PSF model: Rotated elliptical Gaussian function (3D using astigmatism)

3D SMLM imaging can be performed by introducing a weak cylindrical lens into the imaging path to create slight astigmatism in the image [1]. This results in images of molecules with different ellipticity depending on their axial position. When a molecule is in focus, its image appears round. If the molecule is slightly above or below the focal plane, its image appears ellipsoidal. Calibration of the imaging system is needed to determine the orientation of the imaged ellipsoid (the camera chip might not be aligned with cylindrical lens) and the relationships between the axial position and ellipticity of the imaged molecules.

A common PSF model for astigmatic 3D imaging is a rotated, elliptical Gaussian function given by the formula

\mathrm{PSF}_{\mathrm{EG}}\left(x,y\mid\boldsymbol{\theta},\phi\right)=\frac{% \theta_{N}}{2\pi\sigma_{1}\left(\theta_{z}\right)\sigma_{2}\left(\theta_{z}% \right)}\exp\left(-\frac{x^{\prime}{}^{2}}{2\left(\sigma_{1}\left(\theta_{z}% \right)\right)^{2}}-\frac{y^{\prime 2}}{2\left(\sigma_{2}\left(\theta_{z}% \right)\right)^{2}}\right)+\theta_{b}\,,

where \mathrm{PSF}_{\mathrm{EG}}\left(x,y\mid\boldsymbol{\theta},\phi\right) gives the expected photon count at the integer pixel position \left(x,y\right) for the parameters \boldsymbol{\theta}=\left[\theta_{x},\theta_{y},\theta_{z},\theta_{N},\theta_{% b}\right], and

\displaystyle x^{\prime} \displaystyle= \displaystyle\left(x-\theta_{x}\right)\cos\phi-\left(y-\theta_{y}\right)\sin% \phi\,,
\displaystyle y^{\prime} \displaystyle= \displaystyle\left(x-\theta_{x}\right)\sin\phi+\left(y-\theta_{y}\right)\cos% \phi\,.

The entries of the vector \boldsymbol{\theta} are as follows: \theta_{x},\theta_{y},\theta_{z} are the sub-pixel molecular coordinates, \sigma_{1}\left(\theta_{z}\right) and \sigma_{2}\left(\theta_{z}\right) are the imaged widths of the molecule along two perpendicular axes rotated by the angle \phi with respect to xy coordinates, \theta_{N} corresponds to the total number of photons emitted by the molecule, and \theta_{b} is the background offset. The imaged widths \sigma_{1}\left(\theta_{z}\right) and \sigma_{2}\left(\theta_{z}\right) are modeled by a pair of defocusing curves.

References