# Image Analysis Applications for Building Inter-Story Drift Monitoring
## 2 Methods  
The inter-story responses measured in this work include the inter-story displacements \(u\) and \(v\) of the ceiling and the torsion \(\theta\) with respect to the floor. While the columns of a building could deform when they are subjected to lateral forces, it is commonly assumed in structural design that the floors are rigid. For example, for the health monitoring of the first floor, the ground floor and its ceiling, which are transparently colored in Figure 1, are assumed to be rigid diaphragms. In practice, the vertical displacements of columns are relatively small as compared to their lateral displacements as the axial stiffness of a typical column is much larger than its lateral stiffness; therefore, the ceiling and the floor are assumed to be parallel. The proposed method would produce significant errors on inter-story drifts, if the building is collapsed or is partially collapsed, in which the ceiling is not parallel to the floor. Based on the above assumptions, the inter-story responses of each floor are simplified into three degrees of freedom, \(u\), \(v\), and \(\theta\), as shown in Figure 2. The displacements plotted in Figure 2 are scaled for demonstration, i.e., typical structural deformations are normally much smaller than they appear in Figure 2.  
The technical basis for monitoring the inter-story response by a single camera is to monitor the positions of two sets of targets in the images: tracking points and the reference points. It is assumed that the camera is fixed to the ceiling above the story that is being monitored, with the target points referring to objects that are fixed to the floor of the monitored story, and the reference points being objects fixed to the ceiling of the monitored story. When inter-story deformation occurs, the positions of the tracking points in the image taken by the camera change, as shown in Figure 3. By analyzing the movements of the tracking points in the images, the inter-story responses can be estimated. The reference points, which are affixed to the ceiling, are used to estimate the possible unintended movement of the camera. Even a minimal unintended movement of the camera can induce a relatively large change in tracking point and reference point positions in the image. To prevent obtaining incorrect inter-story responses, the movement of the camera needs to be estimated so that a more accurate inter-story response can be obtained. While Figure 3 presents the basic idea of the proposed method in two dimensions, the real world is three-dimensional. Figure 4 demonstrates images taken by a camera installed to align with the ceiling of the monitored story. Figure 4a presents the un-deformed inter-story. The positions of the tracking points and the reference points are continuously tracked by the camera of the single-board computer. Figure 4b-d presents the inter-story responses of \(u\), \(v\), and \(\theta\), respectively, images that represent the proposed inter-story motion model. Figure 4e-g presents the model motion of the unintended camera movement, which include the pitch, yaw, and roll, respectively. While the unintended camera movement can theoretically also have translational movements, only three rotational movements are considered here because images are generally much more sensitive to camera rotations than translations. Even a minimum camera rotation can induce significant image movement. Simplification of the  
Figure 3: A 2D inter-story response monitoring using a single camera. (**a**) Without inter-story displacement. (**b**) With inter-story displacement. Figure 2: The three degrees of freedom of inter-story responses. (**a**) Inter-story displacement \(u\). (**b**) Inter-story displacement \(v\). (**c**) Inter-story torsion \(\theta\). camera movement can also simplify the nonlinear simultaneous equations that will be introduced later, easing the convergence of their numerical solution. An accurate image measurement is based on an accurate camera calibration.