Image Processing Reference

In-Depth Information

Fig. 5.12
A curve and the

tangent at four points

this corresponds to the slope of the tangent at this point. In Fig.
5.12
the tangents of

several different points are shown.

If we represent an image by height as opposed to intensity, see Fig.
5.13
, then

edges correspond to places where we have steep hills. For each point in this image

landscape we have two gradients: one in the x-direction and one in the y-direction.

Together these two gradients span a plane, known as the
tangent plane
, which in-

tersects the point. The resulting gradient is defined as a vector
G(g
x
,g
y
)
, where

g
x
is the gradient in the x-direction and
g
y
is the gradient in the y-direction. This

resulting gradient lies in the tangent plane, see Fig.
5.14
.

We can consider
G(g
x
,g
y
)
as the direction with the steepest slope (or least

steepest slope depending on how we calculate it), or in other words, if you are stand-

ing at this position in the landscape you need to follow the opposite direction of the

gradient in order to get down fastest. Or in yet another way, when water falls at this

point it will run in the opposite direction of the gradient.

Fig. 5.13
A 3D representation of the image from Fig.
5.11
, where the intensity of each pixel is

interpreted as a height