Jgt

argmin

^image(d(k, «„)) + ^internal(k: tn)), d(l,t«),...,d(K,t«) k=l

^internal(d(k, t«)) = fljd(k + l, t«) - d(k, «J^ + fl2^d(k + l, t«) + d(k - l, t«)-2 * d(k, t„)||2.

The two components of this internal energy function approximate the squared magnitudes of the first and second derivatives, respectively, thereby resisting stretching (large first derivative) and bending (large second derivative). The weights a1 and a2 are empirically chosen for good performance. Second, the image energy at the kth point is given by the image brightness at the displaced location

FIGURE 5 Template matching: First a search region, such as the white box, is selected from a tagged image (a), shown here at high magnification. Then, the image intensities of pixels within the search region are fitted to a template (b), which provides the location of the tag line s0. Repeating this process and tracking over time produces a complete set of identified tag points with corresponding values of m (c). Also shown are the identified boundaries of the myocardium.

which is found using gradient descent methods beginning with d(k, tn) = 0.

An example using the active geometry method of Young et al. [7] is shown in Fig. 6, in which 20 snakes with two orientations track a grid of tag lines. Some unique characteristics of their method are, first, they use coupled snakes where intersections are shared by two snakes and are thus subject to two sets of internal forces; second, they add an interactive, user-defined energy to force the snake away from incorrect local energy minima where necessary; and third, they use hand-drawn boundaries to turn off the energy associated with points outside the myocardium (shown as diamonds in Fig. 6b), thereby preventing the snake from being influenced by extramyocardial data. Nevertheless, active geometry methods are generally less sensitive to accurate boundary information and often work well with no boundary information whatsoever. Another variation on this theme is the method of Amini et al. [23] in which the snake is parameterized by a set of B-spline control points. The inherent smoothness properties of the B-spline replace the need for internal snake forces.

FIGURE 6 Using the active geometry method of Young etal. [7], a 10 x 10 grid of coupled snakes is superimposed on the first image in the sequence (a). After energy minimization, the snakes move at a later time to deformed positions (b). The diamonds in the image are snake points that have been turned off because they are outside the myocardium. [Courtesy of A. Young.] From [7] with permission. © 1995 IEEE.

FIGURE 6 Using the active geometry method of Young etal. [7], a 10 x 10 grid of coupled snakes is superimposed on the first image in the sequence (a). After energy minimization, the snakes move at a later time to deformed positions (b). The diamonds in the image are snake points that have been turned off because they are outside the myocardium. [Courtesy of A. Young.] From [7] with permission. © 1995 IEEE.

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