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I need to use this map (PNG file) in which each color is a population density:

enter image description here

I've already located it using Georeferencing. Now I need to convert it to shapefile with polygons, and narrow down its colormap to 5, using ArcMap 10.3.

How can I convert it to polygon, with the right tolerance (because conversion from raster tool just made too may polygons), and how to introduce line as boundaries, not another polygon. And after that, how to classify about 200 colors in the image into 5, and add value to them?

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If this is the only map that you want to process,then the easiest way to convert this raster into polygon in a clean way is to digitize it, since the map that you are showing is not complex map. Create a new field of "population density" and fill in the values based on the information in the legend. If you have an administrative boundary for this shapefile for this region. Then you can use this shapefile as background base to create the population density map.

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