In: Stilla U, Rottensteiner F, Paparoditis N (Eds) CMRT09. IAPRS, Vol. XXXVIII, Part 3/W4 — Paris, France, 3-4 September, 2009 
our five unknowns for a certain building i has been formulated as 
follows: 
em®<) = Y. 
¿=1 
(hrm + h T j (x) - ?f(x)) 2 dx (4) 
Each prior that has been selected for a specific region is forced 
to acquire such a geometry so as at every point its total height 
matches the one from the available DEM. It’s a heavily con 
strained formulation and thus robust. The introduced, here, recog 
nition driven framework now takes the following form in respect 
to (j), %, L and ©,: 
Etotai — E seg ((j)) + [xE2d{4>i Ti, L) + ¡iE3i){Qi) (5) 
2D Quantitative Measures 
Completeness 
Correctness 
Quality 
0.84 
0.90 
0.76 
3D Quantitative Measures 
Completeness 
Correctness 
Quality 
0.86 
0.86 
0.77 
Table 1 : Pixel- and Voxel-Based Quality Assessment 
The energy term E seg addresses fusion in a natural way and 
solves segmentation 0 in both J(x) and H(x) spaces. The term 
E2D estimates which family of priors, i.e which 2D footprint i, 
under any projective transformation T, best fit at each segment 
(L). Finally, the energy E^d recovers the 3D geometry ©, of 
every prior by estimating building’s h rn and h r heights. 
4 QUALITATIVE AND QUANTITATIVE ASSESSMENT 
OF THE PRODUCED 3D MODELS 
The quality assessment of 3D data ((Meidow and Schuster, 2005), 
gent et al., 2007) and their references therein) involves the assess 
ment of both the geometry and topology of the model. During 
our experiments the quantitative evaluation was performed based 
on the 3D ground truth data which were derived from a man 
ual digitization procedure. The standard quantitative measures of 
Completeness (detection rate), Correctness (under-detection rate) 
and Quality (a normalization between the previous two) were em 
ployed. To this end, the quantitative assessment is divided into 
two parts: Firstly, for the evaluation of the extracted 2D bound 
aries i.e. the horizontal localization of the building footprints 
(Figure 3) and secondly, for the evaluation of the hypsometric 
differences i.e. the vertical differences between the extracted 3D 
building and the ground truth (Figure 4). 
In order to assess the horizontal accuracy of the extracted build 
ing footprints the measures of Horizontal True Positives (HTP), 
Horizontal False Positives (HFP) and Horizontal False Negatives 
(HFN), were calculated. 
voxels with an hypsometric difference with the ground truth, con 
taining all the corresponding voxels from the HFP and the corre 
sponding ones from the HTP (those with a higher altitude than the 
ground truth). Respectively, the Vertical False Negatives are the 
voxels with an hypsometric difference with the ground truth, con 
taining all the corresponding voxels from the HFN and the corre 
sponding ones from the HTP (those with a lower altitude than the 
ground truth). To this end, the 3D quantitative assessment was 
based on the measures of the 3D Completeness (detection rate), 
3D Correctness (under-detection rate) and 3D Quality (a normal 
ization between the previous two), which were calculated in the 
(Sar-poiiowing way: 
2D Completeness = 
2D Correctness 
area of correctly detected segments 
area of the ground truth 
HTP 
HTP + HFN 
area of correctly detected segments 
area of all detected segments 
HTP 
2D Quality 
HTP + HFP 
HTP 
HTP + HFP + HFN 
Moreover, for the evaluation of the hypsometric differences be 
tween the extracted buildings and the ground truth the measures 
of Vertical True Positives (VTP), Vertical False Positives (VFP) 
and Vertical False Negatives (VFN) were, also, calculated. The 
VTP are the voxels among, the corresponding Horizontal True 
Positive pixels, that have the same altitude with the ground truth. 
Note that Horizontal True Positives may correspond (i) to voxels 
with the same altitude as in the ground truth (VTP) and (ii) to 
voxels with a lower or higher altitude than the ground truth (VFN 
and VFP, respectively). Thus, the Vertical False Positives are the 
3D Completeness = 
3D Correctness = 
VTP 
VTP + VFN 
VTP 
VTP + VFP 
VTP 
3D Quality = 
VTP + VFP + VFN 
The developed algorithm has been applied to a number of scenes 
where remote sensing data was available. The algorithm man 
aged in all cases to accurately recover their footprint and over 
come low-level misleading information due to shadows, occlu 
sions, etc. In addition, despite the conflicting height similar 
ity between the desired buildings, the surrounding trees and the 
other objects the developed algorithm managed to robustly re 
cover their 3D geometry as the appropriate priors were chosen 
(Figure 1). This complex landscape contains a big variety of tex 
ture patterns, more than 80 buildings of different types (detached 
single family houses, industrial buildings, etc) and multiple other 
objects of various classes. Two aerial images (with a ground res 
olution of appx. 0.5m) and a the coarser digital surface model 
(of appx. 1,0m ground resolution) were available. The robust 
ness and functionality of the proposed method is illustrated, also, 
on Figures 3 and 4, where one can, clearly, observe the Horizon 
tal and the Vertical True Positives, respectively. The proposed 
generic variational framework managed to accurately extract the 
3D geometry of scene’s buildings, searching among various foot 
print shapes and various roof types. The performed quantitative 
evaluation reported an overall horizontal detection correctness of 
90% and an overall horizontal detection completeness of 84% 
(Table 1). 
In Figure 4c, the hypsometric/vertical difference between the ex 
tracted buildings and the ground truth is shown. With a red color 
are the VFN voxels and with a green color the VFP ones. Sim 
ilarly, at Figure 4c where the -corresponding among the HTP 
pixels- VFN and VFP voxels are shown. The performed quan 
titative evaluation reported an overall 3D completeness and cor 
rectness of appx. 86% (Table 1).