Mathematica is a good tool for this problem. I can see two options, whose usefulness will depend on what you want the information for. The main problem is that your initial data is rasterised. In my first suggestion the image is left rasterised, in the second it is vectorised.
1. Morph the image (the easier and less general method). Find a mapping from Mercator[x,y] ->Robinson[x,y], probably using Mercator[x,y] ->(lat,long) ->Robinson[x,y]. GeoGridPosition is a useful mathematica function here. Wikipedia has the mappings if you want to roll your own: https://en.wikipedia.org/wiki/Robinson_projection#Formulation and https://en.wikipedia.org/wiki/Mercator_projection#Mathematics_of_the_Mercator_projection
. Then morph the image using ImageTransformation and your mapping. If you want to overlay this ocean pollution data with other maps (ocean currents etc.), you might want to use the ImageAlign function, depending on the images.
2. Perform some image processing first. Identify regions of interest (lots/absence of pollution etc.) by their colour on the map, using RegionBinarize or similar function. Then vectorise the data. This is key for making the data easy to use elsewhere. You can do it just by creating a polygon between vertices located at the centres of each boundary pixel of the interesting area. Then map the polygon onto the geodetic sphere and then to the Robinson projection. The end result will be a polygon in the Robinson projection, which you can overlay onto a map or other data (ocean currents etc.) in any graphical style you want: colors, opacities, textures etc. If you have multispectral information in your original image (different kinds of pollution being represented by different colours), you can do repeat this process for each colour, ending with a bunch of polygons that you can combine and overlay in different manners. Alternatively you could set different thresholds in RegionBinarize and end up with a kind of contour map of pollution.