The recovering of the epipolar geometry needs to establish some point correspondences between images. Currently, techniques to find candidate matches in a pair of images have been studied in Computer Vision, they mainly belong to two classes:
As this paper deals with point correspondences we are focusing here on iconic methods
which can roughly be separated again into two classes: correlation method and
invariants based methods.
Both of these two classes of methods, can be decomposed into two stages:
a first stage using a process of local comparison involving, as correlation windows extracted
from the images, as comparison of invariants sets, and
a second stage used to to disambiguish the results obtained involving semi-global information,
as for example a relaxation process.
Among iconic methods, the main advantage of correlation methods is that they give very good
results when the images contain
textures and when the transformation between the two images to be matched is small
(mainly small rotations).
But it fails when the transformation between the two viewpoints is more complex
(important rotations, important scale variations).
The other drawback is that correlation is very time consuming when it uses large
correlation windows.
Roughly speaking, if the correlation window is small (the
correlation process is fast) then the correlation scores are not very discriminant and the
correlation process must be followed by a time consuming relaxation process
to disambiguish the matches.
On the contrary if we want to simplify the second stage of the matching process
the size of the correlation windows must increase.
At the contrary, differential invariants based matching methods in gray level images,
are more robust against rotations between images, than correlation based methods.
They are also faster because invariants contain a condensate description of
images around each primitive, so the set of local invariants is generally small.
In gray level images, to obtain a good description of each primitive,
it is often necessary to considerate invariants up to the third order [14]
(5 invariants till second order, 9 invariants till third order).
The problem which then arises, comes from the estimation of the high order
image derivatives.
This problem can be overcome considering color information, we show here
that a set of color invariants involving only first order derivatives
is enough to characterize point primitives.
The description which is then possible to obtain becomes sufficient
to match robustly points of interest even if the images present a wide range of viewpoints,
important rotations and illumination variations.
In the next section we revisit invariant theory in the case of gray level images and we generalize it to color images.