Regression on a Conic

It is clearly possible to generalize the regression of the parabola, the circle, and the ellipse to any conic (section 1.6) that is arbitrarily oriented.

Let $(x_i, y_i)^{\top}$, with $i=1, \ldots, n$, be points affected by noise belonging to the set of points to be estimated.

The equation (1.56) can be rewritten in the form

$$\mathbf{a}_i^{\top}\boldsymbol\beta=0$$ (3.93)

where $\mathbf{a}_i=\left\{ x_i^2, x_i y_i, y_i^2, x_i, y_i, 1 \right\}$ and $\boldsymbol\beta=\left\{ a,b,c,d,e,f \right\}$ from which it is evident that to obtain the parameters $\boldsymbol\beta$ of any conic, one can proceed by solving a homogeneous problem of type $\mathbf{A}\boldsymbol\beta=0$ in 6 unknowns, minimizing a quantity of the form

$$S = \sum_{i=1}^{n} \mathbf{a}_i^{\top}\boldsymbol\beta$$ (3.94)

Such a solution clearly minimizes an algebraic error rather than a geometric one; therefore, this is not the optimal estimator.

An alternative formulation for deriving the parameters of conics can be found in (FPF99).

Finally, to determine whether a point is close to the equation of a conic, or to obtain a geometric approximation of the point-conic distance, one can calculate the Sampson error (section 3.3.7) by leveraging the fact that, for a conic defined by equation (1.56), the gradient of the variety takes on a very simple form to compute:

$$\nabla f (x,y) = \left( 2 a x + b y + d, b x + 2 c y + e \right)$$ (3.95)