The easiest way to do this is to ask the author of the paper for the data. This could take awhile. If you want to do it yourself, you can use GIMP or Fiji to get the pixel location of the points and the axes on your bitmap, then convert the points to actual positions using the axes values. I have done this with GIMP, using its measure function, and it works, but is tedious. Fiji is an image processor, so you should be able to calibrate the bitmap and just read off points. I have never done this with Fiji, but have done similar things.
Note that both GIMP and Fiji (ImageJ) are free and open source. Send the developers a bit of cash if they prove useful, so we can keep enjoying their efforts.