Drift correction for single-molecule imaging by molecular constraint field, a distance minimum metric

Abstract

Background: The recent developments of far-field optical microscopy (single molecule imaging techniques) have overcome the diffraction barrier of light and improve image resolution by a factor of ten compared with conventional light microscopy. These techniques utilize the stochastic switching of probe molecules to overcome the diffraction limit and determine the precise localizations of molecules, which often requires a long image acquisition time. However, long acquisition times increase the risk of sample drift. In the case of high resolution microscopy, sample drift would decrease the image resolution.

Results: In this paper, we propose a novel metric based on the distance between molecules to solve the drift correction. The proposed metric directly uses the position information of molecules to estimate the frame drift. We also designed an algorithm to implement the metric for the general application of drift correction. There are two advantages of our method: First, because our method does not require space binning of positions of molecules but directly operates on the positions, it is more natural for single molecule imaging techniques. Second, our method can estimate drift with a small number of positions in each temporal bin, which may extend its potential application.

Conclusions: The effectiveness of our method has been demonstrated by both simulated data and experiments on single molecular images.

Keywords: Fluorescence microscopy; Image reconstruction techniques; Resolution; Superresolution.

Figure 1

Illustration of the measured position set and drift correction in a structure composed of two emitters. Illustration of the measured position set and drift correction in a structure composed of two emitters. (a) Measured positions of a probe. The black cross denotes the ideal position of the probe. The blue, red and green points denote three measured positions of the probe in different times. (b) The corrected positions of the observed probe positions. The blue, red and green points denote the corrected positions of the corresponding observations in (a).

Figure 2

Flowchart of the incremental drift correction algorithm. Flowchart of the incremental drift correction algorithm.

Figure 3

Simulated data with drift imposed on the data. Simulated data with drift imposed on the data. (a) Super-resolution image of data set Ring, which is degenerated by the drift(10 nm per pixel). (b) Blue solid points denote the exactly simulated values that are imposed on the Ring data set as drift; cyan cycles denote the estimated drifts that are calculated from the degenerated data set by our algorithm. (c) Super-resolution image of data set Grid, which is degenerated by the drift(10 nm per pixel). (d) Solid points denote the exactly simulated values that are imposed on the Grid data set as drift, where the blue one refers to the drift along the x-axis and the red one refers to the drift along the y-axis; cycles denote the estimated drifts that are calculated from the degenerated data set by our algorithm, where the cyan one refers to the drift along the x-axis and the pink one refers to the drift along the y-axis. (e) Super-resolution image of data set Radio, which is degenerated by the drift(10 nm per pixel). (f) Solid points denote the exactly simulated values that are imposed on the Radio data set as drift, where the blue one refers to the drift along x-axis and the red one refers to the drift along y-axis; cycles denote the estimated drifts that are calculated from the degenerated data set by our algorithm, where the cyan one refers to the drift along the x-axis and the pink one refers to the drift along the y-axis.

Figure 4

Super-resolution images of drift corrected data (10 nm per pixel). Super-resolution images of drift corrected data (10 nm per pixel). (a) Super-resolution image of data set Ring which is recovered from the data set illustrated in Figure 3(a). (b) Super-resolution image of data set Grid which is recovered from the data set illustrated in Figure 3(c). (c) Super-resolution image of data set Radio which is recovered from the data set illustrated in Figure 3(e).

Figure 5

Resolution estimation by FRC. Resolution estimation by FRC. The red curve is FRC values and the horizontal line is the 0.143 threshold. (a) FRC curve of data set Ring, which is degenerated by the drift. (b) FRC curve of data set Grid, which is degenerated by the drift. (c) FRC curve of data set Radio, which is degenerated by the drift. (d) FRC curve of data set Ring, of which the drift is corrected by our method. (e) FRC curve of data set Grid, of which the drift is corrected by our method. (f) FRC curve of data set Radio, of which the drift is corrected by our method. (g) FRC curve of data set Ring, which has no drift imposed. (h) FRC curve of data set Grid, which has no drift imposed. (i) FRC curve of data set Radio, which has no drift imposed.

