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The Gradient (also called the Hamilton operator) is a vector
operator for any N-dimensional scalar function
,
where
is an N-D vector variable. For example,
when
,
may represent temperature, concentration, or pressure
in the 3-D space. The gradient of this N-D function is a vector composed of
components for the
partial derivatives:
- The direction
of the gradient vector
is the
direction in the N-D space along which the function
increases
most rapidly.
- The magnitude
of the gradient
is the rate of the
increment.
In image processing we only consider 2-D field:
When applied to a 2-D function
, this operator produces a vector
function:
where
and
.
The direction and magnitude of
are respectively
Now we show that
increases most rapidly along the direction of
and the rate of increment is equal to the magnitude
of
.
Consider the directional derivative of
along an arbitrary
direction
:
This directional derivative is a function of
, defined as the
angle between directions
and the positive direction of
. To find
the direction along which
is maximized, we let
Solving this for
, we get
i.e.,
which is indeed the direction
of
.
From
, we can also get
Substituting these into the expression of
, we obtain its maximum
magnitude,
which is the magnitude of
.
Next: Digital Gradient
Up: gradient
Previous: High-boost filtering
Ruye Wang
2016-10-18