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9.6 Perturbation of generic umbilics

In this section, we give a few numerical examples to demonstrate that generic umbilics are stable with respect to perturbations [257]. The example surface is a wave-like bicubic integral Bézier patch which is illustrated in Fig. 8.11. The surface is anti-symmetric with respect to $ x=0.5$ . There are four spherical umbilics and one flat umbilic point on the surface. We gradually perturb the control points of the surface and observe the behavior of the lines of curvature which pass through umbilics. The control points are perturbed in the following manner. Since the example is a bicubic patch, it has 16 control points. Each control point consists of three Cartesian coordinates $ x,\;y,\;z$ , hence there are 48 components to be perturbed. A random number which varies from $ -1$ to 1 is used to determine the 48 components. Let us denote the randomly chosen numbers for each control point as $ (e_{ij}^x, \; e_{ij}^y, \;e_{ij}^z),\;0 \leq i \leq 3,\;0\leq j \leq 3$ . We normalize the vector and add to each control point as
$\displaystyle \tilde{\bf P}_{ij} = {\bf P}_{ij} + \zeta \frac{(e_{ij}^x, \; e_{ij}^y, \;e_{ij}^z)^T}{\sqrt{{e_{ij}^x}^2 + {e_{ij}^y}^2 + {e_{ij}^z}^2}}\;,$     (9.73)

where $ \zeta $ is a constant. We increase the amount of perturbations gradually by increasing $ \zeta $ from $ 0.02$ by $ 0.02$ up to $ 0.08$ . The curvature value $ \kappa$ , the four coefficients of the cubic terms $ \alpha,\;\beta,\;\gamma,\;\delta$ , angles of the tangent lines to the lines of curvature which pass through the umbilic in the tangent plane of the Monge form $ \theta_1$ , $ \theta_2$ , $ \theta_3$ in the 3-D space, $ \phi_1$ , $ \phi_2$ , $ \phi_3$ in the $ uv$ -space all in radians, index and the type are listed for original surface and two perturbed surfaces ( $ \zeta = 0.04$ and $ \zeta = 0.08$ ) in Tables 9.1 to 9.3. The angles $ \theta_i, \;\phi_i\;(1 \leq i \leq 3)$ are restricted in the range $ -\frac{\pi}{2} \leq \theta_i,\;\phi_i \leq \frac{\pi}{2}$ . Figures 9.6 to 9.8 illustrate how the lines of curvature which only pass through the umbilic behave when the control points are perturbed. The thick solid line represents the lines of curvature for maximum principal curvature, thick dotted line represents the lines of curvature for minimum principal curvature and the thin solid lines are the iso-parametric lines of the wave-like surface.

From the figures and tables we can observe that the umbilic on the upper right jumps off from the domain but the other four umbilics remain inside the domain. All the umbilics which stay in the domain do not change their index nor their type. In Fig. 9.6, when the perturbation is zero, lines of curvature passing through the umbilics at lower left (0.211,0.052) and upper left (0.211,0.984) have a common line of curvature. Similarly lower right umbilic (0.789,0.052) and upper right umbilic (0.789,0.984) have a common line of curvature. As the perturbation gradually increases, they split into two lines of curvature as shown in Figs. 9.7 and 9.8. Note that in Figs. 9.7 and 9.8 the lines of curvature corresponding to the jumped off umbilic (upper right) are not shown, since we cannot compute the initial values for the integration. From these observations we can conclude that the umbilics are quite stable to the perturbation. Also the locations and the angles $ \theta_i, \;\phi_i\;(1 \leq i \leq 3)$ of the umbilics do not move nor rotate too much.

Figure 9.6: Lines of curvature passing through the umbilics, $ \zeta = 0$ (adapted from [257])
\begin{figure}\centerline{\psfig{figure=fig/csin1_loc.PS,height=4.5in}}\end{figure}


Table 9.1: Umbilics of original surface (adapted from [257])
u 0.211 0.211 0.789 0.789 0.5
v 0.052 0.984 0.052 0.984 0.440
$ \kappa$ 1.197 0.267 -1.197 -0.267 0.
$ \alpha$ 4.147 0.926 4.147 0.926 6.514
$ \beta$ -18.306 14.670 18.306 -14.670 0.
$ \gamma$ 0. 0. 0. 0. 4.2763
$ \delta$ -2.337 1.411 2.337 -1.411 0.
$ \theta_1$ 0.671 0.562 0.592 0.638 0.
$ \theta_2$ -0.592 -0.638 -0.671 -0.616 -0.604
$ \theta_3$ 1.571 -1.571 -1.571 1.571 0.604
$ \phi_1$ 0.567 0.562 0.495 0.583 0.
$ \phi_2$ -0.495 -0.583 -0.567 -0.562 -0.752
$ \phi_3$ 1.571 1.571 -1.571 -1.571 0.752

index		 - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ $ \frac{1}{2}$
type star star star star monstar

Figure 9.7: Lines of curvature passing through the umbilics, $ \zeta = 0.04$ (adapted from [257])
\begin{figure}\centerline{\psfig{figure=fig/csin1_pert2_loc.PS,height=4.5in}}\end{figure}


Table 9.2: Umbilics on perturbed surface, $ \zeta $ = 0.04 (adapted from [257])
u 0.190 0.214 0.794 n/a 0.492
v 0.055 0.978 0.081 n/a 0.424
$ \kappa$ 1.293 0.136 -1.458 n/a 0.083
$ \alpha$ 2.390 0.551 5.014 n/a 6.351
$ \beta$ -16.119 13.046 18.926 n/a 0.163
$ \gamma$ 0.563 -1.524 -0.360 n/a 4.666
$ \delta$ -3.182 1.711 3.234 n/a 0.510
$ \theta_1$ 0.658 0.593 0.586 n/a 0.701
$ \theta_2$ -0.623 -0.667 -0.689 n/a -0.055
$ \theta_3$ 1.551 1.509 1.560 n/a -0.644
$ \phi_1$ 0.559 0.549 0.528 n/a 0.857
$ \phi_2$ -0.537 -0.589 -0.596 n/a -0.076
$ \phi_3$ 1.529 1.552 -1.532 n/a -0.811

index		 - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ n/a $ \frac{1}{2}$
type star star star n/a monstar

Figure 9.8: Lines of curvature passing through the umbilics, $ \zeta = 0.08$ (adapted from [257])
\begin{figure}\centerline{\psfig{figure=fig/csin1_pert4_loc.PS,height=4.5in}}\end{figure}


Table 9.3: Umbilics on perturbed surface, $ \zeta $ = 0.08 (adapted from [257])
u 0.167 0.217 0.795 n/a 0.474
v 0.065 0.970 0.113 n/a 0.411
$ \kappa$ 1.426 0.042 -1.779 n/a 0.261
$ \alpha$ 0.701 0.374 6.355 n/a 6.356
$ \beta$ -14.070 11.604 19.405 n/a 0.307
$ \gamma$ 1.621 -2.520 0.727 n/a 5.155
$ \delta$ -4.184 2.057 4.072 n/a 1.273
$ \theta_1$ 0.632 0.573 0.594 n/a 0.773
$ \theta_2$ -0.674 -0.694 -0.692 n/a -0.079
$ \theta_3$ 1.504 1.455 -1.550 n/a -0.655
$ \phi_1$ 0.557 0.534 0.577 n/a 0.928
$ \phi_2$ -0.614 -0.594 -0.631 n/a -0.112
$ \phi_3$ 1.466 1.532 -1.485 n/a -0.841

index		 - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ n/a $ \frac{1}{2}$
type star star star n/a monstar

In computer vision, the geometric information of an object is obtained by range imaging sensors. Generally the data include noise and are processed using image processing techniques to exclude the noise, then the derivatives are directly computed from the digital data to evaluate the curvatures. What we do in the sequel is to fit a surface directly from artificial noisy data and observe the behavior of the umbilics on the fitted surface. The noisy data are produced in the following way. Evenly spaced $ 10 \times 10$ grid points $ (x,y)$ on $ 0 \leq x \leq 1$ , $ 0 \leq y \leq 1$ domain are chosen to evaluate the $ z$ -value of the wave-like bicubic Bézier patch. We add randomly perturbed vectors with $ \zeta = 0.05$ , as introduced in (9.73), to the $ (x, y, z)$ points on the surface as noise. Then the data points $ (x, y, z)$ are fit by a bicubic Bézier patch. Figure 9.9 and Table 9.4 illustrate the results. We observe that all the umbilics stay in the domain with index and types unchanged. Also the locations and the angles $ \theta_i, \;\phi_i\;(1 \leq i \leq 3)$ do not move nor rotate too much. These results provide us confidence for using the umbilics for shape recognition problems.

Figure 9.9: Lines of curvature passing through the umbilics on fitted surface, $ \zeta = 0.05$ (adapted from [257])
\begin{figure}\centerline{\psfig{figure=fig/csin1_fit5_loc.PS,height=4.5in}}\end{figure}


Table 9.4: Umbilics on reconstructed surface, $ \zeta = 0.05$ (adapted from [257])
u 0.198 0.227 0.813 0.796 0.493
v 0.043 0.954 0.025 0.991 0.399
$ \kappa$ 1.278 0.147 -1.005 -0.124 0.090
$ \alpha$ 1.748 0.999 0.678 0.357 6.572
$ \beta$ -16.370 16.161 19.451 -11.981 -0.293
$ \gamma$ -0.054 0.061 4.716 0.536 4.849
$ \delta$ -2.705 2.176 3.082 -1.233 -0.410
$ \theta_1$ 0.656 0.622 0.686 0.622 0.094
$ \theta_2$ -0.616 -0.642 -0.575 -0.634 -0.692
$ \theta_3$ -1.569 -1.569 -1.443 1.547 0.657
$ \phi_1$ 0.532 0.615 0.528 0.543 0.132
$ \phi_2$ -0.535 -0.630 -0.482 -0.554 -0.856
$ \phi_3$ 1.497 -1.562 -1.513 1.544 0.827

index		 - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ - $ \frac{1}{2}$ $ \frac{1}{2}$
type star star star star monstar

next up previous contents index
Next: 9.7 Inflection lines of Up: 9. Umbilics and Lines Previous: 9.5 Local extrema of   Contents   Index
December 2009