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 [40] by means of Michel-Penot

directional derivative ([28]).

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.