How to combine 2 maps in different projections? (new pictures from NASA)?

Question

satellite navigation

NASA has updated its famous Night Lights :
eoimages.gsfc.nasa.gov/images/imagerecords/79000/7... (lowres)
eoimages.gsfc.nasa.gov/images/imagerecords/79000/7... (highres, 300 meters)
I want to overlay a fresh map from NASA on my map:

Above is my map, which I rendered based on geo-coordinates in the mercator projection (there was not enough America in the database, so there is a hole in its place), and below is the NASA map that I tried zoom as close to your map as possible. But even without an overlay, it is clear that closer to the poles it “bends”, in contrast to the rectangular mercator.
The question is how to superimpose these 2 cards with maximum accuracy? You cannot edit your map, since data is plotted on the map based on the coordinates in the mercator.

Answer 1

I would suggest to honestly recalculate the coordinates.
The NASA map appears to be in latitude-longitude projection, i.e.

 x = width / (2pi) * lon
 y = height / 2 - height / (2pi) * lat

where width, height are the width and height of the image, lat, lon are the latitude and longitude in radians.
It remains for you to express lat and lon through your Mercator formulas, I suspect that this will be

 lat = atan(sh(c * (height / 2 - y) / height))
 lon = x / width * (2pi)

where c is your scaling factor, and iterate through all the points on your map, converting their coordinates to lat/lon and those to NASA map coordinates.

Answer 2

Most likely you will find a solution here www.gdal.org , in particular, gdalwarp - image reprojection and warping utility. And here the GdalShell
tool can be useful, but projection parameters are required.

Answer 3

Almost any GIS program will help you (QGIS, ArcGIS, GRASS, MAPINFO, etc.) if you don't understand, ask on the gis-lab ( gis-lab.info/).