Graphics Reference
In-Depth Information
usual solution is to ask for a signal
s
with the property that
L
(
s
)(
v
)
−
h
(
v
)
is as
small as possible, typically by minimizing a sum of squares
E
(
s
)=
v
h
(
v
))
2
.
(
L
(
s
)(
v
)
−
(25.13)
∈
V
Since the Laplacian is invariant under the addition of a constant to the signal, this
minimization must typically be performed with one or more additional constraints,
such as the value
s
(
v
i
)
at some small number
k
of selected vertices
v
1
,
,
v
k
.
While we tend to have many functions defined on meshes—it's common, for
example, to evaluate some lighting model at each vertex of a mesh, and interpo-
late over edges and triangles—the function to which the Laplacian is most often
applied is the one that returns, for each vertex, the coordinates of its location. For
instance, we can regard the
x
-coordinate of each vertex as providing a signal value
at that vertex. The same goes for the
y
- and
z
-coordinates. If we regard
x
:
V
...
R
3
as the vector-valued function that takes each vertex to its
xyz
-coordinates, then it's
common to compute
→
V
. (25.14)
These vectors, one per vertex, are sometimes called the
Laplacian coordinates
or
differential coordinates
for the mesh.
“Laplacian coordinates” is really a misnomer; a coordinate system should have
the property that no two distinct points have the same coordinates (although in
some cases, like polar “coordinates,” we allow a single point to be given multi-
ple coordinates). But it's easy to see that for a regular triangulation of a plane,
the Laplacian coordinates at every vertex are zero, so any two planar parts of
this mesh end up with the same “coordinates.” Perhaps “coordinate Laplacian” or
“coordinate differentials” would be a better term, but “Laplacian coordinates” and
“differential coordinates” are well established already.
Laplacian coordinates have a few obvious properties. First, they are invariant
with respect to translation, that is, if
M
is a translated version of the mesh
M
,
then the Laplacian coordinates at corresponding vertices are identical. Second,
they are equivariant with respect to linear transformations, that is, if
M
is the
mesh resulting from applying a linear transformation
T
to each vertex of
M
(e.g.,
rotating
M
30
◦
in the
xy
-plane), then the Laplacian coordinates at a vertex
v
in
M
can be computed from those at the corresponding vertex
v
by applying
T
. These
facts, taken together, can be summarized by saying that Laplacian coordinates
are equivariant with respect to affine transformations of meshes, as long as we
remember that the action of an affine transformation on a vector ends up ignoring
translations.
In summary: Laplacian coordinates on a mesh provide an affine-
transformation-equivariant description of the geometry of the mesh. A mesh can
be reconstructed from its Laplacian coordinates together with a small number of
known mesh locations. And any vector-valued function of the vertices of a mesh
can play the role of Laplacian coordinates if we reconstruct by solving a least-
squares problem rather than seeking an exact solution.
δ
(
v
)=
L
(
x
)(
v
)
for
v
∈
Nealen et al. [NISA06] describe an approach in which a mesh is “optimized” by
adjusting vertex positions, but retaining the mesh connectivity. The technique is
simple, and its good and bad features are self-evident.
