Minima, Maxima, and Saddle Points

As written, the model presented earlier applies to both minima/maxima as well as saddle points. However, this model does not account for any local rotations that the function may undergo. While such rotation is, in the first approximation, negligible for minima and maxima, it can be quite significant in the case of saddle points.

The version of the equation (1.94) that considers possible rotations of the axes is

$$m_0 x^2 + m_1 x + m_2 y^2 + m_3 y + m_4 x y + m_5 = z$$ (1.97)

The system is fully compatible with that shown in the previous section, with the only difference being that now there are 6 unknowns, thus requiring the processing of at least 6 points in the vicinity of the minimum/maximum/saddle point. Again, there are no remarkable solutions, but it is advisable to factor the matrix of known terms.

The gradient of the function (1.97) vanishes at the point corresponding to the solution of the linear system

$$\left\{ \begin{array}{l} 2 m_0 x + m_4 y = - m_1 \\ m_4 x + 2 m_2 y = - m_3 \\ \end{array} \right.$$ (1.98)

which can be easily solved using Cramer's rule.

Saddle points can be useful, for example, for precisely locating subpixel checkerboard markers.