## 4 Thermographic Analysis on Frescoed Masonry Walls and Results
### From the Masonry Recognition to the Identification of Mechanical Properties  
Non-destructive testing (NDT) techniques can be used for evaluation of mechanical properties as well as for characterization of microstructural features such as grain size, texture, nucleation, defects, deformations, or damages in various contexts, also concerning ancient masonries [27; 28; 29]. In particular, thermographic testing presents the potential to be the solution for studying masonry with both finance and time efficiency, which mainly relies on non-contact optical measurement methods in the sake of safeguard of the parietal art goods. This section discusses the evaluation of structural and mechanical properties starting from thermographic surveys. The subsequent post-processing and numerical analysis techniques are aimed at an identification of the masonry quality and typology as an a priori condition that influences the seismic vulnerability of heritage buildings. The images obtained from thermographic surveys are used to estimate the elastic characteristics of masonry texture through the method of the test-windows [30]. This  
Figure 12: (**a**) Portion of the fresco under investigation, (**b**) highlighting of colours of different gradation, (**c**) thermal image with improved contrast, (**d**) final binary image (see the cones of visions 6 and C in Figure 2). Figure 13: (**a**) Portion of the fresco under investigation, (**b**) thermal image where are highlighted in red the edges of what remains of the frescoes, (**c**) thermal image with improved contrast, (**d**) final binary image (see the cones of vision 7 and D in Figure 2). method estimates the stiffness matrix which relates the sample average of the stress vector and the average of the strain vector as follows:  
\[\left\langle\sigma\right\rangle=C^{Hom}\left\langle\varepsilon\right\rangle, \tag{7}\]  
where  
\[\left\langle\sigma\right\rangle=\frac{1}{\left|\Omega\right|}\int_{\Omega} \sigma\,d\Omega,\quad\left\langle\varepsilon\right\rangle=\frac{1}{\left| \Omega\right|}\int_{\Omega}\varepsilon\,d\Omega \tag{8}\]  
and \(\Omega\) is the region occupied by the masonry sample. It is worth noting that \(\sigma\) and \(\varepsilon\) are the stress and strain vectors according to [PERSON]'s notation, and assuming plane-stress hypothesis they can be written as  
\[\sigma=\left\{\sigma_{xx},\sigma_{yy},\sigma_{xy}\right\}^{T},\varepsilon= \left\{\varepsilon_{xx},\varepsilon_{yy},\varepsilon_{xy}\right\}^{T}. \tag{9}\]  
Therefore, \(C^{Hom}\) is a 3-by-3 matrix. The value of the stiffness matrix \(C^{Hom}\) is estimated by applying two different sets of boundary conditions (b.c. ): (i) essential b.c., where displacements are applied to the boundary of the sample, and (ii) natural b.c., where forces are applied to the boundary of the sample. In particular, in essential b.c. if displacements are applied that would induce uniform strain equals to \(\varepsilon=\{1,0,0\}^{T}\) in a homogeneous material, the first column of the estimate of \(C^{Hom}\) is given by the average of the stresses, \(\{C_{11},C_{21},C_{31}\}^{T}=\left\langle\sigma\right\rangle^{T}\), and so on. The same is true for natural b.c. : if forces inducing uniform stress equal to \(\sigma=\{1,0,0\}^{T}\) in a homogeneous material are applied, the first column of the estimate of \(S=\left(C^{Hom}\right)^{-1}\) is given by the average of the strain, \(\{S_{11},S_{21},S_{31}\}^{T}=\left\langle\varepsilon\right\rangle^{T}\), and so on. By applying essential b.c. an upper estimate of \(C^{Hom}\) is obtained, denoted \(C^{E}\), while employing natural b.c. a lower estimate of \(C^{Hom}\) is obtained, denoted \(C^{N}\). As the size of the sample increases, the range between \(C^{N}\) and \(C^{E}\) decreases, and when it is sufficiently small it is assumed that the Representative Volume Element (R.V.E.)