LOCAL PROCESSING OF 3D IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 65

(the first fundamental form)

1 + f x

f x f y f x f z

1 + f y

F 1 = f y f x f y f z (3.46)

1 + f z

f z f x f z f y

(the second fundamental form)

f xx f xy f xz
f yx f yy f yz
f zx f zy f zz

F 2 = − 1 /D (3.47)

where

D =( 1 + f x + f y + f z ) 1 / 2 . (3.48)
Then, principal curvatures k 1 , k 2 ,and k 3 of a hypersurface S are obtained as
eigen values of a matrix

W = F − 1
1

F 2 . (3.49)

Relations among them are classified as summarized below.

(i) Signs and orders in their sizes ( 20 cases).
(ii) Orders in sizes of their absolute values and zero and nonzero ( 26 cases).
(iii) The sign of the sum k 1 + k 2 + k 3 ( 3 cases).
There exist 1560 cases of all these combinations. Noting that we can assume
|

without loss of generality, consideration of only 20 cases is
enough. They are given in Table 3.4.
These curvatures represent local shape futures of the surface S at a point
P and its vicinity. We could imagine shapes of a hypersurface from the analogy
on a 2D image (= shape of a 3D surface).
Sets of points satisfying various conditions may be utilized as shape
features of surfaces in the 3D space and a hypersurface in the 4D space
such as equidensity surface, a surface of a 3D object and border surface,
for example, [Enomoto75, Enomoto76, Nackman82] for a 2D image, and
[Watanabe86, Thirion95, Monga95, Brechbuhler95, Hirano00].
However, differential features are defined only at a differentiable point for
a continuous image. For a digitized image, the method to calculate difference,
the method of digitization, and the effect of random noise must be taken into
account.

k 1 |≥| k 2 |≥| k 3 |

Remark 3.12. For a 2D continuous image f ( x, y ), consider a curved surface
S 2 : u = f ( x, y ), a point P on it, and a tangent A of the curved surface
(Fig. 3.8). Consider next a plane that includes a tangent A and the normal of
S 2 at P. Then let us obtain an intersecting line C between this plane and the
surface S 2 .Since C is a curve on a 2D plane, the curvature at a point P can
be treated in the frame of 2D geometry. Because the curvature depends on a
tangential line A , we call this curvature the normal curvature of the surface
S 2 at a point P in the direction A (Fig. 3.8).