I'm looking for a function that can perform something similiar to the Euclidean Allocation process in arcGIS (http://resources.arcgis.com/en/help/main/10.1/index.html#/Euclidean_Allocation/009z0000001m000000/) but using open source tools-- ideally Qgis or Gdal.
Alternatively, if there is a function that can take a polygon layer (e.g. map of the world) and run an algorithm similiar to Thiessen polygons-- that is, fill in the gaps in the extent using the nearest polygon that would be most helpful. For this latter solution, I'd prefer if I didn't have to convert boundaries to points.
Answer
Euclidean Allocation
The only tool I know of, that specifically supports Euclidean Allocation based on euclidean distance calculation is Whitebox (see also their Help section).
Region Growing
I don't know the exact problem you are working on, but my guess is you can also solve it with region growing - a process quite common in image segmentation and mostly based on euclidean distance allocation. Here are two tools supporting region growing algorithms.
QGIS: Semi-Automatic Classification Plugin - as well as the source code if you want the Python code
ORFEO Toolbox - as well as their QGIS integration
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