How do I extrude a black & white bitmap with a free 3d modeling program?

The way I learned to do this was in a tutorial for Maya, but surely the principle is the same. You set your bitmap as a background image. You know that you can't directly extrude the bitmap, I gather. How I learned is that you get a mug shot of a face (or draw). A profile and a frontal view. You set one as the background of one plane, and the other as the background of the other plane. Then you draw lines around your frontal view - whatever lines you like, polygon stuff is what I saw. Then you turn to the other plane and extrude those lines out to match your profile view. Basically you draw outlines on the frontal view and then pull them out to match the profile contours and there you have the beginnings of a face. It's just how to start, there is much to do after that.

It would be easier to extrude a vector image - not a bitmap.

You can do the extrusion in Blender, which is free. http://www.blender.org/

To draw the vector shape, you could use Inkscape which is free http://www.inkscape.org/ - you could paste your bitmap image into inkscape, and use it as a guide to draw the vector image. Or you could use inkscape's trace function to generate a vector from the bitmap image. You could then import the vector SVG image produced by Inkscape directly into Blender where it can be extruded.

Tutorial Inkscape SVG import into Blender http://www.youtube.com/watch?v=h78qN3QcF7o