per the approach given in Persad (2012). The location of the
principal point was set to be the orthocenter of the triangle
formed by the three vanishing points. The focal length was es-
timated based on the mutual orthogonal relationship between
any two vanishing point directions. The rotation angles were
determined based on the 3
D
rotation matrix whose column
elements are the coordinates of each vanishing point, respec-
tively. Afterwards, the endpoints of the 3
D
wireframe lines are
back-projected into the 2
D
image using the collinearity equa-
tions (Figure 3 (bottom row)).
The back-projection step enables us to use the 2
D
image as
a common space for establishing the correspondence of pro-
jected 3
D
lines with 2
D
lines. Camera parameters derived from
VP
s result in sub-optimal registration of the back-projected 3
D
wireframe lines to image space. This is due to errors caused
by the
VP
algorithm and line extraction saliency (Figure 3
(bottom row)).
Line-Based Randomized RANdom SAmple Consensus (LR-RANSAC)
Given
N
possible matching image to wireframe line pairs,
some of these line pairs may be erroneous matches, thus nega-
tively affecting the camera parameter solution. The
LR-RANSAC
procedure tries to exclude the erroneous matches using a
RANSAC
-type approach. A minimum sample of random line
pairs are selected and used to generate a camera parameter
solution. The camera solution is treated as a hypothesis and
undergoes a pre-verification test using the minimal sample
only. Samples that pass the pre-verification step are subject
to a second test (i.e., full hypothesis verification) using the
whole line pair set. The details of
LR-RANSAC
’s random sam-
pling strategy, hypothesis generation and pre-verification test
are described in the following sections, while, its hypothesis
verification components are detailed in the Evidence-Based
Hypothesis Verification Section.
Vanishing Point Based-Constrained Random Sampling
Our sampling strategy utilizes a vanishing point-based
orientation constraint approach to reduce outlying line pair
matches. The principal idea is that a significant portion of
outlying matching possibilities can be eliminated by limiting
the random sampling of wireframe-to-image correspondence
candidates to the ones belonging to the same vanishing direc-
tion. This sampling strategy significantly reduces the match-
ing complexity instead of alternatively globally sampling
from the entire line correspondence set. This constraint also
ensures that wireframe-to-image line pairs are sampled in
varying vanishing directions, an important geometric consid-
eration for a stable camera parameter solution.
The selection of the candidate matching pairs is based
on a 2
D
Hough parametric grid whose axes are defined by
the Legoland orientation of the line (i.e.,
θ
) and its distance
from the image center (i.e.,
ρ
) (Hough, 1962). Therefore, the
wireframe-to-image line pairs are projected in the
θ
-
ρ
Hough
parametric space, with each mapped as a point in the 2
D
θ
-
ρ
space (Figure 4). Possible matches of back-projected wire-
frame lines to the image lines will result in similar values of
θ
and
ρ
and thus smaller Euclidean distances between two
θ
-
ρ
points for a wireframe-to-image line pair. The
θ
and
ρ
axes are
subdivided into grid intervals to allow for matching line pairs
to fall into common cells. The
θ
and
ρ
axes are divided into
three intervals each. Along the
θ
axis, the height of each cell
is defined by the values of the
θ
-
angles corresponding to each
of the three vanishing directions ±10 degrees. The width
of each cell is defined as
max min
( )
( )
ρ
ρ
3
−
(Figure 4). Having
a set of cells containing pairs of wireframe and image points,
we randomly select four point pairs from the
θ
-
ρ
cells, corre-
sponding to four line pairs on the image plane. Three of these
four corresponding line pair candidates are randomly selected
from each of the three theta divisions on the grid (i.e., the
three vanishing directions). The remaining fourth pair is cho-
sen randomly from all remaining point pairs in any cell. The
Hough parametric space is used to randomly select matching
line pairs. Using these four randomly sampled wireframe and
image line pair matches, we generate the camera parameter
hypothesis based on the matching between the wireframe and
image line pairs as described in the next Section.
Generation of Camera Parameters Hypothesis
Using the randomly sampled four wireframe-to-image line
pairs, a camera parameters hypothesis is obtained by mini-
mizing two distances,
d
qn
, between each pair of correspond-
ing back-projected 3
D
CAD
model and extracted image lines
(Figure 5), where
n =
{
1,2
}. To initialize the optimization, we
use the initial camera parameters determined in the Initial
Estimation of Camera Parameters Section. A total of eight dis-
tances, resulting from the four line pairs (i.e.,
q
= 1,2,3,4), are
Figure 4. Randomly sampled putative correspondences in θ-ρ
(theta-rho) space.
Figure 5. Residual distance
d
qn
to be minimized between a cor-
responding 3D model and image line pair
q
.
850
November 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
