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where Ni i = 0 m, are polynomial base functions. We use one-dimensional isoparametric finite element shape functions [27] as the base functions in order to control the numerical fluctuation in the x, y and z curves [28]. The normalized indices of the skeleton voxels are selected as the local pn is coordinate r, with 0 < r < 1. Thus, the local coordinate of voxel pj in branch L = p0 p1 r pj = j/n.
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With a standard least squares fitting method, the parameter sets A, B and C in Equation (20.6) can be computed by minimizing the following three merit independent functions:
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where n is the total number of voxels in a branch and ri is the local coordinate of voxel pi = p xi yi zi . These standard merit functions are best suited for least squares data modeling with dense and evenly located discrete data. However, nonuniformity in skeleton voxel location and insufficient input data for a high-order polynomial fitting can lead to artificial fluctuations in the resulting curve. In order to solve this problem, we expand the above merit functions with additional controlling terms:
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Q1 Q2 and Q3 are stabilizing penalty functionals designed to impose smoothness in the fitted curve and are defined as: Q1 A = Q2 B = Q3 C =
1 0 1 0 1 0
xA r r yB r r zC r r
+ + +
xA r r2
dr dr dr (20.9)
yB r r2 zC r r2
where and are predefined weight coefficients. The functionals in Equation (20.9) are analogous to the strain energy of an elastic beam subjected to tension (the first term in the integral) and bending (the second term). Thus, the smoothing weight restricts large stretches in the fitted curve, while restricts excessive bending deformation. The larger the smoothing weights, the stiffer the fitted curve will become (see Figure 20.9). The selection of smoothing weights is heuristic and problem dependent. Based on the classical beam theory, we set = 2 0 . Therefore, only one independent smoothing weight needs to be specified. The saddle points of each merit function Equation (20.8) are obtained through first-order differentiation with respect to the parameter sets A, B and C, respectively. For example, for the solution of parameters A, the condition 1 A / ai = 0 yields the set of linear algebraic equations tij aj = fi for i = 1 m and j = 1 m (20.10)
Topological Segmentation of Discrete Curve Skeletons
40 35 30 25 y 20 15 10 5 0 0 5 10 x 15 20 original data alpha = 0.0 alpha = 0.001 alpha = 0.1
Figure 20.9 Polynomial curves fitted with various smoothing weights. where coefficients tij and fi are given by:
tij =
k=1 n
Ni rk Nj rk + xk Ni rk
Ni r r
Nj r + r
Ni r r2
Nj r dr r2 (20.11)
fi =
The linear system given above can be solved with a standard Gaussian elimination procedure [29]. Similarly, we compute the parameter sets B and C by solving respectively the linear systems obtained from 2 B / bi = 0 and 3 C / ci = 0. As shown in Figure 20.10, our global data modeling
Figure 20.10 Superimposed smoothed skeletons for the object in Figure 20.7 obtained from an image with random noise (dashed black curve) and an image without noise (thick gray curve).
procedure produces a smooth and stable curve representation of the discrete skeleton and effectively corrects local distortions due to image noise.
6. Results
We present the skeleton modeling results for two typical examples of trabecular biological tissue. Shown in Figure 20.11(a) and Figure 20.12(a) are the 3D binary images of trabeculated myocardial tissue in an HH21 chick embryonic heart, and trabecular bone tissue sampled from the human iliac crest, respectively. Figures 20.11(b) and 20.12(b) show the segmented and filtered discrete curve skeletons for the trabeculated myocardium and the trabecular bone. The relevant topological and geometrical
Figure 20.11 Trabeculated myocardium specimen in an HH stage 21 chick embryo. (a) 3D binary image; and (b) computed raw curve skeleton (point representation).