Figure 1.
Synthetic datasets generated in the study for quantitative analysis with known truth. A) Surface rendering of the cell phantom dataset. B) Zero tilt projection of the cell phantom with 16.1% noise contamination according to the coefficient-of-variation test. C) The artificial pattern synthesized to observe missing wedge effects in the frequency domain. D) The spatial domain image corresponding to the artificial pattern.
Figure 2.
Altogether 124 projections of the experimental data were taken approximately from −65° to 58° tilt angle range with 1° increments. The experimental data includes a multivesicular body, intraluminal vesicles and gold particles of two diameters. Projection tilt angles from left to right, and from top to bottom: −64.97°, −46.93°, −29.00°, −11.97°, 6.16°, 24.04°, 41.07°, 58.05°. Projections are presented with inverted intensities to improve the visualization.
Figure 3.
One sequence of the sMAP-EM reconstruction method for a single slice of the sample volume.
f(0) is the initial image, is the reconstruction result of the jth pixel at kth iteration, aij is the system matrix element defining the contribution of the jth pixel to the ith projection, pi is the ith projection, med(⋅) is the median filter with a 3×3 kernel size, and β is the predefined sequence dependent coefficient defining the amount of regularization. The blocks (1), (2), and (3) yield the ML-EM correction factor,
, and the block (5) yields the a priori correction factor
. Two correction factors are combined in the block (4) to update the current estimate.
Figure 4.
Sequential adaptation of the regularization weight, β.
The sequences are initialized with an image of ones (the leftmost image). At the end of sequences with decreased β values, the final reconstruction image is obtained (the rightmost image). In total, 1000 iterations are performed in 11 sequences.
Figure 5.
Illustration of the change of the R-factor throughout the iterations.
R-factor decreases smoothly during the first 10 sequences. At the final sequence, a small increase is observed due to the noise contamination while enhancing the resolution and the contrast.
Figure 6.
Orthogonal x–z slices from the cell phantom reconstructions.
Orthogonal x–z slices from the reconstructions with projections in ±60° tilt angle range having (A) 16.1% (noise level 1, NL1) and (B) 18.0% (noise level 2, NL2) noise contamination according to the coefficient-of-variation test. Slices show gold particles (top row) and virus particles (bottom row). The middle row images present zoomed region of the gold particles in the top row images. The full dynamic range of raw pixel values is presented with pseudo-color. The artifacts caused by missing wedge are clearly present in the WBP and SIRT reconstructions from NL1 projections, whereas sMAP-EM is able to compensate the missing information better. The reconstructions from NL2 projections have lower contrast and as such, also the artifacts are less visible.
Figure 7.
Effects of different missing wedge sizes to absolute z-axis length error and contrast ratio.
Measured means and standard errors of the means of the absolute z-axis length errors and the contrast ratios of 11 gold particles in the cell phantom reconstructions. Measurements from the reconstructions of projections with noise level 1 (solid line) and noise level 2 (dashed line) are shown in the same plot. The tilt angle range is affecting the z-direction resolution with all methods. sMAP-EM is clearly the best and has the least effect of increased missing wedge. sMAP-EM gives the best contrast improving with the angular range while the WBP and SIRT reconstructions have low contrast with all tilt angle ranges.
Figure 8.
3D fitted ellipsoids on a representative gold particle in the cell phantom reconstructions.
2D orthogonal x–y (top) and x–z (middle) slices through the center of a 3D fitted ellipsoid drawn over a representative measured gold particle from reconstructions with projections in ±60° tilt angle range. Surface rendering of the gold particle (bottom) presents overall shape of the reconstructed gold particle. Isosurface threshold value was selected experimentally for the best visualization. All images are in the same scale. The full dynamic range of each image was used for visualization. Intensity inside the gold particle in the sMAP-EM reconstruction appears to have larger variation than in the WBP and SIRT reconstructions. This results partly from higher smoothing especially in the SIRT reconstruction, but also from the fact that the variation of the background intensity in the WBP and SIRT reconstructions is high and the contrast of the gold particle is low making the inside of the gold particle appear more uniform in the visualization. The contrast of the gold particle is high in the sMAP-EM reconstructions from (A) noise level 1 and (B) noise level 2 projections making fitting accurate. The contrast of the gold particle is much lower in the WBP and SIRT reconstructions especially with noise level 2 projections, reducing the accuracy of the ellipsoid fitting as presented quantifically in Figure S1.
