Fast and Accurate ThreeDimensional Point Spread Function Computation for Fluorescence Microscopy
The GibsonLanni ModelThis 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 pointsource 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 GibsonLanni model is expressed by \[ \mathrm{PSF}(\mathrm{r}, \mathrm{z}; z_p, \mathbf{p}) = \leftA\int_0^a e^{iW(\rho,\mathrm{z}; z_p, \mathbf{p})} J_0\left(k\mathrm{r}\mathrm{NA}\rho\right) \rho \mathrm{d}\rho\right^2 \] 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 ApproximationThe 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] \[ \int_0^a t J_0(ut)J_0(vt) dt = a\Big(\frac{uJ_1(ua)J_0(va)  v J_0(ua)J_1(va)}{u^2v^2}\Big). \] 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 \[ \mathrm{PSF}_{\mathrm{app}}(\mathrm{r}, \mathrm{z}; z_p, \mathbf{p}) = \leftA\sum_{m=1}^{M}c_m (\mathrm{z}) R_m(\mathrm{r}; \mathbf{p})\right^2, \mathrm{where\ }R_m(\mathrm{r}; \mathbf{p})=\frac{\sigma_mJ_1(\sigma_ma)J_0(\eta a)a  \eta J_0(\sigma_ma)J_1(\eta a)a}{\sigma_m^2  \eta^2}. \] 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.
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