Fast and High-Accuracy Localization for Three-Dimensional Single-Particle Tracking

We report a non-iterative localization algorithm that utilizes the scaling of a three-dimensional (3D) image in the axial direction and focuses on evaluating the radial symmetry center of the scaled image to achieve the desired single-particle localization. Using this approach, we analyzed simulated 3D particle images by wide-field microscopy and confocal microscopy respectively, and the 3D trajectory of quantum dots (QDs)-labeled influenza virus in live cells. Both applications indicate that the method can achieve 3D single-particle localization with a sub-pixel precision and sub-millisecond computation time. The precision is almost the same as that of the iterative nonlinear least-squares 3D Gaussian fitting method, but with two orders of magnitude higher computation speed. This approach can reduce considerably the time and costs for processing the large volume data of 3D images for 3D single-particle tracking, which is especially suited for 3D high-precision single-particle tracking, 3D single-molecule imaging and even new microscopy techniques.


Supplementary File Title
Supplementary Figure 1 3D scatter plots of the errors illustrating the error ranges of different algorithms in three dimensions.

Supplementary Figure 2
Localization accuracy for different algorithms estimated using simulated 3D CCD images of wide-field microscope.

Supplementary Figure 3
Influence of the lateral size on the accuracy for different localization algorithms.

Supplementary Figure 4
Localization accuracy for different algorithms estimated using simulated 3D images of confocal microscope.

Supplementary Figure 5
Accuracy for different localization algorithms estimated using simulated 3D CCD images of confocal microscope with the S/N ratios of 3 ~ 100.

Supplementary Figure 6
Influence of the axial size on the accuracy for different localization algorithms.

Supplementary Figure 7
Comparation of the precision for radial symmetry and Gaussian fitting method estimated using 3D confocal images of fluorescence beads.

Supplementary Note: 3D radial symmetry localization algorithm
Three-dimensional (3D) diffraction pattern of a single particle is described as a 3D point spread function ( The intensity distribution of 3D diffraction pattern is described as where N is 0 and 1for wide-field microscope and for confocal microscope, respectively.
At the geometrical focus, = = 0 and the intensity is For the points in the focal plane, = 0 and the intensity distribution is The radius of the first dark ring in the focus of 3D diffraction pattern can be calculated from where is also considered as the lateral resolution according to Rayleigh criteria.
Likewise, for the points along the axis, = 0, and the intensity is expressed as The radius of the first dark ring along the axis of 3D diffraction pattern can be calculated from Hence, where R z is also given as the axial resolution.
The ratio of the axial to lateral radius of the 3D diffraction pattern is approximately given as This formula is suitable for both wide-field and confocal microscopy.
Supplementary Figure 8. Illustration of the calculation of the 3D radial symmetry algorithm.
Therefore, we scale the 3D image of a single particle in the axial direction according to the ratio of the axial and lateral resolution. Considering the estimate error and noise, our algorithm determines the particle center ( , , ) as the point having the minimal distance to all intensity gradient lines (Supplementary Figure 8). The distance ( ) of the center to each gradient line can be calculated as follows: gradient magnitude at the point (   ,  ,  ), similarly for ̂ and ̂ in y and z directions. Thus, the distance can be expressed as To calculate the center, we minimize 2 = ∑ 2 , where , a displacement weighting, is the square of the gradient magnitude divided by the distance between the pixel ( , , ) and the particle center evaluated using centroid method. We get the derivative of 2 with respect to c and set it equal to zero, similarly for y c and z c respectively. The equations are as follows: After some mathematical deformation, we solve the following matrix equation to obtain the center (x c , y c , z c ).