# 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

 (a) (b)

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.

## Software

Github repository

 (a) Matlab (b) ImageJ Plugin (C) Icy Plugin (d) Python (by Drs. Kyle Douglass & Hazen Babcock)

Call the PSF generation from the ImageJ-Ops, see more.
        import net.imagej.ops.AbstractOpTest;
import net.imglib2.Dimensions;
import net.imglib2.FinalDimensions;
import net.imglib2.img.Img;
import net.imglib2.type.numeric.real.DoubleType;

double NA = 1.4; // numerical aperture
double lambda = 610E-09; // wavelength
double ns = 1.33; // specimen refractive index
double ni = 1.5; // immersion refractive index, experimental
double resLateral = 100E-9; // lateral pixel size
double resAxial = 250E-9; // axial pixel size
double pZ = 2000E-9D; // position of particle

DoubleType type = new DoubleType(); // pixel type of created kernel
Dimensions dims = new FinalDimensions(256, 256); //dimensions

Img<DoubleType> kernel = ops.create().kernelDiffraction(dims, NA, lambda, ns, ni, resLateral, resAxial, pZ, type);


## Extensions

Non-cylindrically symmetric PSF generation (e.g. possibly with coma, astigmatism aberrations represented in vectorial models)?

In principle, the proposed approach is straightforward to be extended to those cases, provided other models are able to be expressed in the form of Kirchhoff's integral formula. Haeberle [5] showed that the vectorial model can also be combined with the ease of use of the Gibson-Lanni scalar approach, which has the advantage of introducing explicitly the known or sample-dependent parameters.

Here is a sample code which supports angle-dependent PSF generation.

PSF Engineering (e.g. double-helix PSF)?

It would be interesting to include the spherical aberration in the double-helix PSF, see [6] for the discussion.

## References

• [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, pp. 1029-1034, 2017. [PDF][Slides][Matlab Code][ImageJ Plugin][Icy Plugin][Python Code] (Top Downloaded Article in Oct 2017)

• [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]

• [5] O. Haeberle, “Focusing of light through a stratified medium: a practical approach for computing microscope point spread functions. Part I: conventional microscopy”, Opt. Commun., vol. 216, no. 1, pp. 55-63, 2003. [Link]

• [6] S. Ghosh and C. Preza, “Characterization of a three-dimensional double-helix point-spread function for fluorescence microscopy in the presence of spherical aberration”, J. Biomed. Opt., vol. 18, no. 3, pp. 0360101-10, 2013. [Link]