PSF model: Symmetric 2D Gaussian function

A common approximation of the real PSF is a symmetric two-dimensional Gaussian function given by the formula

\mathrm{PSF_{G}}\left(x,y\mid\boldsymbol{\theta}\right)=\frac{\theta_{N}}{2\pi% \theta_{\sigma}^{2}}\exp{\left(-\frac{\left(x-\theta_{x}\right)^{2}+\left(y-% \theta_{y}\right)^{2}}{2\theta_{\sigma}^{2}}\right)}+\theta_{b}\,,

where \mathrm{PSF_{G}}\left(x,y\mid\boldsymbol{\theta}\right) gives the expected photon count at the integer pixel position \left(x,y\right) for a vector of parameters \boldsymbol{\theta}=\left[\theta_{x},\theta_{y},\theta_{\sigma},\theta_{N},% \theta_{b}\right]. The entries of the vector \boldsymbol{\theta} are as follows: \theta_{x} and \theta_{y} are the sub-pixel molecular coordinates, \theta_{\sigma} is the imaged size of the molecule, \theta_{N} corresponds to the total number of photons emitted by the molecule, and \theta_{b} corresponds to the background offset.