Contour recovery in Echocardiography

In clinical cardiac studies the evaluation of cardiac function using parameters of left ventricular contractibility is an important constituent of an echocardiographic examination. These parameters are derived using boundary tracing of endocardial borders of the Left Ventricle (LV). The extraction of the boundary of the LV comprises two steps, once the ultrasound image of a cross section of the LV is given, see Figure 1(a)-(d). First an edge detection is applied to the ultrasound image to detect the boundary of the LV, cf. Figure 1(c).

However this procedure may be hampered by the presence of interfering biological structures (such as papillar muscles), the unevenness of boundary contrast, and various kinds of noise [3]. Thus edge detection often provides only a set of nonuniformly spaced, perturbed boundary points rather than a connected boundary. Therefore a second step is required, to recover the original boundary from the detected edge points, cf. Figure 1(d). Since the shape of the Left Ventricle is definitely smooth, bandlimited functions are particularly well suited to model its boundary.

Figure 1(d)-(f) demonstrate the importance of determining a proper degree for the approximating polynomial. The approximation displayed in Figure 1(e) has been computed by choosing the bandwidth $ M$ (more generally speaking: regularization parameter) too small, we obviously have underfitted the data. The overfitted approximation obtained by choosing a too large $ M$ is shown in Figure 1(f). The approximation shown in Figure (d) has been computed by the multi-level algorithm. This algorithm autmatically adapts to the optimal smoothness, and is at the same time computationally very efficient. For more details see [1,2].

Figure 1: The recovery of the boundary of the Left Ventricle from 2-D ultrasound images is a basic step in echocardiography to extract relevant parameters of cardiac function. The contour in (d) has been computed by the multi-level method, which automatically adapts to the solution of optimal smoothness. The algorithm provides the optimal balance between fitting the data and preserving smoothness of the solution. Figures 1(e) and (f) illustrate ``reconstructions'' obtained when choosing a polynomial degree that is too low (underfitting) or too high (overfitting).
\subfigure[2-D echocardiography]{
...d solution]{


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