Create large amount of random points in binary raster

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I want to create a point vector dataset of 10000 points (or larger) within a binary raster, where the points should be constrained to areas where the raster value is 1.

I tried the following steps.

1. Polygonize raster
2. QGIS: Vector -> Research Tools -> Random Points

This works fine up to 2000 points but anything above just causes QGIS to crash.

Is there a way to create a vector dataset with a large number of features constrained by a binary raster (or the polygonized version of it)?

The following tools are at my disposal, ranked from most to least favourable: QGIS, Python, R, ArcGIS

This is what I'm going for, only with 10x the point features.





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