Hello Alan,
reproject fundamentally isn't that fast, it's doing a lot of
coordinate transforms and interpolation. Also nobody has spent any
significant time profiling it and improving it's performance.
That being said, I think that your main issue here is that you are reprojecting to a very large array, and one which (I think) is oversampling your input image by a significant margin.
If you calculate the angular width of a disk centre pixel in
heliographic carrington coordinates you get approximately 0.12
deg, in your example you are reprojecting to a carrington map with
a pixel width of 0.02deg which is a significant oversampling. If
you reduced the size of your array by about 3x you would get a
significant speed improvement.
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