Given the displacement measurements, several methods have been proposed for estimating the dense displacement field w(x, t) throughout the LV [4,7,40,41]. Each method is unique, but all share the same basic approach, including a similar sequence of steps. In the initial step, the LV is segmented to define its 3D extent and it is registered to a standard coordinate system. Then, a model framework for describing motion is chosen and the parameters within that model are optimized to fit the observed 1D displacements. The approaches used to perform these steps are what distinguishes the various proposed methods.
time frame to time frame to landmarks that move with the LV. A second coordinate system option is a fixed Euclidean system defined by the tag direction unit vectors e1, e2, and e3.
The most significant difference between methods is how the displacement estimate u(x, t) is defined mathematically and fitted to the data. One approach is field fitting, in which scalar functions of 3D space are individually fitted to each of the three components of displacement corresponding to points in T1, r2, and r3. The form of the field is generally a sum of basis functions (x) of the form u(x, t) = ak(Wkm, k=l where (t) are coefficients determined from the observed displacements and K is specified by the method. Such an approach is proposed by O'Dell et al.  using some 50 Legendre polynomial basis functions. Another choice is B-spline basis functions . To find the time varying coefficients of the model, a least squares fit to the data is used that minimizes
Finally, the estimates obtained for each i are compiled into a single 3D estimate u(x, t) = u1(x, t)e1 + u2(x, t)e2 + u3(x, t)e3.
An alternative approach is to use finite element modeling, in which the LV is broken down into individual elements. The movement of material within each element is assumed to be determined entirely by the movement of the element's corners. The displacement estimate is therefore given by
FIGURE 9 Tag line tracking actually amounts to tracking tag planes as they deform. Combining three orthogonal sets of tag planes defines a deforming 3D grid superimposed on the deforming LV (a). From the grid, the 3D motion of points is only known for the grid intersections shown at end-diastole (b) and end-systole (c). Only those points within the LV myocardium are shown.
In the segmentation and registration step, the boundaries of the LV are first identified in the images and subsequently interpolated to approximate the 3D shape of the LV. The accuracy of this segmentation is not critical to the motion estimation process. A coordinate system is also imposed on the LV to identify the segmented points. A common choice of systems is prolate spheroidal coordinates (see [7,40]) that reflect the elliptical shape of the LV. A disadvantage of this choice is that the coordinate system has to be registered from
where x1,..., xK are the corners of the element within which x lies and |k (x) are weights determined by the position of x within the element. In the finite element model of Young et al.  the LV is divided into the 16 elements shown in Fig. 10 and the weights are given by tensor product basis functions. Denney and McVeigh , on the other hand, use numerous small cubic elements (approximately 1 mm on a side) and simple trilinear interpolation weighting. To fit the model to the data, the corner points must be moved such that the model best matches the observations. This is accomplished by minimizing an objective function such as
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