PSF model: Integrated form of a symmetric 2D Gaussian function

The integrated form of a symmetric two-dimensional Gaussian function can be used to help to take into account the discrete nature of pixels present in digital cameras [2, 1]. Assuming a uniform distribution of pixels with unit size, a single molecule intensity profile can be expressed as

\mathrm{PSF_{IG}}\left(x,y\mid\boldsymbol{\theta}\right)=\theta_{N}E_{x}E_{y}+% \theta_{b}\,,

where \mathrm{PSF_{IG}}\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] and

\displaystyle E_{x} \displaystyle= \displaystyle\frac{1}{2}\erf\left(\frac{x-\theta_{x}+\frac{1}{2}}{\sqrt{2}% \theta_{\sigma}}\right)-\frac{1}{2}\erf\left(\frac{x-\theta_{x}-\frac{1}{2}}{% \sqrt{2}\theta_{\sigma}}\right)\,,
\displaystyle E_{y} \displaystyle= \displaystyle\frac{1}{2}\erf\left(\frac{y-\theta_{y}+\frac{1}{2}}{\sqrt{2}% \theta_{\sigma}}\right)-\frac{1}{2}\erf\left(\frac{y-\theta_{y}-\frac{1}{2}}{% \sqrt{2}\theta_{\sigma}}\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.

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