A >
Figure 1: The split-and-merge process: A. The original image;
B. The initial segmentation; C. The result of the
solit-and-merge process.
The result of the split-and-merge process is more likely to be a
correct partitioning of the image where each region associates
with a single surface in object space. However, planar faces
might have been detected also in vegetation regions. To avoid
the influence of vegetation, image data in red and near infrared
channel is used to identify and discard vegetation regions. For
this purpose a normalized difference vegetation index (NDVI) is
computed for each pixel from the following equation:
NIR - RED
I ERE |
NIR + RED (D
NDVI =
where NIR and RED denote pixel values in near infrared and red
channels respectively. NDVI is related to the proportion of
photosynthetically absorbed radiation and its value varies from
—1 to +1. Vegetation is characterized by high NDVI value and a
region is identified as a vegetated region if at least 70% of its
pixels are vegetation pixels.
To determine the final roof planes among the remaining regions
the difference between DSM height and DTM height for height
points belonging to each region is used. A planar surface is
attributed as non-roof if the difference between its average
DSM height and DTM height is smaller than a minimum
threshold; otherwise it is attributed as flat-roof if its largest
slope is smaller than a slope threshold or as slanted-roof if the
largest slope is larger than the slope threshold.
3.3 Reconstruction of vertical walls upon ground plans
The parametric form of a plane passing through a point
ers 2 Gy and perpendicular to a normal vector
X Ly o,
p
ns(a,a,,a,y Can be written as:
a,x+a,y+a,z =k (2)
where:
k=ax,+a,y,+a,z, (3)
For a vertical plane 4. = ( and the plane equation becomes:
ax+a,y=k (4)
where:
k=ax,+a,y,
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004
Therefore, each line segment of the ground plan with endpoints
Pi z(x. yp, Q5 can solve for a, vds and define a
vertical plane as follows:
ax+a,y+a,z+a, =0
where :
a, =5-) (5)
a, =x, —X,
a, =0
a, =x, Vy, =X,
Parameters of the reconstructed wall faces are stored along with
two endpoints used in the calculations. Wall faces and planar
roof faces computed in the split-and-merge process enter the
plane patch reconstruction procedure described below.
4. RECONSTRUCTION OF PLANE PATCHES
A plane patch is defined as a planar polygon in 3D space. So
far, the computed model faces are represented in parametric
form. For graphical visualization, however, one requires plane
patches, which are represented by their vertices. Reconstruction
of plane patches is carried out in three steps: plane intersection,
verification of vertices and sorting of vertices. In the following
these steps are described in more details.
4.1 Plane intersection
Two planes in 3D space intersect in a line if they are not parallel
or coplanar. This line will intersect a third plane in one point if
it is not parallel to it and does not lie in it. Therefore, in regular
case three planes in 3D space intersect in a point if they are not
in a special relation to each other. In algebraic form, a system of
three equations of the form denoted in eq. 2 has exactly one
solution if the equations are linearly independent. More
precisely if normal vectors of the three planes are linearly
independent then equations of the planes form a regular system
of three equations and three unknowns as denoted in eq. 6:
a, x+a,y+a,z =k,
ayx+ayy+anz =k, (6)
a, xta,ytanz =k;
where , are plane parameters and 4. are constants. In order
ij i
to verify the linear independence of normal vectors, let:
|
[4 02-4,
A=| a, dy ay
ay 035 Un
The normal vectors of the three planes are linearly independent
if A20 in which case the set of equations 6 has a unique
solution that can be calculated using Cramer's rule (Pedoe,
1963):
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