Linear Least Squares and SVD

I'm trying to improve alignment between a 3D scan and a virtual model display. Right now I'm simply using the guess-and-check method, which doesn't work well enough. I think there is some shearing (skewing) in the transformation, and the guess-and-check method doesn't account for that. So I'm going to go a little more robust and use SVD. It took some time to find a good explanation of how to use SVD, and here's a nice tutorial with lots of detail. If someone knows of another good one, send it my way (or post a comment).

Link (pdf)

Further than that, I'll be using the basics of this approach to set up the linear system:

http://groups.google.com/group/sci.engr.surveying/msg/45e29b51626626ec

I think his comments about how many points you need are incorrect, especially if one set of points is skewed. I've seen documents that state 6 points are needed and some others that say 7 are needed. At least they all agree that the more you have, the better, so I'll probably sample 10 or more points. However, the basic idea of how to setup the system seems correct. Toward the bottom of the post, there is this line:

R ={INV [P'P]} * [P'U]

Where P' is P transpose. Well, Inv(P'P) * P' is the pseudo-inverse, and we can calculate that using SVD, as shown in the tutorial linked above. So, given a noisy set of points and a corresponding set of points in a different coordinate system, we can find what the (best) transformation is between those points.