Minima, Maxima, and Saddle Points

As written, the model presented above applies both to minima and maxima and to saddle points. However, this model does not account for any local rotations of the function. Although such rotations are negligible to a first approximation for minima and maxima, they may be significant in the case of saddle points.

The version of equation (1.135) that accounts for possible rotations of the axes is

$$m_0 x^2 + m_1 x + m_2 y^2 + m_3 y + m_4 x y + m_5 = z$$ (1.138)

The system is fully compatible with the one presented in the previous section, with the only difference that there are now six unknowns; therefore, at least six points in the neighborhood of the minimum, maximum, or saddle point must be processed. Here too, there are no notable closed-form solutions, but it is convenient to factorize the matrix of known terms.

The gradient of function (1.138) vanishes at the point corresponding to the solution of the linear system

$$\left\{ \begin{array}{l} 2 m_0 x + m_4 y = - m_1 \\ m_4 x + 2 m_2 y = - m_3 \\ \end{array} \right.$$ (1.139)

which can be solved easily using Cramer's rule.

Saddle points can be useful, for example, to locate checkerboard-shaped markers with subpixel precision.