## 2 Planar Deformation Measurement Methods in Masonry Specimens
### Proposed Optical Approach
#### 2.2.1 Measurement Model and Method
In this study, the weak perspective model or the orthographic model with uniform scaling, \(K\), is adopted, allowing to relate the three-dimensional point position (\(X\), \(Y\), \(Z\)) in the world (expressed in millimeters, for example) with the corresponding bi-dimensional position (\(x\), \(y\)) in the image (usually expressed in pixels), i.e.,  
\[\Delta X=K\cdot\Delta x \tag{6}\]  
\[\Delta Y=K\cdot\Delta y \tag{7}\]  
where \(\Delta X\) and \(\Delta Y\) represent the planar displacement of the point between two consecutive time observations, in the \(X\) (horizontal) and \(Y\) (vertical) directions, respectively, of the world coordinate system. In a similar way, \(\Delta x\) and \(\Delta y\) represent the corresponding planar point displacement in the \(x\) and \(y\) directions of the image coordinate system. In the proposed approach, a measurement referential composed of reference targets is placed in front of the observation region in the masonry specimen (with length \(l\) and width \(w\)) at the minimum reasonable distance from the specimen surface in order to minimize the observation depth difference to the monitoring targets fixed and scattered in the observation region (in the inner region of the referential), as shown in Figure 4.  
As shown in Figure 4, the targets positioning on the specimen allows determining horizontal and vertical deformations, including bricks and mortar joints, making the deformation measurement representative of the overall deformation of the masonry specimen. The mentioned referential is subjected, before the specimen testing, to dimensional measurement aiming at the determination of the three-dimensional world position of each reference target. The knowledge of these spatial coordinates supports the determination of the scale coefficient in each acquired image, since the measurement referential is placed in a static region of the specimen experimental setup (assuring that it does touch the specimen and is not subjected to vibrations produced by the testing machine). Through digital image processing, it is possible to obtain the corresponding two-dimensional image positions of the observed reference targets and calculate the scale coefficient between pairs of targets using the expression  
\[K_{i,j}=D_{i,j}/d_{i,j} \tag{8}\]  
Figure 4: Schematic representation of the proposed optical measurement method. where \(i=j=1,2,\ldots,n\), with \(i\
eq j\) and \(n\) being the total number of reference targets in the referential, and \(D_{ij}\) and \(d_{ij}\) are, respectively, the distances between the two targets \(i\) and \(j\) in the world and in the image coordinate system, i.e.,  
\[D_{i,j}=\sqrt{[(X_{i}-X_{j})^{2}+(Y_{i}-Y_{j})^{2}+(Z_{i}-Z_{j})^{2}]} \tag{9}\]  
\[d_{ij}=\sqrt{[(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}]}. \tag{10}\]  
Since the measurement referential can have several pairs of reference targets, an average value of the scale coefficient and the correspondent standard deviation of the experimental sample can be obtained. This calculation procedure (represented in Figure 5) contributes to the reduction of the scale coefficient measurement uncertainty components related to the effect of the lens distortion and planar misalignments between the specimen surface, the measurement referential, and the image plane. It should be noticed that the reference target's image coordinates can also be used to assess the referential static position stability and the turbulence effect on the optical path between the targets and the camera by analyzing low and high-frequency fluctuations in their position during testing.