Based on above deduction, many researchers proposed various stretch metrics by the versatile Γ and γ . For example, Sander et al. [20] define an distortion measure by taking the root-mean-square of
L2
Hormann et al. [8] define their deformation metric as +Γ
= Γ γ γ
Ld , (2.24)
Sorkine et al. [26] define their geometric distortion as }
Khodakovsky et al. [11] define area distortion as γ
⋅ Γ
area =
L , (2.26)
and anisotropic distortion as
γ
=Γ
angle
L (2.27)
2.2 Related work
2.2.1 Parameterization
Over the last years, a lot of research has been done in the area of surface parameterization. Besides, methods that optimize the parameterization for a given surface signal like Sander et al.[19], most approaches aim at minimizing a metric distortion.
In the context of parameterization, harmonic maps were first used by Eck et
al.[4]. To compute harmonic maps, Eck et al. derive appropriate weights for a system of edge springs which can be efficiently solved. However, the texture coordinates for boundary vertices must be fixed a prior and harmonic maps may contain face flips (adjacent faces in texture space with opposite orientation) which violate the bijectivity of a parameterization. Based on earlier work by Tutte[27], Floater[5] proposes a different set of weights for the edge spring model that guarantees bijectivity if the texture coordinates of boundary are fixed to a convex polygon. Desbrun et al.[3]
define a space of measures spanned by a discrete version of the Dirichlet energy, and a discrete authalic energy. While the authalic energy remedies local area deformations, it requires fixed boundaries and results cannot achieve the quality of methods targeted at global length preservation such as Sander et al. [20].
In Hormann and Greiner[8], mostly isometric parameterization are introduced that minimize a non-linear energy. Mostly isometric parameterizations do not require boundary texture coordinates to be fixed and avoid face flips. Furthermore, mostly isometric parameterizations approximate mathematically well studied continuous conformal maps, i.e. maps that perfectly preserve angles.
Another approach to minimize angular distortion is proposed by Sheffer and Sturler[25]. They introduce a angle based flattening method to flatten a mesh to a planar plane so that it minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensional space. Though the minimization problem is linear in the relative distortion of angles, it becomes non-linear as a number of constraints (some of which are non-linar) have to be taken into account to generate the validity of the solution. Levy et al.[15] compute quasi-conformal parameterizations by measuring the violation of the Cauchy-Rieman equation in the least square sense. They also show rigorously that the quasi-conformal parameterization exists uniquely, is invariant by similarity, independent of resolution and preserves orientations. Using a standard numerical conjugate gradient solver they are able to compute least squares approximations to continuous conformal maps very efficiently without requiring fixed boundary texture coordinates. However, in seldom cases triangle flips may occur.
In addition, some methods exist which compute parameterizations over a
non-planar domain. In Lee et al.[12], a mesh simplification is used to parameterize a surface over a base mesh. A similar approach is taken by Khodakovsky et al.[11] but with emphasis on globally smooth derivatives.
Besides angle preserving methods, only a few approaches explicitly optimize global area or global length distortions: Maillot et al.[17] minimize an edge length distortion, but cannot guarantee the absence of face flips. The authors also propose an area preserving energy and combine both energies in a convex combination. Sander et al.[20] minimize the average or maximal singular value of the Jacobian to prevent undersampling of the surface. However, their metric cannot penalize the anisotropic stretching- triangles whose stretch in one direction is significantly higher than in the other direction. As a result of anisotropic stretching, parameterization of [20] has the parametric cracks problem. Yoshizawa et al. [28] improve parametric cracks with adaptively adjusting the spring constants by L2 metric.
2.2.2 Applications of parametrization
Remeshing
Alliez et al.[1] proposed an interative sampling technique. A mesh is decomposed into a set of maps by parameterization and inserted in a pipeline of signal processing algorithms. The output of this pipeline is density map, interatively resampled using an error diffusion technique commonly used for gray level image halftoing.
Geometry video
Briceno et al.[2] present geometry videos based on geometry image to represent animation. Geometry videos resample and reorganize the geometry information, in such a way, that it becomes very compressible. They provide a unified and intuitive method for level-of-detail control, both in terms of mesh resolution (by scaling the two spatial dimensions) and of frame rate (by scaling the temporal dimension). Since geometry videos have a very uniform and regular structure. Their source and computational requirements can be calculated exactly, hence making them also
suitable for applications requiring level of service guarantees.
3D Painting system
Igrashi et al.[10] present a method for dynamically generating an efficient texture bitmap and its associated UV-mapping in an interactive texture painting system for 3D models. They proposed an adaptive and local parameterization mechanism where system dynamically creates a tailored UV-mapping for newly painted polygons during interactive painting process. This eliminates the distortion of brush strokes, and the resulting texture bitmap is more compact because the system allocates texture space only for the painted polygons. In addition, this dynamic texture allocation allows the user to paint smoothly at any zoom level.
Constrained texture mapping
Levy et al.[14] introduce constrained texturing mapping by parameterizing polygonal meshes with minimum deformation between source textures. They enable users to interactively define and edit a set of constraints. Each user-defined constraint consists of a relation linking a 3D point picked on the surface and 2D point of the texture. Moreover, the non-deformation criterion introduced here can act as an extrapolator, thus making it unnecessary to constrain the border of the surface, in contrast with classic methods. To minimize the criterion, a conjugate gradient algorithm is combined with a compressed representation of sparse matrices, making it possible to achieve a fast convergence.
Mesh editing
Levy et al.[16] introduce an idea in editing the mesh on parameter domain. They proposed an objective function of squared curvature. By minimizing this objective function, they extrapolate mesh shape beyond the existing border. In addition, the parameter space provides the user with a new means of controlling the shape of the surface.