Ellipse Regression

As with the circle, it is possible to perform both algebraic and geometric minimization.

The quadratic equation of an ellipse is

$$f(\mathbf{x}) = \mathbf{x}^\top \mathbf{A} \mathbf{x} + \mathbf{b}^{\top} \mathbf{x} + c = 0$$ (3.91)

where $\mathbf{A}$ is a symmetric, positive definite matrix. In this case as well, the solution to the homogeneous problem (3.91) allows us to derive the 6 unknowns (known up to a multiplicative factor) of the system.

The nonlinear solution that minimizes the geometric quantity can be obtained using the parametric representation of the ellipse

$$\mathbf{x} = \begin{bmatrix} x_0 \\ y_0 \end{bmatrix} ... ...{bmatrix} a \cos \varphi \\ b \sin \varphi \end{bmatrix}$$ (3.92)

where $(x_0,y_0)$ represents the center of the ellipse, $(a,b)$ the lengths of the two semi-axes, and $\alpha$ the rotation of the ellipse with respect to the center. As with the circle, the $\varphi_i$ will be auxiliary variables, and the nonlinear problem becomes one of $5+n$ unknowns with $2n$ equations.