>
> surface,
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points by
ent points
dual point
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f its dual
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> pairs. In
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ajdla and
| registra-
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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision", Graz, 2002
(a) (b)
Figure 12: A bitangent point pair (X, X') slides along
the curve pair. The same parameterization can be used
for both curves which simplifies calculation of invariants.
We compute the distance between the bitangent point pair
(X, X^) as it slides along the curves. (b) An invariant sig-
nature of a bitangent curve pair expresses the distance be-
tween the points as function of the arclength of the longest
curve of the pair.
ing point. This introduces the problem that an expression
is only invariant if the same reference point is used. In
the case of bitangent curve pairs, the reference point can
be the corresponding bitangent point on the other curve,
thereby further simplifying the construction of stable in-
variant signatures. Hence, two points are combined that
slide together along the bitangent curves.
In the Euclidean case the distance between bitangent points
is invariant. The Euclidean arclength serves as an invari-
ant parametrization. A single parametrization is used for
both curves, for which we use the arclength of the longest
curve of the pair. By computing the distance as the bitan-
gent point pair slides along the curves, we get an invariant
signature as illustrated in figure 12. These signatures are
well suited for the matching of curves found on different
patches. Rather than trying to directly match different sur-
face patches, the goal is to match their most salient bitan-
gent curves. In order to render the process more efficient
and more robust, the search for matching signatures starts
with the 15 /ongest bitangent curves. For efficiency, this
length is measured as the number of sample points of the
self-intersection curve in the dual space. Only these curves
will be converted into bitangent curve pairs. This proce-
dure does not exactly select the bitangent curves that have
the longest arclength, but it gives a fair approximation.
The matching can be done efficiently by matching their in-
variant signatures. As a criterion, we use the L2-norm.
As it may very well happen that only parts of bitangent
curves are found on each of the patches, the signatures are
divided into segments of equal length and these segments
are matched. For efficient comparison, the signatures on
different surfaces are resampled with the same constant ar-
clength between sample points. Finding the best match-
ing segments between two signatures is done by having a
segment of the first signature slide along the second. No
guarantee exists that corresponding bitangent curve pairs
on different surfaces are parameterized in the same direc-
tion. This means that the starting point and ending point of
the signatures can be inverted. We take this possibility into
account in the matching process by checking whether the
signature increases or decreases over the segment.
A signature segment as defined in the previous paragraph
corresponds to two 3D curve segments, one on each curve
of the bitangent curve pair. Consequently, its endpoints de-
fine four points on the surface patch. If a pair of signature
segments is matched successfully, this suggests a match
between 4 points on the two surface patches. These typi-
cally yield enough information to obtain a crude estimate
for the transformation between the patches. Every match-
ing signature segment provides us with a candidate trans-
formation. Signature matches are ranked according to their
L2-norm. However, only looking at the L2-norm does not
suffice to select the best transformation candidate because
signatures can match exactly without corresponding to the
correct transformation. A typical example is a left-right
symmetric face. Bitangent curve pairs will be symmetric,
and signature segments from the left side can be matched
exactly with the ones on the right side. The transformation
implied by left-right mismatched signatures will result in
noses pointing in the opposite direction. In order to elim-
inate these ambiguities, a verification step is done on the
best matching signature segments. After a good signature
match is found, it is checked by applying the correspond-
ing transformation and by verifying how well the surfaces
fit".
3.3 Approach for texture-based registration
Wide baseline matching between 3D patches can also be
based on their surface texture, rather than their 3D shape.
A direct extension of the previous ideas would be to ex-
tract intensity or colour edges in the texture maps, which
correspond to space curves on the patch surfaces. Such
curves can then be matched, as e.g. proposed by Pajdla and
Van Gool (Pajdla 1995). Here we follow a different and
more robust strategy. The invariant neighbourhoods are
used again. This makes us less dependent on the presence
and clean extraction of edges in the texture maps. This
also renders the features more local and therefore better
suited for cases with limited overlap between patches. The
invariant neighbourhoods can cope with the deformations
that may exist between different texture maps. The actual
matching is simple then. Invariant neighbourhoods are ex-
tracted from the texture maps of the patches, are matched
based on their feature vectors of moment invariants, and
from each of the successfully matched neighbourhoods a
few points are selected (e.g. the center point of the neigh-
bourhood). Next, a 3D Euclidean motion is determined
that minimises the sum of distances between the corre-
sponding points of corresponding neighbourhoods. This
transformation is computed with Horn’s quaternion based
method (Horn 1987).
3.4 Experiments
A first example shows the matching of 3D patches based
on shape. The patches are shown in fig. 13 and belong
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