29 A Comparison of Computer-Vision-Based Structural Dynamics Characterizations 297 Fig. 29.1 (a, b) The CMOS high speed camera and the cantilever beam, (c) field of view for the camera capturing the sequence of images of the vibrating beam I .x C x; y C y; t C t/ DI .x; y; t/ C @I @x x C @I @y yC @I @t t CH:O:T: (29.2) Substituting (29.2) into (29.1) and dividing by t will result the flowing equations for each pixel which is known as the optical flow general equation or referred to as the aperture problem in computer vision community. @I @x x t C @I @y y t C @I @t t t D0: (29.3) It is more conventional to write Eq. (29.3) in the following format in which Vx and Vy are the components of optical flow vectors at each pixel. @I @x Vx C @I @y Vy CIt D0: (29.4) As it is clear there is only one governing equation at each pixel and two unknowns Vx and Vy, as a result there is no unique solution for the aperture problem to estimate the optical flow components. Several approximate methods have been introduced in order to obtain the optical flow vector field including Lucas-Kanade [16] and Horn-Schunck [17]. In this paper iterative Lucas-Kanade method has been used to solve the aperture problem to extract the motion field between two consecutive frames of a sequence of images. The underlying assumption in Lucas-Kanade method is that the nearby pixels have equal optical flow vector components. In this case additional constraints are introduced to the problem and made the aperture problem an over constrained set of equations. This over constraint set of equations are then solved using the least squares algorithm. After estimation of the optical flow vectors at each pixel, interest points are selected on the first frame of the captured video from the structure using Harris corner detection algorithm. The point tracking is performed by moving the interest points with respect to the estimated motion filed provided by Lucas-Kanade algorithm frame by frame. As a result, the interest points will follow the optical flow vectors from one frame to another and point tracking is performed frame by frame. In order to make the results of the Lucas-Kanade algorithm more reliable and robust to noise, disturbances and also to be able to handle large motions in the scene, the algorithm is modified to work in an iterative fashion. The theoretical background of performing iterative Lucas-Kanade is beyond the scope of this paper and the details for the method could be found in references. Figure 29.2 shows three snap shots of the Lucas-Kanade point tracking performance. The points are following the motion pattern as expected. Although the tracking is not performed perfectly, the natural frequencies of the cantilever beam can
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