Fast and Accurate Three-Dimensional Point Spread Function Computation for Fluorescence Microscopy

A realistic and accurately calculated PSF model can significantly improve the performance in deconvolution microscopy and also the localization accuracy in single-molecule microscopy. In this work [1], we propose a fast and accurate approximation to the Gibson-Lanni model [2].

Key points:

  • We express the integral in this model as a linear combination of rescaled Bessel functions, providing an integral-free way for the calculation.

  • Experiments demonstrate that the proposed approach results in significantly smaller computational time compared with the quadrature approach (e.g. PSF Generator [3]).

  • This approach can also be extended to other microscopy PSF models.

The Gibson-Lanni Model

This model is based on a calculation of the optical path difference (OPD) between the design conditions and experimental conditions of the objective. It accounts for coverslips and other interfaces between the specimen and the objective.

\[ OPD(\rho,\mathrm{z}; z_p, {\boldsymbol \tau}) = \left(\mathrm{z} + t_i^* \right)\sqrt{n_i^2 - (\mathrm{NA}\rho)^2}+ z_p\sqrt{n_s^2 - (\mathrm{NA}\rho)^2} - t_i^*\sqrt{(n_i^*)^2 - (\mathrm{NA}\rho)^2} + t_g\sqrt{n_g^2 - (\mathrm{NA}\rho)^2} - t_g^*\sqrt{(n_g^*)^2 - (\mathrm{NA}\rho)^2}, \]

where \(\rho\) is the normalized radius in the focal plane, \(\mathrm{z}\) is the axial coordinate of the focal plane, \(z_p\) is the axial location of the point-source in the specimen layer relative to the cover slip and \(\mathbf{p}=(\mathrm{NA}, \mathrm{n}, \mathrm{t})\) is a parameter vector containing the physical parameters of the optical system: \(\mathrm{NA}\) is the numerical aperture, \(\mathrm{n}\) represents the refractive index and \(\mathrm{t}\) is the thickness of individual layers.

The Gibson-Lanni model is expressed by

\[ \mathrm{PSF}(\mathrm{r}, \mathrm{z}; z_p, \mathbf{p}) = \left|A\int_0^a e^{iW(\rho,\mathrm{z}; z_p, \mathbf{p})} J_0\left(k\mathrm{r}\mathrm{NA}\rho\right) \rho \mathrm{d}\rho\right|^2 \]

where the phase term \(W(\rho,\mathrm{z}; z_p, \mathbf{p})=k\,OPD(\rho,\mathrm{z}; z_p, \mathbf{p})\), \(k=2\pi/\lambda\) is the wave number of the emitted light. \(A\) is a constant complex amplitude, and \(J_0\) denotes the Bessel function of the first kind of order zero.

Bessel Series Approximation

The main idea is based on the fact that the integral \(\int_{0}^{\alpha}tJ_0(ut)J_0(vt)dt\) can be explicitly computed as [4]

\[ \int_0^a t J_0(ut)J_0(vt) dt = a\Big(\frac{uJ_1(ua)J_0(va) - v J_0(ua)J_1(va)}{u^2-v^2}\Big). \]

If we expand the function \(e^{iW(\rho,\mathrm{z}; z_p, \mathbf{p})}\) as a linear combination of rescaled Bessel function and fit the coefficients, then

\[ \mathrm{PSF}_{\mathrm{app}}(\mathrm{r}, \mathrm{z}; z_p, \mathbf{p}) = \left|A\sum_{m=1}^{M}c_m (\mathrm{z}) R_m(\mathrm{r}; \mathbf{p})\right|^2, \mathrm{where\ }R_m(\mathrm{r}; \mathbf{p})=\frac{\sigma_mJ_1(\sigma_ma)J_0(\eta a)a - \eta J_0(\sigma_ma)J_1(\eta a)a}{\sigma_m^2 - \eta^2}. \]

Compared with PSFGenerator


Figure 1. (a) One example PSF (\(256 \times 256 \times 128\)). (b) Comparison of computational time with PSF Generator [3] ('Best’ option) for a variety of image size. The approximation parameters \(M\) and \(K\) in the proposed approach are chosen to result in the same accuracy.


(a) Matlab
(b) ImageJ Plugin
(C) Icy Plugin


  • [1] J. Li, F. Xue and T. Blu, “Fast and Accurate Three-Dimensional Point Spread Function Computation for Fluorescence Microscopy”, J. Opt. Soc. Am. A, vol. 34, no. 6, 2017. To appear. [PDF][Slides][Demo][Matlab Code][ImageJ Plugin][Icy Plugin]

  • [2] S. F. Gibson and F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy”, J. Opt. Soc. Am. A, vol. 9, no. 1, pp. 154-166, 1992. [Link]

  • [3] H. Kirshner, F. Aguet, D. Sage, and M. Unser, “3D PSF fitting for fluorescence microscopy: implementation and localization application,” J. Microsc. vol. 249, no. 1, pp. 13–25, 2013. [Link]

  • [4] G. N. Watson, “A treatise on the theory of Bessel functions”, Cambridge University Press, 1995. [Link]