## Shape Optimization Protocol

In this protocol, we study and analyze the shape characteristics of the lumen with respect to the number of points on the boundary. We know that as the number of

 /\so \ / À-\ \ -•-pou & O 20 40 60 et 100 120 Figure 9.41: Sampling protocol test: Left: Shape optimization test. Right: Concentric shape decagon test. Two circle contours each of radius 60 pixels have their centers separated by 60 pixels. The mean errors given by the PDM and the SDM are plotted against the number of points on the circular contours. As the number of points on each of the contours increases, the difference in the mean errors decreases, and both errors approach an actual value. points increases on the boundary, the boundary becomes more smooth, but we do not know as to how many points are necessary on the boundary to represent the best lumen shape. Figure 9.41 (left and right) demonstrates the mean error around the boundary versus the number of points on the lumen boundary. As the number of points increases from 10 to 120, the mean error drops rapidly using PDM and SDM methods. Using PDM, the mean error drops rapidly when the number of boundary points increases from 10 to 30 and reaches a stage of convergence when the number of points is 50. The same pattern is observed using the SDM method and the mean error falls rapidly from points 10 to 50 and reaches a stage of convergence when the number of points on the boundary is 80. The stage of convergence here means that there is no more change in the mean error, if the number of points increases beyond a certain limit. Lastly, the fall of the errors as the number of points increases is more rapid for SDM compared to that of PDM, and the starting error (when total points are 10) in SDM is much larger compared to that of PDM. A similar experiment was done synthetically when the boundaries are concentric shapes. We took a simple shape of a concentric decagon (with radius 20 and 50 pixels) and increasing the number of points from 10 to 120. Since the boundaries were concentric, the point of convergence was same (70 points) for both PDM and SDM (see Fig. 9.41, right).
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