Graphics Reference
In-Depth Information
5.6 B-Spline Surface
B-spline surface is defined exactly in the same way as the Bezier surface. It is
the Cartesian product surface and is given by
n
q
S ( u, v )=
B i,m B j,p V i,j .
(5.28)
i =0
j =0
5.7 Application
Roberto Cipolla and Andrew Blake [39] used B-spline to measure the differ-
ential invariants of the image velocity field by computing average values from
the integral of normal image velocities around image contours. They showed
how an active observer making small, deliberate motions can use the estimate
of the divergence and deformation of the image velocity field to determine
the orientation of the object surface and time to contact. They tracked arbi-
trary image shapes using B-spline control snakes and computed eciently the
invariants as closed-form functions of the B-spline snake control points. Sub-
sequently, they used this information to guide a robot manipulator in obstacle
collision avoidance, object manipulation, and navigation.
5.7.1 Differential Invariants of Image Velocity Fields
Differential invariants of image velocity fields were originally introduced by
Koenderink and Van Doorn [92, 94, 93] in the context of computational vision
and analysis of visual motion. The image velocity of a point in space due to
motion between the observer and scene [121] is
Q t = ( U
Q )
Q
Ω
Q ,
(5.29)
λ
where U =translational velocity, Ω =rotational velocity around the viewer
center, and λ is the distance to the point. Let us now look at the local variation
of image velocities in the vicinity of the ray Q , and consider an arbitrary co-
ordinary system with the x
y plane spanning the image plane. We assume
that the z-axis is aligned with the ray. With respect to this coordinate system,
let the translational and angular velocity have respectively the components as
shown, U =
. Assume the image velocity
field at a point ( x, y ) in the vicinity of Q is v ( x, y ) with ( u, v )as x and y
components. The image velocity field for a suciently small field of view can
be described by ( u 0 ,v 0 ) and by the first order partial derivatives of the image
velocity, i.e., by u x ,u y ,v x ,v y [171, 122] as
{
U 1 ,U 2 ,U 3 }
and Ω =
{
Ω 1 2 3 }
U 1
u 0 =
λ
Ω 2 .
(5.30)
 
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