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I'm in need of a mapping software which, given a map, lets you calibrate it (choose which geographic coordinates project to which pixels on the map image) and then given a database of cities, you can navigate to the specified city on the map you just calibrated.

To reword the question, I need a mapping software that can calibrate my map image and accepts some kind of database format to display cities on that map.

Where to find such compatible map software and database?

Preferably free and open source software and data.

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Combining spatial maps with databases is the heart of GIS. Practically any GIS will allow you to do both of those things in a variety of ways - though a desktop interface, programatically, served remotely, etc. Here is an excellent list of free and open source desktop GIS packages. QGIS, with a powerful collection of tool and an easy-to-use interface, is particularly popular on this site, and you will be able to find plenty of help for any questions you may have about it.

If you need something for web mapping, another post has a basic introduction, including a list of FOSS tools.

If you need to perform these tasks programmatically, look into GDAL and OGR, two libraries for manipulating geospatial data.

So, nearly any GIS can do those tasks easily. Geospatial data is typically either vector, with lines, points, and polygons, or raster, with a grid of cells. Any of the above GIS tools can handle both, but the process will be different for each type.

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You can download city data from Natural Earth Data site:

You can use all files from that site with QGIS but any other open source GIS should be fine.

There are many possible ways to navigate to a given city. For example, QGIS "Find by attribute" plugin lets you search for a city by it's name and automatically zoom there using a set zoom level.

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I use Autodesk Raster Design It is not free or cheap.
I save my image to geotiff which makes it work in all the gis software I use.
You have to have autocad map 3d or civil3d to be able to geo-reference the images to a coordinate system.
there are several methods for referencing.

Next the city database part. This is tough to answer because of the vague nature of your question.
Generally there are lots of free acccess to point information (which is all it would take to accomplish what your question said) both online and downloadable.
You can access that information in an online fashion as stated or load it into your favorite (as soon as you decide what your favorite is) dbms.
Your favorite dbms is going to interface with your NEW favorite GIS software! (johnny tell him what he won!).
This might be a programming interface like fwtools, python or a full blown UI like qgis, or arcgis.
Decisions decisions.

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