I have an algorithm that segments depth images using surface fitting. At the moment the algothim uses least squares polymonial fitting, but polynomials are not powerful enough to fit the shapes that are in these images. I replaced the explicit polynomial $z = f(x,y)$, with the implicit fitting problem $f(x,y,z) = 0$. Where $x$ and $y$ are the pixels location and $z$ is the value of the pixel and $f$ is a polynomial. A classic problem with many nice linear solutions. This did make the fitting much more powerful, but this left me with a problem I have been unable to solve for a while now. Once I had the least squares solution to the implicit poly, I had to solve for z, not easy at all, but I could search for the minimum (there are only 256 possible pixel values to search through after all). THE problem is that there was more than one solution for z! Where a pixel can only have one value. That is, once the implicit poly was solved for z there were several roots.

So my question is;

How do I formulate a least squares minimization problem for a single root? What contraints must I add?

This might not be completely clear so here is an example: You have a set of noisy data points $\{x,y\}$ that form a semi-circle about the origin in the positive y only. I want to fit $f(x,y) = 0$ to the data points, or to be pricise, a circle. $f == x^2 + y^2 + c = 0$.

The least squares minimization problem is a nice simple linear $\min_c \sum_{i=0}^n (x_i^2 + y_i^2 + c)^2$ but I am only interested minimizing the datapoints distance to the positive half of the circle. Solving for $y$ and taking the positive root gives us $y_i = \sqrt{-c-x_i^2}$. Giving us the acutal minimization problem as $\min_c \sum_{i=0}^n (y_i - \sqrt{-c-x_i^2})^2$ NOT a nice linear problem at all. These two minimisation problems will give differnt results since the negative half of the circle should not try to fit itself to any data points. How can i constrain the linear problem so that it is an equivelent to minimizing to a single root?

Or in general constrin $\min_{ijk} (\sum_{i,j,k}a_{ijk}x^iy^jz^k)^2$ to a minimization to a single root in $z$?

I hope this makes sence and sorry for the length but this has been annoying me for weeks.