Table 1.
Quantitative results for 11 gold particles in the cell phantom reconstruction.
Figure 9.
Observation of the missing wedge in the frequency domain.
Along with the ground truth, the spectra of the synthetic pattern reconstructed by sMAP-EM, WBP, and SIRT are shown. The width of the missing wedge used is ±60°. For better visual comparison, all amplitudes having lower than a threshold value were set to zero and the spectrum was scaled with the logarithm of base 10; the threshold value and scaling factor were adjusted for each reconstruction method. The filling of the missing wedge by sMAP-EM with meaningful information is supported by quantitative results of gold particle studies presented in Figure 7 and Tables 1 and 2.
Figure 10.
3D rendering of the experimental vesicle data reconstructions.
Volume renderings (A) show that the MVB membrane and its breakages (marked with white arrows) are visible with all reconstruction methods. However, large intraluminal vesicle and its membrane (marked with an arrow) is much more clearly visible in the sMAP-EM reconstruction than in the WBP and SIRT reconstructions. The sMAP-EM reconstruction can be directly volume rendered without any thresholding whereas both WBP and SIRT need a threshold to remove the background to make interesting regions visible. The threshold selection is a qualitative process which can lead to the removal of interesting data while reducing the background. For this figure, the thresholds for WBP and SIRT were selected for the best possible presentation. Scale bar 100nm. MVB membrane structure was further studied with surface rendering (B) from the direction of y-axis. The sMAP-EM reconstruction was compared to WBP which was quantitatively 2 shown to produce less elongation than SIRT. It is clearly visible that sMAP-EM is able to distinguish individual particles (regions marked with arrows) where WBP creates elongated connection. Also small particles (marked with arrows) are visible in the sMAP-EM that are absent in the WBP. The isovalues for surface renderings were selected so that the same percentage of the highest density voxels were included in the surface with both methods.
Figure 11.
Comparison of orthogonal slices between sMAP-EM, WBP, and SIRT reconstructions from the experimental dataset.
(Top) The leftmost figure presents the location of the orthogonal x–y and x–z slices over the reconstructed volume. Full dynamic range of each slice is visualized with a pseudo-color. Two boxes below each image are zoomed-up views of red region of interest (ROI) and its corresponding x-z plane. Green box in all slices indicate location of presumable MVB membrane breakage. (Bottom) Only non-negative pixel values of each slice are rendered with the same pseudo-color, by setting zero to all pixels originally having negative values. Two boxes above each image are zoomed-up views of the same ROI as in the images at the top, and associated x–z plane. sMAP-EM yields identical results for full and non-zero dynamic ranges while WBP and SIRT result in different visual impressions. sMAP-EM is superior to WBP and SIRT to reveal association of gold particles (white arrows). Scale bar 100nm.
Figure 12.
3D fitted ellipsoids on gold particles in the experimental vesicle data reconstructions.
Subfigures A, B, and C present reconstructions for three different gold particles. For each gold particle, orthogonal x–y (top) and x–z (middle) slices through the center are given. Surface renderings (bottom) present overall shape of the reconstructed gold particles. Isosurface threshold value was selected experimentally for the best visualization. All images are in the same scale. The full dynamic range of each subimage was used for the best visualization. The z-direction resolution is better in the sMAP-EM reconstruction than in the WBP and SIRT reconstructions. Also the contrast is superior in the sMAP-EM reconstruction making further analysis simpler. The visual impression is supported by quantitative results presented in Table 2.
Table 2.
Quantitative results for 7 gold particles in the experimental vesicle data.