Re: a rigid transformation problem

In article <1142408468.680146.193720@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<jayzhuo@xxxxxxxxx> wrote:
given two point sets {pi} {qi}, determing the rotation matrix R and
translation matrix T, so that the error E is minimized:

E = square sum of d ( R*pi + T, qi)

Do you mean sum of the squares?

the distance is defined as follow:

d ( p, q ) = || p - q|| if || p - q|| < M
M otherwise

where M is a constance.

That could be rather nasty, as it makes the objective function
non-differentiable. Would a smooth cut-off such as
d(p, q) = M tanh(||p - q||/M) work for your purpose?

how to solve this problem?

Using a numeric solver.
You have your choice of parametrizing the rotation matrix, e.g.
in terms of Euler angles, or using constraints R' R = I and
det(R) > 0.

Robert Israel israel@xxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada


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