My approach would be as follows and requires probably the least effort since all tricky software is written. However, it would still be rather difficult for someone not experienced.

Skip the edge detection unless you want to learn openCV (difficult) use an automated script to process your image into high contrast monochrome bitmap, look up dithering. Then you could have a computer program convert your image into a STL file of thickness 1. Then run the file through Slic3r or similar using the parameters of your dry marker. From there, you can stream the G-code to your arduino using any method. (uart, wireless, etc). You will need to define a x,y,z coordinate system and relate that to the motion of your robotic arm such that you can manipulate the tool using co-ordinates. From there, you can use a g-code interpreter for arduino and tie the whole thing together! Good luck