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1
Gaussian sphere 1
Figure 1: The geometry of vanishing points, interpretation
planes, and the Gaussian sphere
tor array, the intersections of interpretation planes with the
sphere (great circles) are histogrammed; maxima in the his-
togram then correspond to orientations shared by several line
segments, and can be hypothesized as vanishing points. The
geometry of vanishing points, interpretation planes, and the
Gaussian sphere is depicted in Figure 1.
A difficulty with the classical approach is its sensitivity to
noise. Texture edges caused by natural features in the scene
can lead to spurious maxima on the Gaussian sphere, result-
ing in incorrect vanishing point solutions. However, these
short edges exhibit greater uncertainty in image position and
orientation, which can be modeled and incorporated into the
sphere histogramming process. In recent work, we have pro-
posed two edge error models which use rigorous camera mod-
eling to treat great circles as swaths of variable width on the
sphere, where the width corresponds to the uncertainty of
the edge. These models locate vanishing points reliably in
the presence of large amounts of noise, and are described in
detail elsewhere [Shufelt, 1996].
Another difficulty with the classical technique is that it makes
no provision for using knowledge about the shapes of objects
of interest to guide the search for maxima on the Gaussian
sphere. Rather than searching for maxima one at a time with
no knowledge of scene structure, we seek a method which
utilizes the expected shape of objects to find vanishing points.
To develop such a method, it is first necessary to make a
choice of representation for buildings.
There exists a wide spectrum of 3D representations, ranging
from CAD-based models, in which shape and size are fixed,
to highly parametric representations such as superquadrics
and physically-based models, in which shape and size are
variable. However, the immense variety of manmade con-
structions renders a model-based representation impractical,
and highly parametric representations have proven difficult
to reliably extract from complex imagery. Instead, we choose
a set of 3D wireframes as "building blocks" for manmade
structures; these units, which have fixed shape and topology
75
Figure 2: Rectangular and triangular primitives and their
vanishing point patterns on the Gaussian sphere
but variable size, are known as primitives [Biederman, 1985;
Braun et a/., 1995]. Geometric constraints provide sufficient
leverage for extracting primitives from aerial imagery, while
the combination of primitives provides representational flexi-
bility for modeling complex manmade structures.
PIVOT currently uses two primitives to model buildings, rect-
angular volumes and triangular prisms, shown in Figure 2.
Rectangular volumes are composed of 3D lines with vertical
and orthogonal horizontal orientations in object space (v, h1,
and h2 respectively); triangular prisms are composed of lines
with two symmetric slanted “peak” lines in a vertical plane
and two orthogonal horizontals (p1, p2, h1, and h2 respec-
tively). Figure 2 also depicts the orientation patterns created
by these primitives on the Gaussian sphere.
Exploiting this knowledge about object shape is now sim-
ple. Rather than scanning the sphere for individual vanishing
point maxima, PIVOT scans for pairs of orthogonal horizon-
tals, with respect to the vertical vanishing point which can be
computed directly from the camera parameters. Once hori-
zontal vanishing points have been located, PIVOT scans for
slanted "peak" lines which lie in vertical planes of the horizon-
tals. This approach leads to robust vanishing point solutions
for all primitive edge orientations [Shufelt, 1996].
3 GEOMETRIC CONSTRAINTS FOR CORNERS
Given a set of vanishing points, each line segment in the im-
age can be tested to see if it lies along a line with a vanishing
point. If so, the line segment can be labeled with the 3D ori-
entation in object space corresponding to the vanishing point.
At the conclusion of this process, each line segment has a set
of vanishing point labels which can be used as a means of en-
suring that PIVOT's intermediate corner representations are
geometrically consistent. This section gives a brief discus-
sion of the use of vanishing point labelings for detecting legal
corners.
Corners are generated in PIVOT by performing a range search
on all line segments, and linking together pairs of segments
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B6. Vienna 1996
