[Insight-users] boundary conditions using non linear optimzers

[Hartmann Philipp]

Hello,

I am trying to fit a parametric curve into a point set using a least
squares cost function. The parametric curve has the following form:


X= a + t*b +(t-l)^2 *c

X,a,b,c are element of R^3

l is element of R
t is element of the interval [tmin, tmax]

the cost function is the sum of all squared distances of points from the
curve

Sum((min(P_i-X(t)))^2)

Parameters to optimize are: tmin, tmax, l, and the components of a,b,c

The Problem is now the boundary conditions which are desired. I need a
maximum extent of the interval [tmin,tmax]

=> abs(tmin-tmax)<=Z

And I need the conditions

Norm(b) = 1 
Norm(c) = 1

Implementing the cost function is easy, but I don't know how to
implement the boundary conditions. Of course I thought about putting a
high "punishing" value into the cost function, if one condition is not
satisfied, but I hoped, there was a "cleaner" way, which allows the
optimizer to anticipate the boundaries and to step more intelligently
through the search space.

Regards,

Philipp Hartmann