Bernstein Polynomial and B´ezier-Bernstein Spline - Bezier and Splines in Image Processing and Machine Vision - page 21

Gaussian Circle

Consider a unit circle in the plane of a curve and draw radii in the direction
of tangents at points P 1 , P 2 ,and P 3 , thus providing points P 1 , P 2 ,and P 3
as shown in Figure 1.9. The process, which assigns P i to P i , is known as the
Gaussian map and the points on the circle are the Gaussian image of the
curve. Therefore, if G is the Gaussian map, then

P i . G ( P i ) −→

G maps every single point P i on the curve to a unique point P i on the circle,
though G − 1 ( P i ) may stand for two or more points on the curve depending
on the directions of tangents at these points. Two points P i and P j appear to
be the same under G if tangents at these points have the same directions. In
other words, it is quite likely that G − 1 ( P i ) equals P i and P j both.
Note that as we move on, from P i to P i +1 and from P i +1 to P i +2 ,itis
not necessary that the same sequential order is maintained by their G -images.
With this effect, we can make the following classification.

The sequential order of the Gaussian image points P i is the same as that
of the points P i of the curve—we get regular points.

•

The sequential order of P i s reverses, whereas that of P i s remains the
same—we get point of inflection.

•

•

The order of P i s reverses, i.e., the direction of the tangents at these points
reverses, whereas that of motion of P i s remains the same—we get cusp of
the first kind.

The order of P i s as well as that of P i s gets reversed—we get cusp of the
second kind.

•

Figure 1.10 shows all these four classifications. In the discrete domain, tangent
to a discrete curve at a point is not defined in the existing literature. Therefore,
it is very dicult to get the Gaussian image of a discrete curve. To detect
between two key pixels on a discrete contour segment, an approximate position
of a pixel as the position of a point of inflection, we first approximate the
contour segment by straight line segments and these line segments are used
to obtain the Gaussian image. If a reversal of order in the Gaussian image
is detected for any line segment, then a point of inflection is marked at the
midpoint of the previous line segment.
The process is repeated for all the pixels between other key pixels. Thus,
all the key pixels and points of inflection can be extracted from the entire
contour. Between any two key pixels or between a key pixel and a point of
inflection or vice versa, the set of pixels can be viewed either as a line or a
convex/concave arc segment.