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Structuring Element or Morphological Op to extract thick lines?

Is the a morphological operation I can use to preserve my thick lines in the below image but also remove thin lines/noise? I have tried the regular approaches such as; close the image, open the image, erode the image n times then dilate the image n times. The approaches either eat into my thick lines too much (creating breaks in them) or don't remove enough noise.

So I am thinking I am maybe not aware of a structuring element or morphological operation that is good for preserving thick lines whilst removing noise/thin lines.

image description

If I threshold I get all the thick lines I want but I also get noise:
image description

My desired result is:
image description

Is there a good structuring element or morphological operation that is good for preserving thick lines whilst removing noise/thin lines?