Tomography for HAADF-STEM Results The method was evaluated on

Figure 4 : Fourier transform of a focal series. (a) The frequencies covered by one focal stack correspond to the shape of a double-wedge of opening semi-angle α . (b) If the tilt increment is chosen as Δ β = 2α , neighboring wedges overlap in a non-trivial shape (orange). (c) When considering a cross section through the origin and perpendicular to the tilt axis, the wedges seem to seamlessly cover the entire frequency space. (d) A cross section shifted along the tilt axis reveals a complex-shaped region in frequency space that contains information from more than one tilt direction. (e) A cross section further along the tilt axis toward highest frequencies exposes that the region containing information from both tilt directions expands toward higher frequencies. Image adapted from [ 17 ].

Figure 5 : Comparison of tilt-series and CTFS reconstructions using SART. (a) Spatial frequency spectrum (Fourier transform) of an xz slice of the tilt-series reconstruction. The orange arrow indicates a single tilt increment highlighted in the image. (b) Spatial frequency spectrum of an xz slice of the CTFS reconstruction. The orange arrow indicates information reconstructed in part of the missing wedge. The white lines mark the border of the missing wedge. Image adapted from [ 15 ].

well-known simultaneous iterative reconstruction scheme (SIRT), but the parallel projection and backprojection models are replaced with weighted integrations of the specimen function over double cones [ 17 ]. T us, the focusing property of a real convergent electron probe is captured in the aberration- corrected STEM. T e forward projection can be computed using a cone-tracing implementation based on stochastic sampling and ray tracing, a technique originally used for the simulation of optical camera systems with limited DOF in movie productions. T e backprojection can be implemented based on a convolution with a kernel, which corresponds to a lateral cut through the double cone, that is, a disc. T e method is described in detail in [ 17 ]. A soſt ware implementation is available as a plug-in for the soſt ware package Ettention [ 21 ].

2016 May • www.microscopy-today.com

a whole-mount macrophage cell with gold nanoparticles coated with native low-density lipoprotein (LDL) taken up into vesicles [ 15 – 22 ] ( Figure 2 ). T e cell had a total thickness of about 1 µm. T e sample was tilted from −40° to +40° at 5° increments, resulting in 17 focal stacks. Each stack consisted of 20 images recorded at focal values 60 nm apart. T e entire dataset therefore consisted of 340 images. Each image was recorded at 160,000 magnifi cation with a pixel size of 2.3 nm and 512 2 pixel resolution. T e dataset was aligned as described in the methods section and reconstructed to a tomogram of 512 3 voxels in size ( Figure 3 ). T e voxel size of the tomogram was 2.3 nm in each dimension. For thick specimens, such as the whole cell investigated in this study, axial elongation as a result of the limited tilt range is the main factor limiting the interpretation of tomograms. In order to quantify axial elongation, we measured the ratio of axial and lateral resolution. Both were determined using the full width at half maximum (FWHM) criterion on a small number of ( n = 17) spherical particles. T e CTFS tomogram had an axial elongation factor e yz = 2.2 ± 0.5. Measured using the same method, a conventional tilt series recorded at the same tilt range and reconstructed using the SART method has an axial elongation factor of e yz = 2.8 ± 0.5 [ 1 ], so the CTFS leads to a signifi cant ( p < 7.0 × 10 −4 ) reduction of the axial elongation factor. In order to evaluate the improved axial resolution, it is of interest which frequencies are covered by a focal series, providing a look at the imaging with limited DOF in Fourier space. For the

case of a parallel projection with unlimited DOF, the Fourier slice theorem states that the Fourier transform of a single projection contains those frequencies that lie on a plane perpendicular to the projection direction in reciprocal space. Comparably, the Fourier transform of the STEM transform is non-zero in a set with the shape of a complementary double cone of semi-angle π /2-α ( Figure 4a ) [ 16 – 17 ]. Considering the focal stacks from two adjacent tilt directions, one can see that the double cones overlap in a complex shape ( Figure 4b ). T is means that certain spatial frequencies are contained in more than one projection, which might be the factor that stabilizes the reconstruction process and, thus, leads to improved resolution in the axial direction. T e reduced axial elongation of the CTFS compared to a tilt series is basically a result of additional data acquired in

