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.
Software
Call the PSF generation from the ImageJOps, 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 = 610E09; // wavelength double ns = 1.33; // specimen refractive index double ni = 1.5; // immersion refractive index, experimental double resLateral = 100E9; // lateral pixel size double resAxial = 250E9; // axial pixel size double pZ = 2000E9D; // 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); ExtensionsNoncylindrically 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 GibsonLanni scalar approach, which has the advantage of introducing explicitly the known or sampledependent parameters. Here is a sample code which supports angledependent PSF generation. PSF Engineering (e.g. doublehelix PSF)? It would be interesting to include the spherical aberration in the doublehelix PSF, see [6] for the discussion. References
