# directional derivative

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## directional derivative

[də′rek·shən·əl də′riv·əd·iv]
(mathematics)
The rate of change of a function in a given direction; more precisely, if ƒ maps an n-dimensional euclidean space into the real numbers, andx= (x1, …, xn) is a vector in this space, andu= (u1, …, un) is a unit vector in the space (that is, u12+···+ un 2= 1), then the directional derivative of ƒ atxin the direction ofuis the limit as h approaches zero of [ƒ(x+ h u) - ƒ(x)]/ h.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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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.

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