Page 291 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes (4) If the data points p are contained in a spherical surface S, the image points b(T ) cover a
i
i
two-dimensional region on B which is contained in a three-space.
In the following, we assume that the data comes from a smooth developable surface. Since the
estimation of tangent planes gives bad results on the boundary of the surface patch and near
measurement errors, there will be outliers in the Blaschke image. To find those, we search for
cells C carrying only a few image points. These cells and image points are not considered for
k
the further computations. The result is referred to as cleaned Blaschke image. In addition, a
thinning of the Blaschke image can be performed.

After having analyzed and cleaned the Blaschke image from outliers we are able to decide
whether the given developable surface D is a general cone or cylinder, a cone or cylinder of
revolution, another special developable or a general developable surface.
So, let T , i = 1, . . . ,M be the reliable estimated tangent planes of D after the cleaning and let
i
b(T ) = b be their Blaschke images. As we have worked out in Section 23.4.3 we can classify the
i
i
type of the developable surface D in the following way.
To check if the point cloud bi on B can be fitted well by a hyperplane H,
H : h + h u + . . . + h u = 0, h + . . . + h = 1. (4)
2
2
4
1
1
1
0
4
4
we perform a principal component analysis on the points b . This is equivalent to computing the
i
ellipsoid of inertia of the points bi. It is known that the best fitting hyperplane passes through
the barycenter c = (  b ) i
i /M of the M data points b . Using c as new origin, the coordinate vectors
of the data points are q = b – c and the unknown three-space H has vanishing coefficient, h = 0.
i
i
0
The signed Euclidean distance d(b , H) of a point qi and the unknown three space H is
i
d(q , H) = h q ,1 + . . . + h q = h . q , i (5)
1 i
i
4 i,4
where h = (h , . . . , h ) denotes the unit normal vector of H. The minimization of the sum of
1
4
squared distances,
M
M
F(h , h , h , h ) = 1  d (q ,H)  1  (q . h ) . (6)
2
2
i
i
i
3
2
1
4
i 1
i 1
M  M 
with respect to h = 1 is an ordinary eigenvalue problem. Using a matrix notation with vectors
2
as columns, it is written as
1 M
T
F(h) = h . C . h, with C : =  q . q . (7)
T
M i 1 i i

The symmetric matrix C is known as covariance matrix in statistics and as inertia tensor in
mechanics. Let i be an eigenvalue of C and let v be the corresponding normalized eigenvector
i
2
(v = 1). Then,  = F (v ) holds and thus the best fitting three-space V belongs to the smallest
i
i
2
i
1
eigenvalue  . The statistical standard deviation of the fit with V is
1
1
 =  1 /(n – 4). (8)
1
The distribution of the eigenvalues     · · ·   of the covariance matrix C (and the
4
1
2
corresponding standard deviations   · · ·   ) gives important information on the shape of the
4
1
surface D:
(1) Two small eigenvalues  ,  and different coefficients h , h , (|h – h | > ): The surface
20
10
20
1
2
10
D can be well approximated by a cone of revolution, compare 6 in section 23.3.1. The
vertex and the inclination angle are computed according to Section 23.3.
284 LOVELY PROFESSIONAL UNIVERSITY