Zernike Analysis and Synthesis Basics

I invested long time in instrument building and zernike graphics are no different from standing wave patterns on a vibrating soundboard. I want to learn are these two different technology have a common in correcting the unwanted deformation and I want to learn how 60 or more different pattern and their each formula could be used to correct everything to a minimum. I found many technical papers which tells the analysis but not told synthesis after analysis to reach a goal. I think there might be a way to control the these different polynomials and there must be a corner stone but I have no idea. Please shed some light.

Thank you ,

Mustafa Umut Sarac

Istanbul