Digital Signal Processing Reference
In-Depth Information
based on approximation can eciently produce the water surface. However, a
large amount of periodic functions is required to represent a complex and re-
alistic water surface. Chen[3] exploited GPU parallelism to eciently process
the periodic functions. The method produces a bump map to represent the wa-
ter surface. LOD approach was also exploited to alleviate the computational
cost[4,5,6].
The motion of water surface can be easily modeled and animated with the pe-
riodic functions. However, the periodic function cannot easily produce complex
surface. In order to represent the complex surface, fast Fourier transform (FFT)
was also exploited[7,8]. However, the method with FFT cannot effectively con-
trol the water surface. Therefore, the methods cannot be employed interactive
application.
In order to enable the interaction with external objects, physically based ap-
proach should be employed. The major categories for physics based water simula-
tion are grid-based Euler method [9,10] and particle-based Lagrange method[11].
However, the physics based approaches require too heavy computation to be in-
tegrated into realtime applications.
Recently, an eciently method based on 'wave particles' were proposed[12].
However, the method cannot produce the wave valleys so that the surface is not
suciently realistic.
3 Basic Concept of Water Surface Animation
Our method links the deformable object to generate wave effect. The physical
property of each object is dominated by restoration force computed by shape
matching. Therefore, each cluster element of water surface is constrained to
maintain original shape. Since each element is overlapped with adjacent element,
the motion of each element is transported to the adjacent elements. Therefore
we can obtain the wave effect along the adjacent object chain.
The shape matching approach produces force to move deformed vertices back
to the target position. The deformable objects try to maintain the original shape,
but the locations of vertices in the objects are not strictly constrained. The
deformable object is deformed when external force is exerted as shown in Fig.
??
.
In this paper,
V
denotes the set of vertices. A vertex
i
is an element of the set
V
. The properties of
i
include the current position
x
i
, the original position
x
,
goal position
g
i
, acceleration
a
i
, velocity
v
i
,andthemass
m
i
. If the object is
rigid, the position of vertex
i
can be expressed with a rotation matrix
R
and
a translation matrix
T
. Once we know the matrix
R
, the goal position can be
easily computed as follows:
0
i
i
0
x
=
m
i
R
(
x
i
− t
0
)+
t − x
i
(1)
where
t
is the current center of mass and
t
0
denotes the original center of mass.
In order to compute the rotation matrix
R
in Eq.1, we employed a optimal
linear transformation matrix as follows:
i
0
i
− t
0
)+
t − x
i
x
=
m
i
A
(
x
(2)
