2.2.2 Reference Configuration This module controls the video system and the DAQ system. It acquires images containing the surface markers from the vertical and inclined cameras and stores their centroid locations. Also it acquires data from DAQ and stores the initial readings of the load cell. All these data namely the centroid positions of the surface markers from both the cameras and the load cell readings are stored in a text file at the end of the recording. 2.2.3 Syringe Pump Module This module controls two stepper motors and the syringe pump. The stepper motors are actuated to axially stretch the blood vessels to a predetermined value then the syringe pump is actuated to inflate the blood vessel. By controlling the speed at which the motors are actuated and the infusion/withdrawal rate of the syringe pump the rate at which the blood vessel is deformed can be altered. Using this module cyclic inflation tests at constant length or cyclic axial extension test at constant pressure with user specified number of cycles can be performed. Apart from this, the module has all the functionalities of the reference configuration module. So as the pump (or motors) load/unload the specimen it also acquires data from the cameras and load cells simultaneously. It then stores the load and pressure values and the location of the centroid of the markers in a text file at the end of the protocol. 2.2.4 Video Calibration Here a calibration wedge (see figure 1b) containing markers separated by 2mm in the horizontal and 1.5mm in vertical direction is imaged in both the vertical as well as the inclined camera. Since, here we need to just record the centroid of stationary markers, which is what we do while recording the reference configuration too, the same reference configuration recording module is used for video calibration. Then, the relationship between the 3D coordinates of these markers and the two projections of these markers onto the imaging plane is established off-line. 2.3 3D Reconstruction from Two Perspective Views The video subsystem has to be calibrated so as to obtain the 3D coordinates of the markers from two perspective views. This is achieved using global reconstruction techniques. The relationship between 3D coordinates of the centroid of the markers referenced with respect to a laboratory coordinate system (X,Y,Z) and its 2D coordinates obtained by projecting it on to the imaging surface and referenced with respect to the camera coordinate system ((uv, vv) or (ui, vi) depending on which camera is being referred to) is assumed to be linear, i.e., PkX = uk – ok , (1) where k can be v or i, referring to the vertical camera and the inclined camera respectively, X denotes the 3D coordinates of the centroid of the marker, uk denotes the 2D projected coordinates of the centroid of the marker on the kth camera, ok denotes the 2D projected coordinates of the origin of the laboratory coordinate system on the plane of the imaging surface of the kth camera, Pk denotes the projection matrix for the kth camera. The projection matrix is a 2 by 3 matrix. Thus, for each camera we have to find 8 constants (6 in the projection matrix(Plm) and 2 corresponding to the projected coordinates of the origin of the laboratory coordinate system). For this we require the coordinates of the centroid of at least four non-coplanar markers both in the laboratory coordinate (X) and the projected camera coordinate system (uk). Usually more than 4 markers is used to find the eight constants and hence these calibration constants are found in a least square sense. During the experiment we need to find the 3D coordinates of the centroid of markers, X, given the two 2D coordinates of the projections of them onto the imaging surface. For this the relation (1) has to be inverted. However, now there are four equations with just three variables. Hence, these equations are also solved in least square sense to obtain the 3D coordinates of the centroid of the markers. The assumption that the relationship between the laboratory coordinates and the projected camera coordinates is linear holds only if the camera lens provides uniform magnification over the depth of view and there is no distortion in the field of view. The validity of these assumptions is verified by examining if no strain is developed due to rigid body translation along any axis. It is found that the invariants deviate from the expected value by ±10í2 due to these rigid body translations, irrespective of the axis, as long as one is in the depth and field of view. Hence, it is concluded that the relationship between the laboratory coordinates and the projected camera coordinates is linear and that the accuracy of the measured invariants is ±10í2. 2.4 Determination of Deformation Gradient In this work, 9 to 12 markers are placed on the surface of the vein and are tracked as the specimen is loaded. In general, it is not possible to experimentally determine the exact deformation field by tracking 12 markers (or for that matter even a larger number of markers), we can at most get a reasonable approximation of the deformation field and its gradient locally. However, we can verify if the observed deformation field is consistent with that theoretically predicted. Since, we are going 81
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