## Results: 10

Fig. 3. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 2. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

*δ*, involves only a 1-D slant, which is very fast and can be done repeatedly to all ring differences after a single rotation from Step 1. This makes the rotate-and-slant projector computationally efficient for fully-3-D projections to a large number of ring differences.

Fig. 10. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

*ϕ*. The area-of-overlap between LOR A and image voxel 1 is indicated (left diagram) and can be computed by the length-of-overlap between the LOR edges and voxel as indicated. The area-of-overlap in the axial plane between LOR C and image voxel 5 is similarly indicated (right diagram), and is computed based on the length-of-overlap as shown (which introduces a slight approximation in some cases as discussed in the text).

Fig. 7. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 8. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 6. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 9. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 5. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

*δ*= 20. Projection times dropped quickly with compression factors up to about 8, whereas accuracy was largely unaffected for compression factors of 8 and below. These results indicate that using a compression factor of 4 or 8 (for a 128 × 128 image) offers significant speedup of the rotate-and-slant projector without a significant concomitant loss of accuracy.

Fig. 4. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

Fig. 1. From: Rotate-and-Slant Projector for Fast LOR-Based Fully-3-D Iterative PET Reconstruction.

*s*,

*ϕ*,

*z*,

*δ*) used to parameterize an LOR for a generic ring PET tomograph. The

*z*coordinate describes the axial position of the midpoint of the LOR, which falls at the point of closest approach to the central axis of the tomograph. Note that the length of the LOR (here denoted Δ

*y*, where the

*y*-direction is defined perpendicular to

*s*within the transaxial plane), is dependent upon

*s*. As a result, the polar angle

*θ*is not only dependent upon the ring difference

*δ*, but there is also a secondary dependence upon

*s*. For cylindrical ring PET tomographs, the LORs grouped into a parallel projection at a given angle are unevenly spaced in

*s*. The diagram at right also shows interleaving of LORs from adjacent azimuthal angles (solid and dashed lines), which have been merged into a single angular bin, effectively halving the number of angular samples but doubling the transverse sampling in each.