rapidSTORM can fit both astigmatic and biplane 3D data by introducing an explicit Z parameter into the point spread function. The Z parameter modifies the width parameters of the PSF according to Equation 1, “Polynomial 3D PSF Width”. In other terms, we model the variance of the PSF in the lateral directions (σ_x) as a polynom of the axial offset from the best-focused plane. The necessary parameters are the axial positions of the best-focused planes (z_x and z_y), the standard deviation of the PSF in the best-focused plane (σ_0,x and σ_0,y) and the effective focus depths for the polynomial terms (Δσ_i,x and Δσ_i,y). The point spread function model has been adapted from [Huang2008] and expanded with the natural linear term. However, rapidSTORM improves upon it by fitting the Z coordinate directly instead of using the complicated variance-space distance determination presented in the paper.

Equation 1. Polynomial 3D PSF Width

These parameters are normally determined externally from calibration samples. For astigmatic imaging, the best-focused planes z_x and z_y are set to different values. While the distance between the planes is crucially important for 3D localization, the absolute values and relative sign of the best-focused plane coordinates determine the direction and offset of the Z axis in the results. For biplane imaging, z_x and z_y are set equal to each other, but take different values for each plane.

Traditionally, rapidSTORM supported two PSF models called "Parabolic 3D" and "Holtzer 3D". Both of these models are subsets of to the polynomial model, and their parameters can be converted. For the Holtzer model, only the second derivative needs to be given as

, where ω denotes the Holtzer widening constant. For the parabolic model, the second and fourth derivatives must be given as

, where ω denotes the parabolic widening constant.