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Figure 4 : Fourier transform of a focal series. (a) The frequencies covered by one focal stack correspond to the shape of a double-wedge of opening semi-angle α . (b) If the tilt increment is chosen as Δ β = 2α , neighboring wedges overlap in a non-trivial shape (orange). (c) When considering a cross section through the origin and perpendicular to the tilt axis, the wedges seem to seamlessly cover the entire frequency space. (d) A cross section shifted along the tilt axis reveals a complex-shaped region in frequency space that contains information from more than one tilt direction. (e) A cross section further along the tilt axis toward highest frequencies exposes that the region containing information from both tilt directions expands toward higher frequencies. Image adapted from [ 17 ].

Figure 5 : Comparison of tilt-series and CTFS reconstructions using SART. (a) Spatial frequency spectrum (Fourier transform) of an xz slice of the tilt-series reconstruction. The orange arrow indicates a single tilt increment highlighted in the image. (b) Spatial frequency spectrum of an xz slice of the CTFS reconstruction. The orange arrow indicates information reconstructed in part of the missing wedge. The white lines mark the border of the missing wedge. Image adapted from [ 15 ].

well-known simultaneous iterative reconstruction scheme (SIRT), but the parallel projection and backprojection models are replaced with weighted integrations of the specimen function over double cones [ 17 ]. T us, the focusing property of a real convergent electron probe is captured in the aberration- corrected STEM. T e forward projection can be computed using a cone-tracing implementation based on stochastic sampling and ray tracing, a technique originally used for the simulation of optical camera systems with limited DOF in movie productions. T e backprojection can be implemented based on a convolution with a kernel, which corresponds to a lateral cut through the double cone, that is, a disc. T e method is described in detail in [ 17 ]. A soſt ware implementation is available as a plug-in for the soſt ware package Ettention [ 21 ].

2016 May • www.microscopy-today.com

a whole-mount macrophage cell with gold nanoparticles coated with native low-density lipoprotein (LDL) taken up into vesicles [ 15 – 22 ] ( Figure 2 ). T e cell had a total thickness of about 1 µm. T e sample was tilted from −40° to +40° at 5° increments, resulting in 17 focal stacks. Each stack consisted of 20 images recorded at focal values 60 nm apart. T e entire dataset therefore consisted of 340 images. Each image was recorded at 160,000 magnifi cation with a pixel size of 2.3 nm and 512 2 pixel resolution. T e dataset was aligned as described in the methods section and reconstructed to a tomogram of 512 3 voxels in size ( Figure 3 ). T e voxel size of the tomogram was 2.3 nm in each dimension. For thick specimens, such as the whole cell investigated in this study, axial elongation as a result of the limited tilt range is the main factor limiting the interpretation of tomograms. In order to quantify axial elongation, we measured the ratio of axial and lateral resolution. Both were determined using the full width at half maximum (FWHM) criterion on a small number of ( n = 17) spherical particles. T e CTFS tomogram had an axial elongation factor e yz = 2.2 ± 0.5. Measured using the same method, a conventional tilt series recorded at the same tilt range and reconstructed using the SART method has an axial elongation factor of e yz = 2.8 ± 0.5 [ 1 ], so the CTFS leads to a signifi cant ( p < 7.0 × 10 −4 ) reduction of the axial elongation factor. In order to evaluate the improved axial resolution, it is of interest which frequencies are covered by a focal series, providing a look at the imaging with limited DOF in Fourier space. For the

case of a parallel projection with unlimited DOF, the Fourier slice theorem states that the Fourier transform of a single projection contains those frequencies that lie on a plane perpendicular to the projection direction in reciprocal space. Comparably, the Fourier transform of the STEM transform is non-zero in a set with the shape of a complementary double cone of semi-angle π /2-α ( Figure 4a ) [ 16 – 17 ]. Considering the focal stacks from two adjacent tilt directions, one can see that the double cones overlap in a complex shape ( Figure 4b ). T is means that certain spatial frequencies are contained in more than one projection, which might be the factor that stabilizes the reconstruction process and, thus, leads to improved resolution in the axial direction. T e reduced axial elongation of the CTFS compared to a tilt series is basically a result of additional data acquired in

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