estimated.
In this paper we explore the matching problem based on featu-
res. For feature extraction we use the point operator proposed
by Fórstner (1987, 1989). The points selected in the different
images have to be matched e.g. to solve the problem of point
transfer or the problem of DTM reconstruction. Today image
analysis and matching are based on pyramids and some strate-
gies, which guide the process usually from coarse to fine resolu-
tion (for examples cf e.g. Ackermann and Hahn, 1991). For the
analysis of the single image, tracking of features through scale
space is proposed by Bergholm (1987).The scale space Witkin
(1983) is a kind of an image pyramid which consists of an infi-
nite number of smoothed image levels. The characteristic of this
representation is that the scale (smoothness) parameter is conti-
nuous which is of benefit for tracking. In the matching case,
e.g. in DTM reconstruction, it is usual to work with standard
image pyramids, almost with a Gaussian image pyramid. The
smoothing levels of this pyramids are fixed and the spatial
resolution between two consecutive levels decreases from the
bottom to the top of the pyramid. In this case not the tracking
idea is dominant but the questions due to approximate values,
efficiency of the algorithm and reliability of matching are ad-
dressed.
The concept of this investigations and according to this the
organization of the paper is as follows: (1) We want to find out
characteristics of the point operator in scale space. This implies
questions due to the tracking of the point location of the interest
point from fine to coarse and vice versa. Moreover the signifi-
cance of the selected features in scale is important. The example
used in section 2 is a synthetic image. (2) The scale space
tracking of features in real images is discussed in section 3.
Influences due to physical (illumination, etc.) and geometric
(perspective projection) aspects can be observed. Mainly the
consequences for the image location of the features and for the
stereo displacements are of interest. For the synthetic image as
well as for the real images we want to restrict ourself to one
dimension. The one-dimensional real signals are taken from a
epipolar stereo pair.
1.1 RELATED WORK
Related work which has not been addressed up to now mainly
concerns the representation and reasoning about features in scale
space. Since the early days of computer vision it was quite clear
that high-level processes need and have to use a lot of different
knowledge for reasoning. For low-level processes such as edge
detection a common belief was that they are simply data driven
without use of explicit knowledge. This assessment today chan-
ges. A lot of operators for edge detection have been proposed in
parts but research has clearly demonstrated that the edge detec-
ted by these techniques do not give satisfying results (Lu and
Jain, 1992; Bergholm, 1987). Because of this the role of reaso-
ning in low level processing comes into the center of interest.
As one of the first Witkin (1983) analyzed thoroughly the be-
havior of edges in scale space. He reflected work of Marr
(1982), who argued "that physical processes act on their own
intrinsic scales". A scale-structured representation, called the
interval tree, he introduced to describe contours over scale. This
organization characterizes the information over a broad range of
scale, that means, between a coarse resolution level with a small
number of edges and a fine resolution level with usually signi-
ficantly more edges. This organization is expected to be useful
Witkin for matching or object reconstruction tasks. The sym-
bolic image description over scale is generated by the zero-
crossings of a Laplacian of Gaussian (LoG) convolved image, in
which the Gaussian c is addressed as scale. Three typical edge
behaviors in Gaussian scale space were observed by researchers:
(1) The locations of edges in filtered images using different
scale parameters can (and in general will) be different. (2) in
scale space zero crossing occurring at finer scales can disappear
at coarser scales. (3) Spurious edges are those that occur at a
coarser scale but have no corresponding edges at a finer scale.
For more details cf. Lu and Jain (1992). This two authors pre-
sented the most sophisticated algorithm up to now, called RESS,
which stands for reasoning about edges in scale space. The
knowledge about edge behavior in scale space is explicitly
formulated in 35 rules and is used in RESS to select proper
scale parameters, to correct dislocation of edges (1), to recover
missing edges (2), and to eliminate noise or false edges (3). The
separation of significant edge information from noise mainly has
been also the aim of Bergholm (1987) in his multiscale tracking
procedure, called edge focusing, as well as Canny (1986), who
proposed a multiscale edge detector.
Finally we want to address the work of Heikkilà (1989), which
has some similarity to our investigation because he used also
the Fórstner point operator. In the one-dimensional case the
point operator coincides with an edge detector. The estimated
point position locates the edges with subpixel accuracy, at least
in theory. Therefore the estimation of the edge location in scale
space will be interesting. In the paper of Heikkilä the properties
of the operator by varying the scale parameter of the integrating
window are investigated. The integration works on the squared
gradient image. Consequently the interest operator is a nonlinear
edge operator. So far this is presumably the main difference to
other edge operators like the linear LoG operator mentioned
before. Our interest is not the problem of a varying window
size. We investigate the behavior of the operator applied to a
series of Gaussian smoothed images. In the 2 D case the opera-
tor can be formulated as
[XG,. * f) VG, * f) * G, (1)
With o, the scale space image of f(x,y) is generated, whilsto,
is responsible for the size of the weighting in the window. the
dyadic product vv" indicates the nonlinearity of the operator. If
C, is constant and o, varies mainly the following characteristic
for edges can be observed:
With increasing c, the number of edges decreases, i.e. the
edges fuse or disappear with coarser scale o,. Just invert is the
situation when the scale space parameter c, varies and o, is
constant:
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