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“Hierarchical Least Squares Conformal Maps”
Nicolas Ray and Bruno Lévy
11th Pacific Conference on Computer Graphics and Applications, Canmore, Canada, 2003

Abstract: A texture atlas is an efficient way to represent information (like colors, normals, displacement maps ...) on triangulated surfaces. The LSCM method (Least Squares Conformal Maps) automatically generates a texture atlas from a meshed model. For large charts (over 100k facets), the convergence of the numerical solver may be slow. It is well known that the conformality criterion, minimized by LSCM, also corresponds to a harmonicity condition, meaning that barycentric coordinates are locally preserved through the parameterization. This has two different consequences \,: -cascadic multigrid methods (coarse to fine) are well adapted to this criterion, and dramatically speed up the convergence of the numerical solver \,; - the obtained parameterization naturally minimizes texture swimming when used to texture-map a progressive mesh. In this paper, we introduce HLSCM (Hierarchical LSCM), a cascadic multigrid version of LSCM. As an example of possible applications, the paper shows how normal maps and simplified models can be automatically generated from large scanned meshes. Using these normal maps, the visual appearance of the model can be preserved even when 90% of the vertices are removed from the initial model.

BibTex reference 

@INPROCEEDINGS{ray:HLS:2003,
   AUTHOR     = "Ray, Nicolas and Lévy, Bruno",
   TITLE      = "Hierarchical Least Squares Conformal Maps",
   BOOKTITLE  = "11th Pacific Conference on Computer Graphics and Applications, Canmor
                   e, Canada",
   YEAR       = "2003",
   PAGES      = "263-270",
   MONTH      = "Octobre",
}

Supplemental material, links, hindsight ...

Hierarchical Least Squares Conformal Maps

Generating a normal-mapped simplified model (Stanford Bunny)

  • Harmonicity and conformal maps: A minimizer of the conformal energy is also an harmonic map, which means local barycentric coordinates are preserved through the mapping. Note the stability of the iso-parametric curves while the model is simplified. This has two consequences:
    • cascadic multigrid is a good numerical strategy to minimize the LSCM criterion;
    • a model texture-mapped using a conformal parameterization can be simplified without introducing any noticeable texture deviation.

  • normal map generation: using a texture atlas, it is easy to generate a normal map , i.e. an image which colors encode the normals of the object. To avoid artifacts at chart frontiers, it is necessary to fill-in the empty regions using a push-pull algorithm, or mathematical morphology (like here). When the mesh is simplified, thanks to the harmonicity property, the linear interpolations performed by texture hardware correctly put the normal map in correspondence with the simplified model;

  • normal mapped simplified model: The initial Stanford Bunny model has 70K faces. This video shows the simplified model (4K faces) displayed without and with its associated normal map. Using this representation, modern graphics hardware can combine textures with per-pixel diffuse and specular shading in real time;

  • corresponding normal map.

Generating a normal-mapped simplified model (Hand)

Generating a normal-mapped simplified model (Stanford Buddha)

  • normal-mapped simplified model: The initial Stanford Buddha model has 1.087M faces. This video shows the simplified model (46K faces) displayed with its associated normal map, using per-pixel real time shading. Specular shadings shows that small-scale details are well reproduced by the normal map.

  • corresponding normal map.

Numerical Stability: single-chart 'Hand' model

  • multiresolution analysis of the 'Hand' model (73K faces);

  • parameter-space of this model. This model has been parameterized using a single chart. The high differences of scaling is a challenge for numerical solvers. As can be seen in the video, HLSCM remains stable even in this context. Clearly, this parameterization cannot be used for texture mapping (it is better to create a texture atlas, as in previous videos). However, this type of parameterization is very useful for remeshing applications, where local isotropy is more important than stretching (stretching can be taken into account by the remeshing metric, see e.g. Alliez's work).