Definition of the fitting region

Given the approximate position of a molecule \left(\tilde{x}_{p},\tilde{y}_{p}\right) and a user-specified fitting radius r>0, we define \mathcal{D}=\left\{-r,\ldots,r\right\}\times\left\{-r,\ldots,r\right\} as a set of integer \left(x,y\right) coordinates, and \tilde{I}\left(x,y\right)=I\left(x+\tilde{x}_{p},y+\tilde{y}_{p}\right) as intensity values of an l\times l sub-image centered at the point \left(\tilde{x}_{p},\tilde{y}_{p}\right) of the raw input image I, where l=2r+1 is the size of the sub-image. The desired sub-pixel coordinates of the molecule are obtained as \hat{x}_{p}=\hat{x}_{0}+\tilde{x}_{p} and \hat{y}_{p}=\hat{y}_{0}+\tilde{y}_{p}, where \hat{x}_{0} and \hat{y}_{0} define the sub-pixel refinements of the coordinates obtained by the methods described below.