Protocol for Spline Fitting Over Boundaries

The last protocol consists of fitting a Bezier curve (spline) to the boundaries. There are two-fold purposes in our protocol: (a) To make the boundary curves smoother and (b) to make the points on the boundary curves equidistant. We use the methodology discussed in Graphics Gems for spline-fitting, and we used curve interpolation for making the curve equidistant. Both these effects show a reduction in the mean error. Figure 9.42 shows the effect of

Figure 9.42: Effect of fitting splines over the estimated boundaries. Top left: MRF, PDM, with and without splines. Top right: MRF, SDM, with and without splines. Bottom left: FCM, PDM, with and without splines. Bottom right: FCM, SDM, with and without splines.
Figure 9.43: Optimization curves. Left: a2 = 500. Right: a2 = 1000.

fitting splines over the estimated boundaries. There are four parts in this figure showing the effect of splines over two classification systems, using two distance methods: (a) MRF using PDM, (b) MRF using SDM, (c) FCM using PDM, and (d) FCM using SDM. In all four subprotocols, we find the same behavior that the spline-fitted mean errors are lower than nonspline-fitted mean errors. We also observed that there is a very consistent standard deviation error for all four subprotocols. We also did lumen shape optimization on fitted spline shapes, and this can be seen in Fig. 9.43 (left and right). As the number of points on the boundary increases, the mean error drops and reaches a stage of convergence.

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