Figure 1.
Empirically measure point spread functions (PSF).
110 nm Tetraspeck beads (Invitrogen) are suspended in ethanol. The solution is then applied to ultra-thin (70 nm) tissue arrays and let dry. The arrays are then mounted in mounting medium and imaged. Special care is made to ensure that no more than a single pixel per bead is saturated. Beads from multiple images across the entire field is registered and averaged to produce a single PSF. (A) PSF at 488 nm. Average of 268 beads. (B) PSF at 594 nm. Average of 282 beads. (C) PSF at 647 nm. Average of 335 beads. (d) Cross-sectional plot of each PSF. Note that the width of the PSF increases slightly with increased wavelength.
Figure 2.
Two-dimensional Richardson-Lucy deconvolution of array tomographic images.
(A) Comparison of RL deconvolution with empirical PSF versus blind deconvolution using an initial Gaussian PSF on a single sub-diffraction bead. Both methods are intensity conserving, while RL using an empirical PSF does a much better job of returning the light back to the central pixel. The graph is a plot of the sum of intensities across each column of pixels in the images. Note the conservation of intensity and the dramatic increase in central intensity in the RL plot. (B) Single section and Max projection of AT reconstructed YFP dendrite before and after two-dimensional deconvolution. Spines are clearly resolved in the deconvolved image. Scale bar = 1 um.
Figure 3.
Deconvolution improves the resolvability of adjacent point sources.
Here we plot the central cross-sectional profiles of an empirically measured point source, duplicated to simulate two adjoining point sources. Through the linear addition of intensity profiles after the two point sources were shifted in space, we demonstrate the improved resolution of the point sources after deconvolution. Note: 0 along the x axis denotes the center of the two point sources. (A) The centers are shifted by 1 pixel (100 nm) apart, and predictability one cannot resolve the two points in either case, because the centers occupy adjacent pixels. (B) The centers are shifted by 2 pixels (200 nm) apart, and now clearly the deconvolved point sources are resolvable. (C) The centers are shifted by 3 pixels (300 nm) apart. The situation is not different from the 2pixel shift. (D) The centers are shifted by 4 pixels (400 nm), and finally, the non-deconvolved point sources are resolvable. Thus, deconvolution decreases the threshold of resolvability for 4 pixel shift to 2 pixel shift.
Figure 4.
Deconvolution improves computational resolution of fine cellular structures.
(A–C) Comparison of microtubules before and after deconvolution. Images are max projection of AT volumes composed of twenty 70 nm sections. (A–B) It is clear that there is more contrast and higher frequency information is more visible in the deconvolved image. Scale Bar 10 um. Blow-up: Scale Bar 2 um. (B–C) We quantify two parallel microtubules separated by one pixel distance at the cross sections marked in the blow-up images in (B). Scale Bar = 1 um. Intensity cross sections (C) along the length of the microtubules show that the peaks of the microtubules are clearly resolved in the deconvolved case, as compared to the original image, where the peaks are barely separated. (D) Deconvolution of Synapsin puncta, a presynaptic protein, makes individual puncti more readily resolvable. Images are max projection of 15 AT sections. More importantly, computationally calculated 3D centers of mass are more accurate and better represent the number of puncta visible after deconvolution. Scale Bar = 1 um.
Figure 5.
Comparison of microtubules imaged using SIM versus deconvolution.
(A) Max projection of identical tissue volumes (ten 70 nm sections) imaged in SIM, ATW and ATD. Scale Bar = 10 um. (B–C) We quantify two parallel microtubules at the cross sections marked in the blow-up images in (B). Intensity cross sections (C) along the length of the microtubules show that the peaks of the microtubules are clearly resolved in the deconvolved and SIM scenario, whereas the peaks in the wide-field images are barely separated. The intensity of deconvolved image is higher, because it represents the computationally returned light at that pixel from the more blurred original image. For a more quantitative measurement of the similarity between SIM, ATD and ATW images, the Pearson's correlation coefficient is calculated using the images, which is: SIM to ATW = 0.8394. SIM to ATD = 0.8862, and ATW to ATD = 0.8982. The numbers confirm that in reference to the SIM, ATD is a closer match than ATW.
