Roughness of 3D Surface

Hello everyone,

I would like to calculate the salience (or roughness) from a 3D object recovered from spacial cordinates (x,y,z) cloud. The data represents a closed 3D rough surface object, meaning that the noise/roughness is not unique to the z component, so it cannot be represented just as a projected z(x,y) surface. It is a true closed shell data set in euclidian space.

The objective is to watermark zones where there would be high local noise value. I've done this with z(x,y) surfaces, but I could not find, at first, if there are any efforts in that direction within the community.

If anyone have pointers regarding this assessment I thank you in advance !

[KSSV]

Any pictorial example?

[Matheus de Lorenzo]

Hello, not a matlab imaging, but here there are two images of the same surface. The one on the left is meshed in such a way that the data can be represented as a z(x,y) function, while the one on the right is a true 3D convex surface where if we tried to do a z(x,y) function we would have more than one point in z present for the same (x_i, y_i) plane coordinate (and other projection losses). The characterization of roughness of the case on the right is on point to what I am looking for.

Answer 1

[Bjorn Gustavsson]

There are a number of hits returned when searching for "curvature" on the file exchange. The gptoolbox seems to have a bunch of tools for topographical(topological?) tasks. Some might be useful for your task. Perhaps this one: curvature-estimationl-on-triangle-mesh - if you have a general surface you ought to be able to approximate it with a triangle-mesh (not topology-general, but "every-day general"...)

HTH

[Matheus de Lorenzo]

Hello Bjorn, changing the keywords like roughness/salience to curvature indeed brought a new perspective of this problem not only in mathematical functions but also in the articles found. Thank you very much.

[Bjorn Gustavsson]

My pleasure. I guessed that your roughness/salience would be some kind of local average of some kind of function (absolute value?) of the local curvatures - happy I guessed right enough...

[Bruno Luong]

Not really roughness is defined as statistical quantities of the distribution of z(x,y). It does not required the surface has the curvature (eg step surface has roughness defined but not the curvature).

[Bjorn Gustavsson]

@Bruno Luong: I learnt something today. For what I could gather all different definitions seem to require the surface be single-valued (z(x,y)) and not wrap over itself - which to me seems like a curiously rigid design choise, but that is the industry-standard. To me it seems more general to use some local measure of the curvatures. (and on a discrete sampling there's no such thing a a step-function?)

[Bruno Luong]

Industrie standard defines roughness for various measure instruments (optical / mechanical). Mostly those instruments measure profile z(x,y), I'm not aware any instrument that can currently measure a close surface in a single shot. The surface low frequency must be filterred out for roughness calculation, thus the over shape of the object doesn't matter in the final result.

One of the most used roughtness measuremen is the first moment of z(x,y), it's called Ra/Sa, or more or less

Sa = integral abs(z(x,y)-zmean) ds / area

after detrend z(x,y). You can google "surface roughness" to learn more.

[Bjorn Gustavsson]

So in the case where one has a more general surface that might "wrap over itself" one would then make some kind of re-interpolation into the form z(x,y) where the z-values are the max of the possibly multiple values at each x,y? It seems that would imply that such a mathematical surface might have different roughness from either side.

[Bruno Luong]

The roughtness is a statistical metrology quantites, there is nothing strange where you have different roughtness in either side, or even change wrt the macro-region under consideration.

[Bjorn Gustavsson]

If you were to have a boundary like this (trivially extended to 2-D for rigors sake):

Then the roughness-measures would certainly differ.

[Bruno Luong]

???? differ to what?

The roughness is measured by an probe that scan (x,y) and for each position the probe measure the heigh z(x,y). The roughness is derived from the data z(x,y) and intended to be used to access the quality/characteristics of the polishing subject by surface such as optical surface, tribology surface , etc.... It's an industry metrology standard, not a rigouruous math definition that defines for fancy surface like yours.

[Bjorn Gustavsson]

That is not lost on me.