Image Processing Reference
In-Depth Information
obtained by computing the rate of change of edge direction. In this figure, curvature is
defined only at the edge points. Here, by its formulation the measurement of curvature κ
gives just a thin line of differences in edge direction which can be seen to track the
perimeter points of the shapes (at points where there is measured curvature). The brightest
points are those with greatest curvature. In order to show the results, we have scaled the
curvature values to use 256 intensity values. The estimates of corner points could be
obtained by a uniformly thresholded version of Figure
4.32
(b), well in theory anyway!
(a) Image
(b) Detected corners
Figure 4.32
Curvature detection by difference
Unfortunately, as can be seen, this approach does not provide reliable results. It is
essentially a reformulation of a first-order edge detection process and presupposes that the
corner information lies within the threshold data (and uses no corner structure in detection).
One of the major difficulties with this approach is that measurements of angle can be
severely affected by
quantisation error
and accuracy is
limited
(Bennett, 1975), a factor
which will return to plague us later when we study methods for describing shapes.
4.6.2
Approximation to a continuous curve
An alternative way to obtain a measure of curvature is to evaluate Equation 4.34 for small
continuous curves
that approximate curves in discrete data (Tsai, 1994), (Lee, 1993).
Continuous curves are estimated by fitting a curve to points given by the known position
of image edges. A reliable value of curvature is obtained when the fitting process gives a
good approximation of image segments. The main advantage of this approach is that it
reduces (or at least averages) bias due to small variations between the true position of the
points in the curve and the discrete position of the image pixels. That is, it reduces digitisation
errors.
Small segments are generally defined by cubic polynomial functions. Cubic polynomials
are a good compromise between generality of the representation and computational complexity.
The fitting can be achieved by considering a parametric, or implicit, fitting equation.
However, implicit forms do not provide a simple solution leading to excessive computational
requirement. This is an important deterrent if we consider that it is necessary to fit a curve
for each pixel forming an edge in the image. In a parametric representation, the contour
v
(
t
) can be approximated by the two polynomials given by,
