218 H. Brand et al. In the method, local phase oscillations are extracted according to the method discussed in [9]. Complex steerable pyramid filters are applied to a time series of images [11]. Each level of the pyramid filter are Gaussian windowed sinusoids (at certain spatial frequencies) which allow local amplitude and phase to be characterized at a finite spatial extent. Utilizing this filter over a time series of image data allows for local amplitude and phase over time to be characterized. The object will likely exist as a relatively constant contours in phase space [10]. Showed that the motion field can be estimated by tracking contours in phase space and that phase is more robust to illumination conditions and object distance than intensity-based tracking. This allows individual changes in phase corresponding to complex in-plane motion to be characterized on a local pixel basis. For oscillating responses, a temporal filter is employed to extract pixel level periodic oscillations in phase while removing the DC component. Assuming the dominant direction of the deflection of the material is horizontal the filters are oriented to describe phase changes in the horizontal direction. The extracted phase oscillations therefore, describe the materials response to perturbations. Assuming the physical load of the material remains in the linear elastic regime, the overall response of the material can be well described according to a linear combination of modes excited by the load. In practice there are more pixels capturing the in-plane motion then there are active modes. PCA is used to for dimension reduction of the response. Additionally, PCA is able to retain the frame’s high spatial-resolution. The principle components of the system are related to the modal components by linear mixing transformation due to non-uniformities is the system’s mass distribution. Complexity pursuit is used to estimate the mixing transform and modal coordinates by exploiting that predictability of statistical independent source signals should be greater than that of their linear mixtures. The complexity pursuit algorithm is an efficient method of separating modes from superposed signals for close, highly damped signals. For a linear system the highest predictable fundamental components should be single sinusoids at natural frequencies corresponding to mode shapes. After modal coordinates are found, a simple Hilbert transform is used to calculate the damping ratios for the video data. This method utilizes an envelope to estimate the damping of the system as the specimen oscillates. Additionally, the displacements of the pixels are found using the superposition of the mode shape multiplied by the time series. The actual metric deformation of the specimen is calculated by multiplying the pixel displacement by the spatial resolution of the lens. The spatial resolution is found by using the pinhole camera model and by using the focal length of the lens, along with the sensor size and the number of pixels in the image. In exploring the use of this technique for analyzing modal properties of polymers at intermediate strain rates, there were a set of challenges that needed to addressed by the experimental setup: 1. Excitation: The excitation strategy needs to be capable of exciting high frequency modes which can be challenging for stiff polymers. Additionally, the imager method needs to observe a decaying response within these modes. Dynamics must be maintained in the linear regime. 2. Imaging: the viscous properties of the material create highly damped responses. This provides challenges to imaging in the following ways: (a) Rapidly decaying responses require high frame rates. (b) The material undergoes a very low-amplitude response requiring high spatial resolution. 3. Illumination: the extremely high time resolution of the imaging process requires a really high energy and stable light source. 22.3 Experimental Setup A10mm×10mm×85 mm rectangular prism, Teflon specimen was used for the experiment. Markings were applied to the specimen to assist in the phased-based optical flow motion detection technique. The specimen was clamped down using a small vice which was then connected to an aluminum block mounted on a linear guide. A Labworks ET-132-2 shaker was attached to the aluminum mount and used to excite the specimen in the kHz range. The shaker was suspended from a frame by bungee cords to allow for adjustment during testing. This experimental setup is portrayed in Fig. 22.2.
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