Gaussian rendering

This method draws a normalized symmetric 2D or 3D Gaussian function integrated over the voxel volume for every localized molecule, with a standard deviation equal to the computed, or user-specified localization uncertainty. The visualized molecules are added sequentially to the final super-resolution images. The contribution of one molecule to the voxel intensity at the integer position $\left(x,y,z\right)$ is computed as

$v\left(x,y,z\mid\boldsymbol{\theta}_{p}\right)=E_{x}E_{y}E_{z}\,,$

where indexes the molecules, and the parameters $\boldsymbol{\theta}_{p}=\left[\hat{x}_{p},\hat{y}_{p},\hat{z}_{p},\hat{\sigma}% _{xy,p},\hat{\sigma}_{z,p}\right]$ . Here $\hat{x}_{p},\hat{y}_{p},\hat{z}_{p}$ is the estimated position of a molecule, $\hat{\sigma}_{xy,p}$ is the corresponding lateral localization uncertainty, $\hat{\sigma}_{z,p}$ is the axial localization uncertainty,

$\displaystyle E_{x}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\erf\left(\frac{x-\hat{x}+\frac{1}{2}}{\sqrt{2}\hat{% \sigma}_{xy}}\right)-\frac{1}{2}\erf\left(\frac{x-\hat{x}-\frac{1}{2}}{\sqrt{2% }\hat{\sigma}_{xy}}\right)\,,$
$\displaystyle E_{y}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\erf\left(\frac{y-\hat{y}+\frac{1}{2}}{\sqrt{2}\hat{% \sigma}_{xy}}\right)-\frac{1}{2}\erf\left(\frac{y-\hat{y}-\frac{1}{2}}{\sqrt{2% }\hat{\sigma}_{xy}}\right)\,,$
$\displaystyle E_{z}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\erf\left(\frac{z-\hat{z}+\frac{\Delta_{z}}{2}}{\sqrt{% 2}\hat{\sigma}_{z}}\right)-\frac{1}{2}\erf\left(\frac{z-\hat{z}-\frac{\Delta_{% z}}{2}}{\sqrt{2}\hat{\sigma}_{z}}\right)\,,$

and $\triangle_{z}$ is the size of a voxel in the axial direction. Contributions from one molecule are limited to an interval given by a circle with radius of $3\hat{\sigma}_{xy,p}$ around the molecule position in lateral dimension and by $3\hat{\sigma}_{z,p}$ in axial direction. For data visualization in the 2D case, and the term $E_{z}=1$ .

Gaussian rendering

See also