Practical Quad Mesh Simplification

Just a shameless plug to our last EG paper that will find is way inside MeshLab:

Marco Tarini, Nico Pietroni, Paolo Cignoni, Daniele Panozzo, Enrico Puppo
Practical Quad Mesh Simplification
Computer Graphics Forum, Volume 29, Number 2, EuroGraphics 2010

In our community it is well know the old religious war between quad vs. triangle meshes, each approach has its own merits and I not discuss them here.
Moving back and forth between the two approaches is often useful but the issue of getting a good quad mesh from a highly irregular tri mesh is a tough one.

In the above paper we present a novel approach to the problem of quad mesh simplification, striving to use practical local operations, while maintaining the same goal to maximize tessellation quality. We aim to progressively generate a mesh made of convex, right-angled, flat, equally sided quads, with a uniform distribution of vertices (or, depending on the application, a controlled/adaptive sample density) having regular valency wherever appropriate.

In simple words we start from a tri mesh, we convert into a dense quad mesh using a new Triangle-to-Quad conversion algorithm and then we simplify it using a new progressive quad simplification algorithm. The nice side is that the quad simplification algorithm actually improves the quality of the quad mesh. Below a small example.

We are currently adding this stuff inside MeshLab. The first things that will appear are the triangle to quad conversion algorithms and some functions for measuring the quality of a quad mesh according to some metrics. More info in the next posts....


(2/1/10 edit: if the above link for the paper does not work try this:  Practical Quad Mesh Simplification)

On the storage of Color in meshes

A very short post to clarify a bit the ways in which people can store color (and other) information on a mesh. AFAIK there are mostly three ways to keep color:

• Per-vertex: each vertex stores a color. A triangular face can have vertexes of different color and inside a triangle the color is linearly interpolated. Color is smooth across the surface.
  
• Per face: each face has a distinct color. You can easily see the discontinuity of colors among the faces (no interpolation is usually done.
• As texture: the most general way that is reasonably decoupled from the mesh itself. You only need a good parametrization.

In MeshLab most of the algorithms (painting, color processing etc.) manage per-vertex color, but a few conversion tools are provided.

• per-vert -> per -face
  
• per-face -> per-vert
• texture -> per-vert

A per-vert -> texture is strikingly missing :). To be added in the very near future. Obviously assuming the previous existence of a parametrization...
A fourth technique could be mentioned, keeping stuff per wedge, i.e. for each corner of the face we can store different colors (or other attributes), but this approach is rather unused (it can be simulated by duplicating vertices).
Below a few images showing the difference between the three modes on a small (40k tri) mesh; respectively: no color (to give you an idea of mesh density), color by texture, per vertex color, per face color.

Creating Interactive 3D objects inside a PDF

One of the nice feature of MeshLab is its ability of saving meshes, in a variety of formats. Support for saving meshes in U3D format is useful for creating, using LaTeX, cool appealing PDFs with embedded 3D models.

Yes, that means that when, using a plain Acrobat Reader, you open a pdf like this one, you will be able to freely interact with the 3D model, directly inside the text.

To generate such a pdf you simply need to convert your mesh into u3d format, and include the small snip of latex code generated by MeshLab with the right viewing parameters, in your latex document and simply compile it with pdflatex. Thanks to the Movie15 latex package, you will have your u3D embedded in the pdf. Note that the u3d file format is quite compact; for an example you can look at this pdf that contains the 50k triangle mesh of the Laurana's bust squeezed to less than 250 kb. A zip with sources (latex and u3d file) can be found here.

A couple of notes. The conversion process is done through the use of the Universal 3D Sample Software by directly using the IDTF converter provided with the sample library. The conversion process can take a fairly long processing time (many seconds for a mesh composed by 50k triangles) so be patient! Moreover be careful that the process can fail when involving pathnames with non trivial chars. Moreover very large meshes take a LONG time to be converted, be patient... Acrobat reader support this kind of files since ver. 7.