## Radial-Symmetry-Based Particle Localization in 3D

There are imaging applications for which it's useful to localize spherically symmetric objects in three dimensions. The radial-symmetry-based algorithm I developed ([1] and [2]) is derived for two dimensional images. It is very easy, however, to extend it to three dimensions, calculating intensity gradients and the point of minimal distance to them in a three-dimensional space. A scan of my inelegant notes from March, 2012, deriving the localization expression is here; on page 4 is a simple matrix equation that gives the center of a 3D cloud of intensity.

A MATLAB function I wrote that implements this is here. Feel free to use it! It allows for the pixel size to be different in "z" than in "x" and "y," as is typically the case in confocal and light sheet microscopies Please note that unlike the 2D case, I have not extensively tested the 3D algorithm, and I haven't looked at issues like sensitivity to asymmetry. These are interesting, but I never found the time or inspiration to investigate them and think about this thoroughly! Thankfully, the world is a large place: in 2013, Liu et al. published a paper on the 3D radial symmetry algorithm, which I think does the same thing my extension does. (I haven't checked the math). The link is Liu, S.-L. et al., *Sci. Rep.* **2013**, 3.
If you use my function: I recommend spending some time testing simulated images with asymmetry or other properties you might expect. I hope you find it function useful, and I'd be very interested in knowing if it works (or doesn't work) for your applications!

Feel free to email with any questions or comments: raghu [at] uoregon.edu