Confidential. Not to be copied, distributed, or reproduced without prior approval. Divergent-beam backprojection-filtration formula with applications to region-of-interest imaging Aymeric Reshef , Cyril Riddell , Yves Trousset , Saïd Ladjal , and Isabelle Bloch contact: aymeric.reshef@ge.com The Fifth International Conference on Image Formation in X-Ray Computed Tomography May 20-23, 2018 Fort Douglas/Olympic Village, Salt Lake City, Utah, USA http://www.ucair.med.utah.edu/CTmeeting/index.html

X-ray tube Table X-ray detector C-arm RAO/LAO Large display monitor 2D field-of-view (FOV) Source-to-image distance (SID) Lift

Confidential. Not to be copied, distributed, or reproduced without prior approval. C-arm cone-beam computed tomography (CBCT) May 21, 2018 BPF formula for ROI imaging 3 Soft-tissue imaging Vascular imaging Advanced applications Stereo, Segmentation, Registration/Fusion, Detection, Tracking… Brain soft-tissue protocol: ➢ Signs of bleeding  Low-contrast detection (LCD) ➢ LCD sensitive to artifacts ➢ Diagnostic CT is the gold standard for LCD

Confidential. Not to be copied, distributed, or reproduced without prior approval. Cone-beam X-ray projection May 21, 2018 BPF formula for ROI imaging 5 3×3 homography matrix Write a 3D point as the combination of ➢ a 2D point on a plane ⊥ ∈ ² ➢ a plane location = ⋅ Tomographic reconstruction problem: Given a collection of projections ∈Θ, reconstruct image ➢ Fan-beam geometry with linear detector  Midplane ➢ Parallel-beam geometry  move source to infinity Riddell, C., & Trousset, Y. (2006). Rectification for cone-beam projection and backprojection. IEEE Transactions on Medical Imaging, 25(7), 950-962.

Confidential. Not to be copied, distributed, or reproduced without prior approval. Filtered backprojection (FBP) May 21, 2018 BPF formula for ROI imaging 6 Parallel-to-fan-beam change of variables Approximate extension to cone-beam case Parallel-beam Fan-beam Cone-beam Ramp filter: Feldkamp, L. A., Davis, L. C., & Kress, J. W. (1984). Practical cone-beam algorithm. JOSA A, 1(6), 612-619.

Confidential. Not to be copied, distributed, or reproduced without prior approval. Hilbert-transformed differentiated backprojection (DBP-HT-1) May 21, 2018 BPF formula for ROI imaging 7 Parallel-beam case: Hilbert transform DBP of a uniform disk Truncated Hilbert lines Actual support of the true inverse Hilbert transform The result is derived only in the continuous, parallel-beam geometry… How to derive a (discrete) fan-beam DBP-HT formula? Finite inverse Hilbert transform Noo, F., Clackdoyle, R., & Pack, J. D. (2004). A two-step Hilbert transform method for 2D image reconstruction. Physics in Medicine & Biology, 49(17), 3903.

Confidential. Not to be copied, distributed, or reproduced without prior approval. 1 2 DBP-HT-1 : sampling issues May 21, 2018 BPF formula for ROI imaging 8 Parallel-beam case: This line must be well sampled! Parallel-to-fan-beam change of variables: A dense angular sampling is required for both 1. DBP computation 2. Finite Hilbert transform inversion

Confidential. Not to be copied, distributed, or reproduced without prior approval. Intrinsically fan-beam Hilbert-transformed DBP May 21, 2018 BPF formula for ROI imaging 9 projection (1) filter (non-local) (2) backproject projection (1) filter (local) (2) backproject (3) mono-directional 2D filter (non-local) FBP Proposed

Confidential. Not to be copied, distributed, or reproduced without prior approval. Intrinsically fan-beam Hilbert-transformed DBP (cont’d) May 21, 2018 BPF formula for ROI imaging 10 Single projection Full K-pass Hilbert-transformed DBP (DBP-HT-K) Angular subset ✓ Full split ( = ), =  FBP ✓ = 1, parallel-beam geometry  DBP-HT-1 ✓ = 2, frontal/lateral splitting  Fourier-based filtering along rows / columns ✓ Intrinsically view-wise formula and algorithm  as good as FBP whatever the angular sampling ✓ Immediate extension to cone-beam reconstruction

Confidential. Not to be copied, distributed, or reproduced without prior approval. Cone-beam artifacts May 21, 2018 BPF formula for ROI imaging 11

Confidential. Not to be copied, distributed, or reproduced without prior approval. May 21, 2018 BPF formula for ROI imaging 12 DBP-HT-2 FDK RE() = − FDK FDK Mean relative error in the head: 0.43% Window width: 50 HU

