The above-mentioned results serve as a formal and empirical justification for
using Gaussian filtering followed by derivative computations as
initial steps in early processing of image data.
More important, a catalogue is provided of what smoothing kernels
are natural to use,
as well as a framework for relating filters of different types
and at different scales.
(Figure 4 shows a few examples of
filter kernels from this filter bank.)
Linear filtering, however, cannot be used as the only component
in a vision system aimed at deriving symbolic representations
from images; some non-linear processing steps must be introduced
into the analysis.
More concretely, some mechanism is required for combining the
output of these Gaussian derivative operators of different orders and
at different scales into more explicit descriptors of the image geometry.

An approach that has been advocated by Koenderink and his co-workers
is to describe image properties in terms of differential geometric
descriptors, i.e., different possibly non-linear
combinations of derivatives.
Since one would typically like image descriptors to possess
invariance properties under certain transformations
(typically, rotations, rescalings and affine or perspective deformations),
this naturally leads to the study of differential invariants [].
A major difference compared to traditional invariant theory, however,
is that the primitive derivative operators in this case are
smoothed derivatives computed from the scale-space representation.
In this section, a few examples will be given of how
this framework of multi-scale differential geometry
can be used for expressing various types of
multi-scale feature detectors.
The output from these feature detectors is in turn intended to be used as
input to higher-level visual modules,
for task such as object recognition,
object reconstruction/manipulation and robot navigation.