Information Technology Reference
In-Depth Information
r
j
1
þ
j
2
2
C
¼
ð
4
Þ
where
j
1
and
j
2
are the principal curvatures. Curvedness is a local measure that can
be computed at each vertex on the mesh. The larger C is, the more curved the local
surface is. The speed function, constructed using Eqs.
2
and
3
but with curvedness
replacing grey level gradients, ensures that the level set contour expands in
the center of the end plates (low curvedness) and stops at the ridgelines (high
curvedness) [
32
].
The level set evolution equation (Eq.
1
) can be implemented on a mesh with two
important adjustments relative to level sets in rectangular grids: (1) Gradients and
curvatures have to be computed in local coordinate systems de
ned around each
vertex as small enough neighborhoods can reasonably be considered planar. (2)
Gradients and curvatures have to be computed using least square estimation
methods rather than
finite differences [
32
].
We use the following de
nitions and notations for level sets on mesh. A function
fV
ðÞ
de
ned on a mesh associates to each vertex V the quantity fV
ðÞ
. A vertex V is
de
ned by its three coordinates (x, y, z) which can be relative to a global or a local
orthonormal frame. V can therefore also be seen as a vector. By immediate neighbor
of vertex V, we mean a vertex linked to V by an edge. The 1-ring neighborhood of
V is the set of immediate neighbors of V. The 2-ring neighborhood of V consists of
its 1-ring neighborhood and all the immediate neighbors of the vertices in the 1-ring
neighborhood. The process can be iterated. Thus, the n-ring neighborhood of V is
comprised of its (n
1)-ring neighborhood and all the immediate neighbors of the
−
vertices in the (n
1)-ring neighborhood.
To implement Eq.
1
, the gradients of the distance function
w
and the speed
function g have to be evaluated. We do this locally on the mesh in a 1-ring
neighborhood around each vertex. The components of
−
r
fV
ðÞ
the gradient of any
function f at vertex V can be evaluated by minimizing:
X
N
i
¼
1
r
2
¼
ðÞ~
n
i
r
ðÞ
i
ð
Þ
E
fV
fV
5
The summation is over the N immediate neighbors of V. The ith neighbor V
i
of
V de
nes the unit directional vector
~
n
i
:
V
i
V
n
i
¼
ð
6
Þ
j
V
i
V
j
The quantity
r
fV
ðÞ
i
is the
finite difference of function f in the direction of the
ith neighbor Vi:
i
:
