ESTIMATION OF THE SLOPE OF A PLANE LAYER VOLUME

If a given image volume is approximated by a plane layer volume, we can find the normal vector ${\bf n_1}$ by minimizing following expectation over the volume:

 

$$\min_{\bf n_1} E(({\bf n_1} \times \nabla f({\bf x}))^2) = \min_{\bf n_1} E((\nabla f \sin \theta)^2)$$ (2)

where $\theta$ denotes the angle between $\nabla f$ and ${\bf n_1}$.We will have to decide on an orientation for ${\bf n_1}$, since both ${\bf n_1}$ and ${\bf -n_1}$ minimize the expectation.

Since ${\bf n1}$ consists of only two independent variables, small expectation volumes will yield reliable estimates of the normal vector, or the corresponding dip. In particular, the minimum is independent of amplitude and polarity changes of the gradient $\nabla f$. For example, in Figure 3, the gradient varies within the window while the cross product ${\bf n_1} \times \nabla f$ is zero everywhere.