Figure 6.
Comparison of Synapsin puncta imaged using CWSTED versus deconvolution.
(A) Image of a single thin section imaged in CWSTED, ATW and ATD image. (B) Overlay of CWSTED (red) to ATW (green), CWSTED (red) to ATD (green) and ATW (red) to ATD (green). Note the high correspondence between the images, especially that of the CWSTED and ATD. The arrows in B point to apparent points in the ATD image (green) that is not in the CWSTED image (red). These few discrepancies are most likely due to the loss of primary antibody in a viscous mounting media overnight, because the CWSTED images were taken the day after the AT images, mostly because of the tendency of the CWSTED to bleach the sample during imaging. R value is the calculated Pearson's correlation coefficient between the two images. (C–E) Max projection of identical tissue volumes (ten 70 nm sections) imaged in CWSTED, ATW and ATD. Note that the CWSTED images were acquired using a 100× objective with 50 nm pixels whereas the AT images were acquired using a 63× objective with 100 nm pixels, thus the images were scaled and aligned to maintain the correct aspect ratios. Scale Bar = 5 um. (F) A distance histogram of centers of mass calculated from the CWSTED volume as compared to the ATW volume or the ATD volume. (G) A bar graph representing the total number of object centers found in the CWSTED volume, the ATW volume and the ATD volume.
Figure 7.
(A) Top row: the PSFs of ATW, AT with blind deconvolution, STED and ATD. Bottom row: the calculated MTF of the above modalities, which we accomplished by performing a FT on the PSF and graphing the modulus or the real component of the FT as a 2D intensity plot. (B) A plot of the rotationally averaged MTFs of ATW, AT with blind deconvolution, ATD and CWSTED. Yellow lines denote the theoretical cut-off frequency for a 1.4NA objective is calculated using 2NA/λ (λ = wavelength-488 nm, NA = numerical aperture-1.4). The grey region represents the frequency domain that is between the diffraction limit and the pixel limit of AT images. The frequencies in this region are not present in the actual recorded image, and ATD's MTF extension into this region must be accounted for purely through analytical continuation by RL. In contrast, blind deconvolution does not apply analytical continuation and remains bandwidth limited by the cut-off frequency. CWSTED on the other hand clearly surpasses the diffraction limit and has 50 nm pixels. Further tests using ATD with 50 nm pixels (63× objective with 1.6× optivar) reveal that the MTF extension can be further pushed by allowing analytical continuation to continue further using a smaller pixel size.
Figure 8.
The effect of noise on the spatial frequency recovery of RL deconvolution.
(A) This graph clearly illustrates the effect of Poisson noise on the fidelity of RL deconvolution. Poisson noise was artificially generated and added to the same image stack, and the RL was performed on those images, then the MTF of the images were calculated to demonstrate that even with a 5% injection of noise the spatial frequency recovery was significantly degraded, and for a 10% noise increase for most frequencies of the MTF the ATD is not better than ATW. Pixel size of the images were 100 nm.
Figure 9.
RL deconvolution of confocal images.
(A, C, E, G) Images of Confocal (CF) PSFs at different pinhole sizes for different pixels sizes and after deconvolution. Each PSF is an average of ∼200 individual images from 110 nm fluorescent beads. (B) MTF plot of CF with 100 nm pixels at different pinhole sizes, compared to ATW with 100 nm pixels. Note the increase in MTF magnitude for all CF cases as compared to ATW. (D) MTF of RL deconvolved confocal images (CFD) at 100 nm pixels as compared to ATD with 100 nm pixels. The magnitude of the MTF is comparable at the high spatial frequencies with CFD performing better at lower spatial frequencies. (F) MTF plot of CF with 50 nm pixels, which is very similar to (B), with a slight increase in MTF frequency. (H) MTF plot of CFD at 50 nm pixels, as compared to ATD at 100 nm and 50 nm pixels. It is clear that CFD at 50 nm completely out performs ATD at 100 nm. ATD at 50 nm perform closely with CFD in the high spatial frequencies, with the exception of CFD at 0.5 airy unit (au). At lower spatial frequencies, as seen in (D) as well, CFD out performs ATD.