For working with raster images, get a piece of GIS software that supports reprojection of images (The GDAL package has good ones, but it's no GUI: QGIS has a GUI). This piece of software will get you into and out of your stereographic projection. Then find a package that will invert your image with respect to a circle ( https://github.com/synesthesiam/invert claims to be able to do this sort of thing, but I have not used it ) and use that on your intermediate image.

Vector data would be somewhat more complex because changing the points in this manner would most likely require a high-density data set and/or interpolation along the segments to avoid significant visual distortion.