Non-linear approximation of an image of a geometry
I am trying to approximate different shapes of a weld bead geometry cross section in additive manufacturing with a graph or ideally (but not necessarily) a function. The regions are the outer shape as well as the individual layers. (see following images)
Therefore, I applied some pre-processing methods to extract the relevant pixels which represent the geometry of a weld bead which are shown as white pixels. (see third image) I derived this image with canny edge detection and multiple morphological operations such as closing erosion and dilation prior to that and of course converting it into grey-scale.
The "noisy" areas are the transition areas between individual layers of metal and only show up in this way, so in general there is not a "better" or "sharper" transition in thus less "noise". Pictures 3 and 4 are an example of some of the image pre-processing methods I used.
My main approach to treat the inner geometry so far was to split up the image in several sub-images and perform least squares regression on each individual one by interpreting the white pixels as data points. Afterwards I've stitched all those little approximation functions back together to form the image of the original size. I've tried it with different sizes of those sub-images. (see pictures 5 and 6)
However, this approach produces jumps between the functions as well as functions next to each other where the pixels or data points in my case should only be approximated with one function (see attached image). My next approach would be to use multivariate adaptive regression on the sub-images.
Thus, I'm asking if anybody knows a better solution for my problem, maybe even for an approximation on global scale without splitting the image into the sub-images. The approximation does not need to be a polynomial function, piece wise linear but connected functions are totally sufficient. I would be thankful if anybody knows a method that is at least capable of achieving what I want to do. Whether a pure non-linear regression method. Unfortunately I don't have many images (only 64), hence I don't think I can use an ANN. (please correct me if I'm wrong)
If you need to take a look at my code, just let me know. Thank you! :)
The best I could obtain is with bilateral filtering for denoising, then adaptive binarization.
And on a reduced image:
