The correctness of the uncertainty analysis is demonstrated in this section on a number of examples. It is shown that the ground truth measurements always lie with the estimated error bounds. Furthermore, the utility of the analysis is illustrated. The covariance expression predicts uncertainty given the number of image-world computation points and their distribution. It is thus possible to decide where correspondences are required in order to achieve a particular desired measurement accuracy.
In order to illustrate the accuracy of this first order error propagation analysis for the planar case we show, in figure 2 , a comparison between the covariance ellipse obtained by the first order analysis and the one obtained by a Monte-Carlo evaluation of the actual non-linear homography mapping. The predicted ellipse and the simulated one are almost overlapping. These figures are obtained using parameters related to a real situation.
Figure:
(a)
H
is computed from the theory of sections
3.1
and
5
using only 4 computation points. 10000 image points are randomly
generated from a Gaussian distribution centered on an image test point
and then backprojected onto the world plane. The statistical covariance
ellipse of the world points is computed and plotted together with the
predicted one. 3 std. dev. are visualised for each uncertainty ellipse -
(b) The first order and simulated uncertainty ellipses areas decrease as
the number of computation points increase from 4 to 5 to 10 as expected
from the theory.
Figure
3
shows an example of an indoor scene. Figure
3
a is the original image and in figures
3
b-d this image is back-projected onto the world plane by the computed
homography. The
H
matrix is computed using four computation points in all cases. The
standard deviations used for the computation are
and
.
Three test points are shown with their uncertainty ellipses. Note that as the distance of the test point to computation points increases the uncertainty increases. More spatially homogeneous uncertainties are achieved by distributing the computation points across the scene.
Figure:
(a) Original image. (b-d) length measurements based on a homography
computed from the points marked by black asterisks. The uncertainty
bound is
std. dev. The actual width is 139 cm.
Figure:
(a) Original image. (b-d) projectively unskewed back-projection. The
computation points used to estimate
H
are marked by white asterisks. The uncertainty ellipses shown for test
points are at the 9 std. dev. level for clarity.
Figure
4
b-d shows length measurements for a homography computed from four, six
and eight correspondences. The standard deviations used for the
computation are
and
. Again measurements further from the computation point have a larger
uncertainty. Increasing the number of computation points decreases all
the uncertainties. Note that all the measurement ranges returned by the
system include the actual window width (139 cm).
Figure 5 illustrates that the uncertainties also depend on the viewpoint. In both cases the ground truth lies within the predicted measurement range, but this range is larger in the view with more severe perspective distortion.
The figure also illustrates the computation of parallel world lines. In the image a line is selected, and then the one parameter family of lines parallel to it on the world plane are computed from the estimated H .
Figure:
Keble College Oxford. The computation points are the same, but the
viewpoint distortion is more severe in (b). This is reflected in the
larger (1 std. dev.) uncertainties. The actual width of the upper
windows is 176 cm. Note the computed parallel lines.
Antonio Criminisi