Bringing images into alignment I observe a scene with two cameras, c1 and c2, that produce two images i1 and i2, respectively. What I now want is to bring i1 and i2 into alignment, that is I want to know where pixel (x,y) in i1 is in i2. This "alignment-matrix" should be valid in general, that is not only for i1 and i2, but also for further images of the cameras c1 and c2 as long as they haven't been moved. I believe this problem is called "correspondence problem". 

I have now read quite a bit about camera calibration, image rectification, essential and fundamental matrix. But there are still open questions and I would not know how to achieve my task.

So, the question basically is:

The essential matrix E and the fundamental matrix F both only give constraints for the correspondence problem (point p in i1 must lie on line l in i2). How do I actually solve the problem?

Thanks a lot for your time and answers!
For 2D cameras, you can't.  There's a reason why the matrices only give constraints that let you say that point p in i1 must lie on line l in i2 -- that's because that's all you can say.

In particular, for a particular point p in image i1, with 2D cameras, you don't know the depth of the corresponding object.  The depth affects where it appears in image i2, so without knowing the depth, you can't know which pixel in i2 corresponds to p.  If the object is very close, then there might be a very large offset; if the object is very far away, there might be a very small offset.

This is in fact the basis of binocular vision.  Some 3D sensing methods work by somehow finding the correspondence using various heuristics (e.g., image similarity metrics), and then using this to infer the depth of the object.

So, the intrinsic and extrinsic parameters are not enough to uniquely deduce the correspondence, if both cameras are 2D cameras.  With a 3D camera, this problem can be solved.