 2000 2500 3000 3500 4000 number of measurements used for reconstruction Alexander Hartl contributed. 16

Experiments: Real World Setup Result 17

Conclusion ◮ Sensor trajectory optimization for tomographic reconstruction ◮ General approach directly using the mathematical framework ◮ Real-time implementation available 18

Acknowledgements ◮ DFG SFB 824 ◮ DFG Cluster of Excellence MAP ◮ European Union FP7 grant N o 25698 ◮ Looking forward to meet you at our poster ( Th-2-AG-01 ), today 3:00 – 4:30 pm 19

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 522 measurements, case id 94 21

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 522 measurements, case id 171 21

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 853 measurements, case id 190 21

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 853 measurements, case id 249 21

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 939 measurements, case id 167 21

Singular Value Spectrum: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 939 measurements, case id 233 21

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 522 measurements, case id 94 23

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 522 measurements, case id 94 23

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 853 measurements, case id 190 23

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 853 measurements, case id 190 23

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 939 measurements, case id 167 23

SVD vs. QR Spectra: Sample Cases 1 10 0 10 singular value intensity −1 10 −2 10 −3 10 −4 10 singular value index 700 basis functions, 939 measurements, case id 167 23

SVD vs. QR Spectra: Random Paths SVD QR 0.25 0.2 energy 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 iterations 25

SVD vs. QR Spectra: Random Paths SVD QR 0.25 0.2 energy 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 iterations 25

SVD vs. QR Spectra: Random Paths SVD QR 0.25 0.2 energy 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 iterations 25

SVD vs. QR Spectra: Random Paths SVD QR 0.25 0.2 energy 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 100 iterations 25

Robot Control: Correlation Path – Energy 27

Robot Control: Correlation Path – Energy 27