# 3D Monitoring of Buildings Using TerraSAR-X InSAR, DInSAR and PolSAR Capacities
## 5 Polarimetric Analysis
### Benefit of Polarimetry for Building Monitoring
#### 5.2.2 Dedicated Ground Truth  
This study has been performed on all buildings presented in Section 2. The five Front-de-Seine buildings are represented in Figure 11. The towers are represented first by their faces in the line of sight of the radar in Figure 10(a). In Figure 10(b), we represent the faces that are not seen by the radar. In this view, the buildings appear in the same order of appearance as in the polarimetric radar image represented in Figure 10(c).  
The particularities of these polarimetric images reside in the wavelength and the particular spatial resolution. These features create new processing difficulties. At this wavelength of 3 cm, scattering mechanisms are very anisotropic, and it is very hard to predict which kind of elements will dominate the signal response, because several interactions can occur with very small objects. However, it is interesting to check how and to what extent we can deduce information about the orientation of the main structuring element such as main beams of the Eiffel Tower or main fronts of more conventional towers, from the TerraSAR-X polarimetric mode.  
To this aim, 3D representation of the structures under study were used to constitute a ground truth of the orientations of our structures. From these models, we extract the orientation vector \(\mathbf{n}\) of uni-axial object:  
* For the five towers, we consider any horizontal vectors belonging to each vertical face including window sills, vertical ducts, floor separations, balcony edging profile, etc.
* For the Eiffel tower, we consider all principal beams of the structure.  
In Figure 12, we explain the notations for a horizontal element \(\mathbf{n}\) of a wall oriented with an angle \(\phi\) with respect to the trajectory.  
To calculate the theoretical orientation of any vector \(\mathbf{n}\) of our ground truth in the wave plane, it is sufficient to first calculate its projection \(\mathbf{n}^{\prime}\) in this wave-plane by \(\mathbf{n}^{\prime}=\mathbf{n}-<\mathbf{k}.\mathbf{n}>\mathbf{k}\). In the particular case represented in Figure 12 where \(\mathbf{n}\) is horizontal:  
\[\mathbf{k}=\begin{pmatrix}\sin\theta\\ 0\\ -\cos\theta\end{pmatrix},\mathbf{n}=\begin{pmatrix}-\sin\phi\\ -\cos\phi\\ 0\end{pmatrix},\mathbf{n}^{\prime}=\mathbf{n}-<\mathbf{k}.\mathbf{n}> \mathbf{k}=\begin{pmatrix}-\sin\phi\cos^{2}\theta\\ \cos\phi\\ -\sin\phi\cos\theta\sin\theta\end{pmatrix}.\]  
Figure 11: The buildings analyzed using a dedicated 3D modeling **(a)** The ground Truth with the buildings as seems as the sensor; **(b)** The Ground Truth with the far side of the buildings; **(c)** Polarimetric images of the considered buildings.  
Then the orientation angle \(\alpha\) formed between this projection and the horizontal polarimetric vector \(\mathbf{h}\), can be found by \(\cos\alpha=\frac{\mathbf{n^{\prime}}.\mathbf{h}}{\|\mathbf{n^{\prime}}\|}\). It should be noted that only a vertical vector has a simple expression of its orientation, which remains vertical. The orientation of any horizontal vector projected in the wave plane is not exactly its azimuth angle, as soon as the incidence is not zero. Indeed, the literal expression leads to an influence of the angle of incidence \(\theta\):  
\[\cos\alpha=\frac{\mathbf{n^{\prime}}.\mathbf{h}}{\|\mathbf{n^{\prime}}\|}= \frac{\cos^{2}\phi}{\sqrt{\sin^{2}\phi\cos^{2}\theta+\cos^{2}\phi}}.\]  
However, the difference between azimuth and polarimetric angle is not very important for low incidence we are considering. The relation between \(\alpha\) and \(\phi\) has been plotted in Figure 13 for several incidence angles ; the incidence angle of the images is \(34^{\circ}\).  
Figure 12: Geometrical notation for computing the relation between azimuth orientation and polarimetric orientation.  
Figure 13: Link between azimuth angle \(\phi\) and polarimetric orientation angle \(\alpha\) for an horizontal vector and several incidence angles.