Confidential. Not to be copied, distributed, or reproduced without prior approval. Applications: dual-rotation C-arm CBCT May 21, 2018 BPF formula for ROI imaging 13 Reshef, A., Riddell, C., et al. (2017). Dual‐rotation C‐arm cone‐beam computed tomography to increase low‐contrast detection. Medical physics, 44(9). Dual-rotation C-arm CBCT consists of 2 tomographic acquisitions 1. One truncated, high-dose (low-noise) acquisition with dense angular sampling 2. One un-truncated acquisition either a) low-dose (high-noise) with dense angular sampling b) high-dose (low-noise) with angular subsampling Applications: • low-dose CT / IGRT • Virtual bow-tie • region-of-interest (ROI) imaging How do we merge dual-rotation acquisitions? Merge the projections, then reconstruct? Reconstruct images separately, then merge the reconstructed images? Merge data in the image domain but prior to full reconstruction

Confidential. Not to be copied, distributed, or reproduced without prior approval. Image-based data merging ➢ Backprojection is correct everywhere up to subsampling streaks or noise ➢ Backprojection is correct only within the truncated field-of-view Ω′ but with excellent angular sampling Un-truncated Truncated Key: Backproject locally processed projections and merge data prior to applying a non-local filter! BPF formula for ROI imaging 14 May 21, 2018

Confidential. Not to be copied, distributed, or reproduced without prior approval. May 21, 2018 BPF formula for ROI imaging 15 Window width: 50 HU FDK fails at reconstructing low- contrast structures FDK from un-truncated, high-dose, subsampled data (90 views) FDK from un-truncated, low-dose, densely sampled data (1440 views) FDK from truncated, high-dose, densely sampled data (1440 views) (with ad hoc data extrapolation)

Confidential. Not to be copied, distributed, or reproduced without prior approval. May 21, 2018 BPF formula for ROI imaging 16 Window width: 50 HU Proposed (subsampling) Proposed (noise) FDK from un-truncated, high- dose, densely sampled data Mean relative error in ROI: 0.50% Mean relative error in ROI: 0.44% Reference

Confidential. Not to be copied, distributed, or reproduced without prior approval. Scatter reduction in dual-rotation C-arm CBCT May 21, 2018 BPF formula for ROI imaging 17 Single-rotation BPF Dual-rotation BPF Residual Dual-rotation BPF reduces the cupping effect induced by scattered radiations Window width: 50 HU

Confidential. Not to be copied, distributed, or reproduced without prior approval. Speeding up the reconstruction May 21, 2018 BPF formula for ROI imaging 18 Reference image 37 un-truncated projections 37 un-truncated projections 37 un-truncated projections Grid size outside ROI / Grid size inside ROI 1 1/4 1/16 Mean residual error (%) 0.15 0.18 9.27 Window width: 50 HU

Confidential. Not to be copied, distributed, or reproduced without prior approval. 2-view extrapolation for ROI imaging May 21, 2018 BPF formula for ROI imaging 19

Confidential. Not to be copied, distributed, or reproduced without prior approval. Conclusion May 21, 2018 BPF formula for ROI imaging 20 Intrinsically fan-beam Hilbert-transformed DBP reconstruction ➢ Exact in the fan-beam geometry and approximate in the cone-beam geometry ➢ Adapted to geometric non-idealities (calibration matrices) and short-scan geometries ➢ Based on the properties of the Hilbert transform ➢ Translates into a view-wise algorithm ➢ Extends the approach of “traditional” DBP-HT-1 ➢ Provides flexibility regarding the choice of angular subsets and the filtering directions Applications to dual-rotation C-arm CBCT ➢ For both virtual bow-tie and ROI imaging ➢ Uses the unfiltered backprojection domain as a merging space ➢ May be embedded in iterative reconstruction schemes to better correct for additional artifacts ➢ Reduces the influence of scattered radiations by design and may help improve scatter correction ➢ Further speed-up strategies available using coarser grid sizes outside the ROI ➢ Towards fast 2-view extrapolation for ROI imaging?

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Confidential. Not to be copied, distributed, or reproduced without prior approval. Side dish: scattered radiations May 21, 2018 BPF formula for ROI imaging 22 Scatter correction (Siewerdsen et al. 2006) I = P + S Additional signal corrupting intensity measurements Detector Source line integrals are under-estimated Scatter rejection ✓ Air gap ✓ Field of view ✓ Anti-scatter grid P+S Intensity projection S S S S Sest Pest ~ 0 ~ 0 Scatter estimate Scatter- corrected - = I = P

Confidential. Not to be copied, distributed, or reproduced without prior approval. May 21, 2018 BPF formula for ROI imaging 23 Lateral views Frontal views

Confidential. Not to be copied, distributed, or reproduced without prior approval. May 21, 2018 BPF formula for ROI imaging 24