Finding the surface RMS error using Zernike coefficients

[Monsterboy]

Hello,

I am given 200,000 surface points in terms of x,y and z coordinates which is supposed to represent a circular paraboloid antenna reflector surface and another 200,000 points when the antenna surface is subjected to wind loads and gravity. I am supposed to find the surface RMS error of the surface using Zernike coefficients. I have looked through a lot papers but i am not getting a clear methology to do this, can anyone help me out ?

I am new this Zernike cofficients and polynomials and stuff and i don't know whether i am upto this work. Perhaps i need to study right from the basics but i don't know where to start. Suggestions will be appreciated.

 

[Mech_Engineer]

I've seen Zernike polynomials used for describing optical surface error in the past, this sounds very similar. A recent paper I saw published on the topic goes through fitting methods for Zernike polynomials for point clouds as well as shows some examples of some super high-order polynomials for diamond-turned optics. Take a look and see if it helps.

SPIE Digital Library- PROC SPIE 99610P: Zernike polynomials for mid-spatial frequency representation on optical surfaces

Abstract
Mid-spatial frequency structure on freeform optical elements induces small-angle scatter and affects performance. Fabrication techniques involved in making freeform surfaces leave tooling marks on the surface due to the sub-aperture nature of the fabrication process. In recent years, there has been a growing need for specification and characterization of the mid-spatial frequencies for freeform surfaces. There are a range of methods to consider for representing the midspatial frequency content: the power spectral density (PSD), the structure function (SF) and a polynomial basis representation such as Zernike and Forbes Q-polynomials, as examples. In this paper, we investigate a Zernike polynomial representation for quantifying the mid-spatial frequency content in height maps. We will show fit coefficients of synthesized and real data sets to Zernike polynomials from low orders to very large orders. We also illustrate how this polynomial representation captures certain characteristics of the mid-spatial frequency error. The results are analyzed and compared with Forbes gradient orthogonal polynomials. Finally, limits of Zernike polynomials for representing mid-spatial frequency content of the surface are discussed.

 

[Ranger Mike]

My question to you is Why are you measuring RMS? This surface texture parameter was dominant in the 1950s manufacturing process. It is geometrically weighted parameter that does not tell the whole story of anyone particular surface. It was replaced by AA or Arithmetic Average that eventually became Ra or Roughness Average. Look up the ANSI B46.1 standard called Surface Texture (Surface Roughness,Waviness and Lay) available from American Society o Mechanical Engineers. The RMS is now called Rq by the way. Rq typically is weighted and runs about 10% higher than the Ra calculation.

Man, this is a hoot. The longer you live the more things don’t change. In the 50s and 60s the machine tool world was starting to produce machined surfaces to the point they could not be measured effectively with your finger nail or optically. The first mass produced instrument to do this was the Profilometer invented in Ann Arbor by Doctor Abbot in 1938.

To use a current trendy term…. Long story short - the above cloud of points seems very similar to discussion we had back then to try to describe a squiggly line that represented the actual Profile of the surface.

We used a LVDT gage head that would output to a thermal strip chart recorder so we had a 2D graph of the surface. Imagine this scenario but with hundreds of profiles graphed out.

The whole point back then was “ How do you qualify a profile for the application” “ How do you calculate the proper squiggly line for the assembly fit or function of the part you just machined”?

This is why industry developed over 120 different surface texture parameters. We had RMS that was replaced with CLA, AA and finally Ra. Rt, Rz, Rtm… 120 of these parameters that would apply to the application.Looks like the point cloud situation brings up the same discussion. How to capture the unrendered cloud of points and mathematically qualify it?We did not have computers back then so we worked on the strip chart “assuming “ the rest of the surface was going to be similar. The surface had peaks and valleys and the data could be assessed as the Least Squared center line that went thru each data point.

This was the easiest and most common evaluation. We could calculate from two parallel lines that captured all the point cloud data and look at the distance between these two lines. There are as many algorithms as you can imagine

How do you describe a cloud of points? Same as you describe a squiggly line but there’s more of them.Metrology Humor…its and inside joke for us metrologists.