Radial symmetry

This algorithm finds the sub-pixel position of a molecule by determining the point with maximal radial symmetry in the data as described in [1]. The general idea is to find the origin of radial symmetry (i.e., the center of a molecule) as the point with the minimum distance to gradient-oriented lines passing through all data points. The calculation of each molecular position is very fast due to an analytical solution, but the algorithm does not estimate the intensity or imaged size of a molecule. Radial symmetry is a robust feature in SMLM data, making the algorithm resistant to noise.

The calculation starts by determining the intensity co-gradient

\nabla\tilde{I}_{x,y}=\left[\frac{\partial\tilde{I}\left(x,y\right)}{\partial u% },\frac{\partial\tilde{I}\left(x,y\right)}{\partial v}\right]^{\top}

for every integer point \left(x,y\right) from the set \mathcal{D}. Here uv coordinates are rotated by 45^{\circ} from the xy coordinate system of the image, because partial derivatives are determined using the Roberts cross operator [2] as

\displaystyle\frac{\partial\tilde{I}\left(x,y\right)}{\partial u} \displaystyle= \displaystyle\tilde{I}\left(x+1,y+1\right)-\tilde{I}\left(x,y\right)\,,
\displaystyle\frac{\partial\tilde{I}\left(x,y\right)}{\partial v} \displaystyle= \displaystyle\tilde{I}\left(x,y+1\right)-\tilde{I}\left(x+1,y\right)\,.

The computed co-gradient \nabla\tilde{I}_{x,y} corresponds to the point A_{x,y}=\left(x+0.5,y+0.5\right) and the slope of a gradient-oriented line passing through the point A_{x,y} is, in a xy coordinate system, given by

\mathit{s_{x,y}}=\left(\frac{\partial I\left(x,y\right)}{\partial u}+\frac{% \partial I\left(x,y\right)}{\partial v}\right)\left(\frac{\partial I\left(x,y% \right)}{\partial u}-\frac{\partial I\left(x,y\right)}{\partial v}\right)^{-1}\,.

The origin of the radial symmetry \left(\hat{x}_{0},\hat{y}_{0}\right) can be determined as the point that minimizes the sum of weighed distances of all considered lines to that point. It can be shown [1] that the problem has an analytical solution. For simplicity, let index j=x\left(2r+1\right)+y, where x,y\in\mathcal{D}, thus s_{j}=s_{x,y}\,, w_{j}=w_{x,y}\,, and b_{j}=b_{x,y}\,. The analytical solution is given by the following equations

\displaystyle\hat{x}_{0} \displaystyle= \displaystyle a^{-1}\biggl[\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}b_{j}}\Bigr)% \Bigl(\sum_{j\in\mathcal{D}}{w_{j}}\Bigr)-\Bigl(\sum_{j\in\mathcal{D}}{w_{j}b_% {j}}\Bigr)\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\Bigr)\biggr]\,,
\displaystyle\hat{y}_{0} \displaystyle= \displaystyle a^{-1}\biggl[\Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}b_{j}}\Bigr)% \Bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\Bigr)-\Bigl(\sum_{j\in\mathcal{D}}{w_% {j}b_{j}}\Bigr)\Bigl(\sum_{j\in\mathcal{D}}{s_{j}^{2}w_{j}}\Bigr)\biggr]\,,

where a=\bigl(\sum_{j\in\mathcal{D}}{s_{j}w_{j}}\bigr)^{2}-\bigl(\sum_{j\in\mathcal{% D}}{s_{j}^{2}w_{j}}\bigr)\bigl(\sum_{j\in\mathcal{D}}{w_{j}}\bigr)\,, b_{j}=y-s_{j}x\,, and w_{j}=\frac{\tilde{w}_{j}}{s_{j}^{2}+1}\,. The point-to-line distances are weighted by the factor \tilde{w}_{j}=\tilde{w}_{x,y}=\frac{\bigl|\nabla\tilde{I}_{x,y}\bigr|^{2}}{d% \bigl(A_{x,y},C\bigr)}\,, where d\bigl(A_{x,y},C\bigr) is the Euclidean distance of point A_{x,y} to the centroid C. The centroid is computed analogously to centroid estimator but from the gradient magnitudes \bigl|\nabla\tilde{I}_{x,y}\bigr|.

The implementation of the radial symmetry method is taken from [1], where the partial derivatives are additionally smoothed by a 3\times 3 averaging filter in order to reduce noise and improve the accuracy of the results.

References

  • [1] R. Parthasarathy(2012) Rapid, accurate particle tracking by calculation of radial symmetry centers, Nature Methods 9 (7), pp. 724–6. External Links: Document. Cited by: Radial symmetry, Radial symmetry, Radial symmetry.
  • [2] M. Šonka, V. Hlaváč and R. Boyle(2007) Image Processing, Analysis, and Machine Vision, 3rd edition edition, Cengage Learning. Cited by: Radial symmetry.