10 Extension of the Monogenic Phasor Method to Extract Displacements and Their Derivatives from 3-D Fringe Patterns 73 x1 x2 x3 dp12 dp13 u1 u2 u3 A B B A C D Iv Iu P P P Displacements P Long axis Iw1 Fig. 10.11 The figure illustrates the process of retrieving the displacements components at a given point of the left ventricle. The vectors converge to the point P indicating compressive displacements Fig. 10.12 Displacement components (expressed in m) of the left ventricle J D 2 6 6 6 6 6 4 @u1 @x1 @u1 @x2 X @u2 @x1 @u2 @x2 X X X @u3 @x3 3 7 7 7 7 7 5 (10.34) These missing derivatives are computed utilizing the relationships among derivatives. After all the components of Eq. (10.34) are computed, the simplified Almansi strain tensor is obtained. For example, eE x1 D @u1 @x1 1 2 " @u1 @x1 2 C @u2 @x1 2 C @u3 @x1 2# (10.35) eE x2 D @u2 @x2 1 2 " @u1 @x2 2 C @u2 @x2 2 C @u3 @x2 2# (10.36) eE x3 D @u3 @x3 1 2 " @u1 @x3 2 C @u2 @x3 2 C @u3 @x3 2# (10.37) Figure 10.13 shows the Eulerian components of the simplified Almansi strain tensor along the coordinate axis.
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