## 2 Planar Deformation Measurement Methods in Masonry Specimens
### Proposed Optical Approach  
#### 2.2.1 Measurement Model and Method  
The proposed optical measurement solution consists in the use of a single camera with a spatial position and orientation that allows the visualization of a set of targets distributed in a balanced way by different regions (in the static region surrounding the specimen and in the dynamic tested specimen surface). The use of a single camera instead of multiple cameras is justified by its simplicity, numerical stability, and less costly experimental apparatus, without compromising the intended measurement accuracy level. In the geometrical modeling of the bi-dimensional image formation process in optical systems [6], it is possible to find two main categories of camera models--perspective and affine models--which can potentially be applied to the proposed optical measurement solution. The perspective camera model (see Figure 3a) is usually applied in observation contexts that demand a highly rigorous geometrical accuracy, being related to complex and nonlinear mathematical formulation, for example, related to the lens distortion correction. The affine camera models are characterized by a higher simplicity, since they are approximations to the perspective model; therefore, they have a lower geometrical accuracy. Figure 2: Functional diagram of the quantities and mathematical models related to the compressive mechanical testing of masonry specimens. Within the generalized affine camera models, it is possible to distinguish three particular cases: orthographic, weak perspective, and paraperspective. The orthographic model (see Figure 3b) is the simplest affine camera model, where the depth of the three-dimensional component is ignored, and the optical center is located in infinity. This type of projection is also designated as parallel projection, since the projection lines established between points in space and in the image are parallel to each other and orthogonal to the image plane. It assumes a unitary scale coefficient and corresponds to an exact solution for camera modeling when using a telecentric lens (originates a constant magnification of objects seen in the field depth range, independent of the observation distance), or an approximate solution in the case of tele-objective lens (with a high focal length) and object visualization where the depth variation is quite reduced when compared with the observation distance). The addition of a non-unitary uniform scale coefficient for both image axis to the orthographic model, relating the object dimension in the world (expressed in millimeters, for example) with the corresponding image dimension (usually expressed in pixels), originates the weak perspective model, as represented in Figure 3c. In this type of projection, object points are projected to the average depth plane, \(Z_{c}\), parallel to the image plane, followed by a perspective projection with isotropic scaling to the image plane. For this reason, the weak perspective is also designated orthographic projection with uniform scaling, where a depth average value is used instead of the depth individual values of each object point (individual scaling), as it is of the case of the perspective projection. The projection error of the weak perspective model relative to the perspective model is minimized in the case of the use of a reduced focal length camera, narrow field-of-view, and reduced depth variation relative to the average depth plane. Figure 3: Geometrical camera models: (**a**) perspective; (**b**) orthographic; (**c**) weak perspective; (**d**) paraperspective (adapted from reference [6]). In the paraperspective, as shown in Figure 3d, object points in space are projected to the average depth plane (parallel to the image plane) following a direction parallel to the line that passes through the object's centroid and the camera optical center.