U is a linear continuous form in [theta]; it represents a directional derivative
in the direction [theta].
However, in the process of image denoising and enhancement, these classic geometrical regularization methods, based on operators in differential geometry such as gradient, divergence, and directional derivative
, often tend to modify the image towards a piecewise constant function and blur fine features of the image, particularly the image's details and layered structures.
A Neumann boundary condition in the Laplace or Poisson equation imposes the constraint that the directional derivative
of \phi is some value at some location.
First-order necessary conditions for problem (P) have been obtained by Ye in  by means of Michel-Penot directional derivative
The edges are determined in every direction, as opposed to the operator using the directional derivative
of the first order;
In analogy with the directional derivative
of a convex function, we introduce the notion of the generalized directional derivative
of a locally Lipschitz function f at x [member of] X in the direction h [member of] X by
The directional derivative
becomes [phi]'([x.sub.k])d = (A[x.sub.k] - b, d} for arbitrary directions d [member of] X.
It binary encodes the relationship between the center (or referenced) pixel and its neighbors characterized by transformation consistency statistics of directional derivative
in horizontal and vertical direction.
Then the directional derivative
of [[PHI].sup.u](L, F) in the direction ([DELTA]L, [DELTA]F) satisfies
[alpha] and [beta] are constants and [partial derivative]u/[partial derivative]n represents the directional derivative
in the direction normal n to the boundary [partial derivative][OMEGA] which by convention points outwards.
In the first phase, the networks of linear objects (morpholineaments) were created based on interpretation of the models inferred from individual DEM morphometric methods (models of terrain slope, terrain aspect, first directional derivative
, gradient operator and combination of altitude and shaded relief).
Secondly, the potential's directional derivative
in a direction crossing S' and parallel to [??] will be discontinuous on S', since V exhibits an angle point at (0,0,0) in this case.