Figure 6

Drift correction of the data set where there are 150 molecule positions in each temporal bin. Drift correction of the data set where there are 150 molecule positions in each temporal bin. (a) Super-resolution image of data set Grid, which has no drift imposed. (b) Super-resolution image of data set Grid, which is corrupted by drift. (c) Super-resolution image of data set Grid, of which the drift is corrected by our method. (d) FRC curve of the data set in sub-fig(a). (e) FRC curve of the data set in sub-fig(b). (f) FRC curve of the data set in sub-fig(c). (g) Super-resolution image of data set Radio, which has no drift imposed. (h) Super-resolution image of data set Radio, which is corrupted by drift. (i) Super-resolution image of data set Radio, of which the drift is corrected by our method. (j) FRC curve of the data set in sub-fig(g). (k) FRC curve of the data set in sub-fig(h). (l) FRC curve of the data set in sub-fig(i).

Figure 7

The values of the mean Δ _d and the standard deviation σ _d of residual over the T=40 time intervals for the data set Grid, as a function of molecule number per temporal bin n and localization precision ξ. The values of the mean Δ _d and the standard deviation σ _d of residual over the T =40 time intervals for the data set Grid, as a function of molecule number per temporal bin n and localization precision ξ . (a) Curve of residual Δ _d in different localization precisions ξ, as a function of molecule numbers n. (b) Curve of residual’s standard deviation σ _d in different localization precisions ξ, as a function of molecule numbers n. (c) Curve of residual Δ _d in different molecule numbers n, as a function of localization precisions ξ. (d) Curve of residual’s standard deviation σ _d in different molecule numbers n, as a function of localization precisions ξ.

Figure 8

The values of the mean Δ _d and the standard deviation σ _d of residual over the T=40 time intervals for the data set Radio, as a function of molecule number per temporal bin n and localization precision ξ. The values of the mean Δ _d and the standard deviation σ _d of residual over the T =40 time intervals for the data set Radio, as a function of molecule number per temporal bin n and localization precision ξ . (a) Curve of residual Δ _d in different localization precisions ξ, as a function of molecule numbers n. (b) Curve of residual’s standard deviation σ _d in different localization precisions ξ, as a function of molecule numbers n. (c) Curve of residual Δ _d in different molecule numbers n, as a function of localization precisions ξ. (d) Curve of residual’s standard deviation σ _d in different molecule numbers n, as a function of localization precisions ξ.

Figure 9

Experimental data. Experimental data. (a) Super-resolution image of experimental data set, of which the drift has not been corrected. (b) Super-resolution image of experimental data set, of which the drift is corrected by our method. (c) Super-resolution image of experimental data set, of which the drift is corrected by method based on beads. (d) FRC curve of the data set in sub-fig(a). (e) FRC curve of the data set in sub-fig(b). (f) FRC curve of the data set in sub-fig(c).

Figure 10

Drift correction of the data set where there are 30 molecule positions in each temporal bin. Drift correction of the data set where there are 30 molecule positions in each temporal bin. (a) Super-resolution image of data set Grid, which has no drift imposed. (b) Super-resolution image of data set Grid, which is corrupted by drift. (c) Super-resolution image of data set Grid, of which the drift is corrected by our method. (d) FRC curve of the data set in sub-fig(a). (e) FRC curve of the data set in sub-fig(b). (f) FRC curve of the data set in sub-fig(c). (g) Super-resolution image of data set Radio, which has no drift imposed. (h) Super-resolution image of data set Radio, which is corrupted by drift. (i) Super-resolution image of data set Radio, of which the drift is corrected by our method. (j) FRC curve of the data set in sub-fig(g). (k) FRC curve of the data set in sub-fig(h). (l) FRC curve of the data set in sub-fig(i).