Hormann et al. [11] defines a deformation metric as Ld= Γ
γ + γ Γ Sorkine et al. [24] defines a geometric distortion as
Lg = max
½ Γ, 1
γ
¾
Khodakovsky et al. [15] defines an area distortion as Larea = Γ · γ and an anisotropic distortion as
Langle= Γ γ
2.2 Mesh Parameterization
Over the last years, a lot of research has been done in the area of surface parameter-ization. In the context of parameterization, harmonic maps were first used by Eck et al. [5]; see Figure 2.5. However, the texture coordinates for boundary vertices must be fixed a prior and harmonic maps may contain face flips, which violate the bijectivity of the parameterization.
Based on earlier work by Tutte [25], Floater proposed a specific weight based on the barycentric maps to obtain a mapping that is shape-preserving [6]. It guarantees the embedding for convex boundaries and find a parameterization by solving a linear system.
The first step of the method is to specify the parameter points ψ(v) of the boundary vertices v ∈ VB. Then, set each interior vertex v ∈ VI to be a convex combination of its neighbors. For each interior vertex v, a set of strictly positive convex weights λvw, w ∈ Nv, is chosen such that
X
w∈Nv
λvw = 1.
2.2 Mesh Parameterization 11
For all interior vertices, the mapping ψ(v), v ∈ VI, is determined by solving the follow-ing linear system of equations:
ψ(v) = X
w∈Nv
λvwψ(w), v ∈ VI.
This equation can be rewritten as
ψ(v) − X
The problem remained is how to determine the value of λvw. There are two cases needed to be discussed. The first case is that v is inside of a triangle as shown in Figure 2.3. We can represent v as
v = area(v, w1, w2)
The second case is that v has more than three neighbor vertices. As shown in Figure 2.4, since 4w1w4w5 encloses v, we can solve λvw1, λvw4 and λvw5 using the previous method. Similarly, for each triangle k that has an end point wi and encloses v, we compute λkvwi and set λvwi as
Floater later proposed an improved method, called mean value coordinates, which derives the weights by using the mean value theorem [7]. The method also guarantees
2.2 Mesh Parameterization 12
Figure 2.3: Determine λvwfor v that is inside of a triangle.
Figure 2.4: Determine λvwfor v inside a n-sided polygon [6].
the existence of bijective mapping and is faster than the shape-preserving parameteriza-tion.
Desbrun et al. defined a space of measures spanned by a discrete version of the Dirichlet energy and a discrete authalic energy [4]. While the authalic energy remedies
2.2 Mesh Parameterization 13
local area deformations, it requires fixed boundaries and, moreover, produces results that are no better than the one computed by the parameterization using global length preservation.
Figure 2.5: A cat head model and its harmonic map [5].
Hormann and Greiner proposed MIPS (Most Isometric Parameterizations) algo-rithm, which attempts to preserve the ratio of singular values over the parameterization using a hierarchical solver [11, 12]. The method finds a parameterization with “natural boundary” that minimizes the highly non-linear stretch metric. Figure 2.6 shows the MIPS parameterization with natural boundary. However, the metric disregards absolute stretch scale over the surface. As a result, a small domain area can map to a large region on the surface.
Sander et al. proposed a non-linear stretch that integrate the sum of squared singular values over the map. We refer to this metric as geometric stretch. The parameterization is derived by a coarse-to-fine optimization scheme that minimizes the geometric stretch over the map. Note that the resulting parameterization may encounter parametric crack problem.
Sander et al. developed a signal-stretch metric that combines both surface area and surface signal bandwidth [18]. It is shown that the stretch metric is related to SAE
2.2 Mesh Parameterization 14
(a) Most Isometric Parameterization (b) Discrete Harmonic Parameterization
Figure 2.6: Gray-coded deformation energy of different parameterizations [11].
(Signal-Approximation Error) - the difference between a signal defined on the surface and its reconstruction. Sander’s signal stretch can be seen as the extension of the geome-try stretch. Figure 2.7 shows the results of the parameterization using geometric-stretch and signal-stretch parameterizations.
(a) Geometric-stretch parameterization (b) Signal-specialized parameterization
Figure 2.7: Examples of geometric-stretch and signal-stretch parameterizations [18].
Parameterization using either geometric stretch or signal stretch involves a process of expensive non-linear optimization. Yoshizawa et al. developed a simple and fast method that computes the parameterization of low geometry stretch [26]. Floater’s shape preserving parameterization [6] is used as an initial parameterization, which is then optimized gradually. At each step, the parameterization generated at previous step is optimized by updating the set of positive convex weights λvw for each interior vertex
2.2 Mesh Parameterization 15
v ∈ VI using geometry stretch. The formula is as follows:
λnewvw = λoldvw
σw , w ∈ Nv where
σw =qX
A(Tu)σ(Uu)2/X
A(Tu), σ(U) =p
(Γ2+ γ2)/2, U = hu1, u2, u3i in the parametric plane,
and A(Tu) is the area of triangle Tu and the sums are taken over all triangles Tu sur-rounding the vertex. After the weights are updated, the new parameterization is com-puted by solving a linear system, which is fast. Moreover, the parametric cracks that may happen in Sander’s global optimization will not be encountered. This method is, however, heuristic and lack of rigorous mathematical support. Nevertheless, it is fast and powerful for generating parameterization with low geometry stretch. See Figure 2.8 for the comparison.
Figure 2.8: Comparison of two mesh parameterization schemes [26]. Top : Stretch minimization of Sander et al [17], Down : Yoshizawa et al. [26]
Another approach that minimizes angular distortion is proposed by Sheffer and Sturler [21]. The parameterization is derived by minimizing the relative distortion of the