ıral perspective
ts. In this paper
"he intersection
which provide
on. Triplets of
zation tasks by
ple junctions in
introduction of
ng of adjacent
h can occlude
cuboids in the
rvention it is
mation about
their incidence
ied in terms of
ojective flags".
f an incidence
ition relative to
tive lines) and
). The relative
d to the image
nd T-type) and
€ intersections
val assigned to
rom a Deriche's
s in automatic
ractive visual
management of
ion of pairs of
lines. Both of
collinearity and
itic grouping of
pective lines in
void sensitivity
ted by a robust
ice perspective
way, by taking
| are grouped
n or RANSAC
: of perspective
F least squares
mini-segments,
highly textured
to recover and
primitive-based
yed a primitive-
based approach for perspective models because it supports
better vectorized information. Robust and accurate estimation
of vanishing points is obtained integrating different
methodologies belonging to Photometry and Computer Vision
with robust procedures. Parameters linked to transformations
between views are expressed in terms of affine maps between
quadrilaterals determined by intersections of perspective lines.
Planar collineations between pairs of views are lifted to
collineations in the ambient 3D space by constructing cuboids
from the third vanishing point. We have developed a symbolic
management of information based on propagation models from
an initial cuboid located around the focal point and generated
from pencils of lines through vanishing points. Perspective lines
provide the main directions for propagation phenomena in our
model. Thus, the robust identification of vanishing points
allows to avoid "tedious" local compatibility and global
coherence conditions for homologue cuboids in different
constructions, and simplify the tracking in presence of self-
motion [Fin02]. We have restricted ourselves to simple indoor
and outdoor scenes with a static camera, by avoiding occlusion
problems and possible cumbersome alternance between
concave/convex regions for evaluating the goodness of our
results.
2. METHODOLOGY
With the goal of investigating and testing the behavior of our
robust methodology, we have developed a program that allows
us to test and analyze under different geometric and stochastic
conditions several perspectives models and oblique images. The
next scheme shows the steps of the methodology developed:
| SINGLE IMAGE |
Canny's feature
extraction | |
v
Automatic or
| Image preprocessing |
Human assistance
v
Labelling and |
grouping
Compute of
Rd ^ Support by Area
vanishing points
Minimization and Robust E.
Quadrilaterals and
cuboids decomposition
Y
Perspective
reconstruction
Fig.1: Methodology developed.
2.1 Automatic extraction of basic elements
Automatic grouping in Computational Geometry is usually
performed by using auxiliary constructions such | as
decompositions in triangles, trapezoids or, more generally,
convex sets associated to a multivectorized treatment of data
[Ber97] Minisegments are obtained from Canny's detector.
Minisegments are grouped along lines to determine ordinary
multiple junctions. Large segments are grouped depending on
the slope and collinearity restrictions. In this way, we attenuate
the striped effect which appears from boundaries and
minisegments obtained from the application of Canny's
detector. Furthermore, we eliminate short segments with length
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004
under a selected threshold. In indoor scenes, we apply an active
discrimination for vertical lines, due to the characteristics of
scene and bad illumination (reflectances, irradiance of the
ground and floor, etc). Large segments make part of putative
candidates to become perspective lines. To validate them as
perspective lines, we must have a robust an accurate estimation
of the vanishing points, and verify incidence conditions of large
segments through vanishing points.
2.2 Elements of perspective
Vanishing points are computed from bundles of perspective
lines in the framework of Projective Geometry [Har00]. The
relative location of vanishing points w.r.t. the image determines
the perspective model (frontal view, angular perspective, and
oblique perspective, respectively) which is useful for
visualization tasks (linked to one, two or three vanishing points
to finite distance, respectively). Selection of constraints in
manually driven reconstruction is typical of photogrammetry,
and it depends on the perspective model chosen. In frontal and
angular perspective, rigidity constraints relative to motions and
objects are translated to parallelism and perpendicularity
constraints for typical representations of architectural scenes.
The projective extension of Euclidian approach is clearly very
useful for oblique perspective. It is based on maps of cuboids
appearing as the image of parallelepipeds by a projective
transformation. This kind of volumetric representation has been
extensively developed and applied for reconstruction from a
single view and their applications [Wil02]. :
2.3 Robust estimators for vanishing points
In scenes with bad illumination or annoying architectural
elements, sometimes it is difficult to identify perspective
elements to visualize 3D scenes. In this case, we have
developed a real-time implementation based on the intersection
of putative perspective lines obtained by regression and Hough
transform [Fin02]. A weighted average around a vanishing line
allows to identify a coarse approach to a putative vanishing
point 7, and retracing perspective lines through V. The high
rate of errors (around a 59$) can be corrected in mobile
navigation [Fin02]. However, this error is not allowed in more
accurate 3D Reconstruction. Thus, we have concentrated our
attention in the implementation of algorithms for robust
estimation of vanishing points. In this section we compare
different several estimators for vanishing points: Danish
method, Minimal Sum, Huber, German-McClure. The main
goal of all of them is the automatic elimination of lines which
are not adequate for the "right" determination of vanishing
points. The rightness is measured in terms of the area
minimization. In typical architectural scenes we have a high
redundancy degree, more precisely, we have enough elements
arising from an automatic vectorization to provide bundle of
errors in perspective lines.
Robust estimators provide an adjustment method, and detect
wrong observations appearing as outliers in an efficient way.
Robustness corresponds to their independence w.r.t. the errors
distribution. Robust estimators are based on applying variable
weight functions P(w) , and they can be written as Pow!)
where i is a residual number and £ the iteration number. The
function P allows to modify original weights in order to reject
observations having blunders errors in the adjustment. We have
considered the following robust estimators:
