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
of the umbilics do not move nor
rotate too much.
u | 0.211 | 0.211 | 0.789 | 0.789 | 0.5 |
v | 0.052 | 0.984 | 0.052 | 0.984 | 0.440 |
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1.197 | 0.267 | -1.197 | -0.267 | 0. |
![]() |
4.147 | 0.926 | 4.147 | 0.926 | 6.514 |
![]() |
-18.306 | 14.670 | 18.306 | -14.670 | 0. |
![]() |
0. | 0. | 0. | 0. | 4.2763 |
![]() |
-2.337 | 1.411 | 2.337 | -1.411 | 0. |
![]() |
0.671 | 0.562 | 0.592 | 0.638 | 0. |
![]() |
-0.592 | -0.638 | -0.671 | -0.616 | -0.604 |
![]() |
1.571 | -1.571 | -1.571 | 1.571 | 0.604 |
![]() |
0.567 | 0.562 | 0.495 | 0.583 | 0. |
![]() |
-0.495 | -0.583 | -0.567 | -0.562 | -0.752 |
![]() |
1.571 | 1.571 | -1.571 | -1.571 | 0.752 |
index | -
![]() |
-
![]() |
-
![]() |
-
![]() |
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type | star | star | star | star | monstar |
u | 0.190 | 0.214 | 0.794 | n/a | 0.492 |
v | 0.055 | 0.978 | 0.081 | n/a | 0.424 |
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1.293 | 0.136 | -1.458 | n/a | 0.083 |
![]() |
2.390 | 0.551 | 5.014 | n/a | 6.351 |
![]() |
-16.119 | 13.046 | 18.926 | n/a | 0.163 |
![]() |
0.563 | -1.524 | -0.360 | n/a | 4.666 |
![]() |
-3.182 | 1.711 | 3.234 | n/a | 0.510 |
![]() |
0.658 | 0.593 | 0.586 | n/a | 0.701 |
![]() |
-0.623 | -0.667 | -0.689 | n/a | -0.055 |
![]() |
1.551 | 1.509 | 1.560 | n/a | -0.644 |
![]() |
0.559 | 0.549 | 0.528 | n/a | 0.857 |
![]() |
-0.537 | -0.589 | -0.596 | n/a | -0.076 |
![]() |
1.529 | 1.552 | -1.532 | n/a | -0.811 |
index | -
![]() |
-
![]() |
-
![]() |
n/a |
![]() |
type | star | star | star | n/a | monstar |
u | 0.167 | 0.217 | 0.795 | n/a | 0.474 |
v | 0.065 | 0.970 | 0.113 | n/a | 0.411 |
![]() |
1.426 | 0.042 | -1.779 | n/a | 0.261 |
![]() |
0.701 | 0.374 | 6.355 | n/a | 6.356 |
![]() |
-14.070 | 11.604 | 19.405 | n/a | 0.307 |
![]() |
1.621 | -2.520 | 0.727 | n/a | 5.155 |
![]() |
-4.184 | 2.057 | 4.072 | n/a | 1.273 |
![]() |
0.632 | 0.573 | 0.594 | n/a | 0.773 |
![]() |
-0.674 | -0.694 | -0.692 | n/a | -0.079 |
![]() |
1.504 | 1.455 | -1.550 | n/a | -0.655 |
![]() |
0.557 | 0.534 | 0.577 | n/a | 0.928 |
![]() |
-0.614 | -0.594 | -0.631 | n/a | -0.112 |
![]() |
1.466 | 1.532 | -1.485 | n/a | -0.841 |
index | -
![]() |
-
![]() |
-
![]() |
n/a |
![]() |
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
grid
points
on
,
domain are
chosen to evaluate the
-value of the wave-like bicubic Bézier patch.
We add randomly perturbed vectors with
, as
introduced in (9.73), to the
points on the surface as noise. Then the data points
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
do not move nor rotate too much. These results
provide us confidence for using the umbilics for shape recognition problems.
u | 0.198 | 0.227 | 0.813 | 0.796 | 0.493 |
v | 0.043 | 0.954 | 0.025 | 0.991 | 0.399 |
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1.278 | 0.147 | -1.005 | -0.124 | 0.090 |
![]() |
1.748 | 0.999 | 0.678 | 0.357 | 6.572 |
![]() |
-16.370 | 16.161 | 19.451 | -11.981 | -0.293 |
![]() |
-0.054 | 0.061 | 4.716 | 0.536 | 4.849 |
![]() |
-2.705 | 2.176 | 3.082 | -1.233 | -0.410 |
![]() |
0.656 | 0.622 | 0.686 | 0.622 | 0.094 |
![]() |
-0.616 | -0.642 | -0.575 | -0.634 | -0.692 |
![]() |
-1.569 | -1.569 | -1.443 | 1.547 | 0.657 |
![]() |
0.532 | 0.615 | 0.528 | 0.543 | 0.132 |
![]() |
-0.535 | -0.630 | -0.482 | -0.554 | -0.856 |
![]() |
1.497 | -1.562 | -1.513 | 1.544 | 0.827 |
index | -
![]() |
-
![]() |
-
![]() |
-
![]() |
![]() |
type | star | star | star | star | monstar |