Allen Cell Discussion Forum

Spherical harmonics for hexagonal cells

I have tried using the spherical harmonics method for retinal pigment epithelial tissue cells . The cells are hexagonal and spherical harmonics seem to not capture this shape easily due to the existence of sharp edges. Is there a modification to the method that can capture this information?

Also, 3-5% of our cells may have multiple nuclei. How do you deal with the parametrization since the method is based on single nucleus?

I would greatly appreciate if there are any suggestions or leads that you may share!

Hi Pushkar.

Thank you for your message.

The first thing to try is to increase the spherical harmonics expansion degree. Currently we use L=16 for cell and nuclear shape but you can increase this number.

Other than that, there is a generalized version of spherical harmonics expansion in which not only the radial coordinate R(theta, phi) is parameterized as a function of theta and phi, but instead, all the three cartesian coordinates X(theta, phi), Y(theta, phi) and Z(theta, phi) are parameterized. This is not currently implemented in cvapipe_analysis but I can point you to some resources if you like,

Sorry, I forgot to reply the 2nd part of your message.

The SHE parameterization is not suitable for objects with multiple connected components. I would suggest you to either discard those cells since they are a small fraction of the whole population, or keep only the largest nucleus in those cells.


Hi Matheus,
Thanks for your replies. I have tried upto L = 50 and overall the reconstructions definitely get quite a bit better, but still struggle with edges. The primary concern stemmed from the fact that the average cells do not capture the polygonal nature of cells. I can see how the components can build up to an almost polygonal cell but the mean cell seems quite round. This in my opinion is both because of spherical harmonics struggling with sharper edges and rotation method in our lower resolution images (due to discrete pixelated nature of data).

Cartesian coordinate based SHE seems to be an interesting approach. I would really appreciate if you can point to some resources on that topic for me!

I agree with the methods you described for multiple connected components. This was what I was thinking but wanted to make sure.