Barycentric Coordinates for Arbitrary Polygons in the Plane

14 K. Hormann
This is due to the large linear system that needs to be solved and the large number
of basis functions that have to be evaluated. In general, if n is the number of vertices
in Ψ and k is the number of samples per edge, then solving the linear system with a
standard method is an O(k 3 · n 3 ) operation and evaluating the m pixels of Î is of order
O(m·k·n). Of course there exist specialized solvers that can reduce the cost for solving
the linear system, but the advantage of the barycentric warp is that it does not require
any system to be solved and the whole computation is of order O(m · n) only.
We conclude that the barycentric warp is particularly useful whenever straight edges
need to be preserved. For example, making the boundary of the rectangular image
one of the source and the target polygons guarantees that the warped image will be
rectangular too; see Figure 9.
5.3 Transfinite Interpolation
Our generalized barycentric coordinates also provide an efficient solution to the following
interpolation problem. Given a set of closed planar curves c j and some
d-dimensional data over these curves d j : c j → IR d we would like to have a function
F : IR 2 → IR d that interpolates the given data, i.e. F (v) = d j (v) for any v ∈ c j .
The obvious approach is to approximate the given curves c j with a set of polygons
Ψ whose vertices v i lie on the curves c j . Then the function F in Equation (12) with
f i = d j (v i ) clearly is an approximate solution of the stated interpolation problem. And
as the polygons Ψ converge to the curves c j by increasing the sampling density, so does
the function F converge to the desired solution.
The interesting fact now is that the structure of our generalized barycentric coordinates
allows an efficient update of the solution if the sampling density is increased.
Assume that we computed F (v) and also W (v) from Equation (9) for some polygon Ψ
and then refine Ψ to ̂Ψ by adding a vertex ˆv with data value ˆf = d j (ˆv) between v j and
v j+1 . Then it follows from Appendix A that the refined interpolation function ̂F (v) can
be written as
with
̂F (v) = F (v) + ŵ(v)
Ŵ (v) ( ˆf − ρ(v)F (v) − σ(v)f j − τ(v)f j+1 ) (14)
Ŵ (v) = W (v) + ρ(v)ŵ(v) (15)
where ŵ(v) is the new homogeneous coordinate function for ˆv and ρ(v), σ(v), and τ(v)
are the normalized barycentric coordinates of ˆv with respect to the triangle [v, v j , v j+1 ].
Note that ρ, σ, and τ can be computed from the same ingredients as ŵ.
In the example in Figure 11, we exploited this recurrence relation as follows. We
first sampled the three given curves uniformly with 100 vertices v i and computed the
curve normals n i at v i . Then we evaluated the interpolating function F as well as W
on a regular 512×512 grid, yielding values F kl and W kl with 0 ≤ k, l ≤ 511. We
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