10 Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns 71 Fig. 10.9 Heart-long axis cross section (top) and short-axes cross sections (bottom) To facilitate the understanding of the deformation analysis of the heart, it is necessary to provide a brief description of the heart anatomy, Fig. 10.9. The heart is a complex pumping system, consisting of two main vessels, right ventricle (RV), left ventricle (LV), a common wall, the septum. At the upper end of the ventricles are located valves controlling the inflow and outflow of blood regulated by electric signals. At each stage of the heart cycle the walls contract or expand to receive the blood inflow and to propel the blood output. The composition of the heart is complex because it has muscles, veins, arteries, nerves and connecting tissues. For the deformation analysis, the Continuum Mechanics model is utilized and the continuum replaces the actual complex configuration of the heart. If the MRI is recorded from a leaving organism there are problems that must be solved. These problems are consequence of rigid body translations and rotations inherent in a leaving organism. The detailed description of the required correction of the observed images is beyond the scope of this paper. In the images analyzed corrections have been introduced. Figure 10.10 shows tagged images of the RV and LV of a human heart, it shows the tagged images of the hearts’ RV and LV in two subsequent times of the contraction period at mid-systole. The corrections caused by rigid body motions have already been mentioned. The presence of the tagged lines on the images provided the necessary information to make small corrections for scale and rotation of the images. The reference tagging planes are captured at the instant of time that the heart begins to contract. For data processing, digital moiré is applied and to get accurate values at boundaries of the ventricles fringe patterns are extended to the full field of view [8]. To complete the data processing [2], additional carrier lines are added to the image. Since the digital moiré is utilized, the fringes in the reference configuration provide the pitch “p” of the gratings utilizing to tag the images. It should be mentioned that orthogonal carriers are not present in the vertical plane, due to the time limitations imposed by the tagging process. As indicated before about Fig. 10.3, only one projection is required. Furthermore, as indicated in the analysis of the derivatives, the missing derivatives can be computed by the corresponding eqs.
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