AS-RIGID-AS-POSSIBLE SURFACE EDITING

In Houdini we are mainly working with meshes and quite often these meshes need to be deformed. For this reason we can use several tools such as cages or proxies together with the LatticeSop or the newer PointDeformSop. However, sometimes it’s handy to deform the shape directly. Of course, there is the EditSop but getting smooth deformations can be tricky. Even if soft radius is activated, you might end up with unsatisfying results, especially when rotations are involved. This might be the case because the EditSop uses just a spherical distance falloff and no geometry-aware distance, or maybe because local details don’t get maintained during editing, or maybe it’s because the deformed shape looses to much local volume/area compared to the original. Anyway, one of the best techniques to avoid pretty much all of these problems is “as-rigid-as-possible” deformations and it is described in detail in the paper: “As-Rigid-As-Possible Surface Modeling” by Olga Sorkine and Marc Alexa.

A perfect mesh editing tool should enable us to select some vertices to move and rotate them around. The deformation of the handle should get propagated smoothly to the rest of the geometry. And of course, this should happen without removing or distorting surface details and obviously as fast as possible. In other words, the deformation should be smooth and properties such as area, curvature, distance and so on, should be preserved as much as possible. Well, the problem is that if local mesh properties should be preserved, only rotation and translation should be involved since scaling would destroy the local structure. This however is clearly not possible in the case of mesh editing because as soon as we select a vertex and transform it, some parts of the mesh will get scaled and/or shifted. As consequence we´ll end up with distorted surface details.  “As-rigid-as-possible” however, presents a nice solution to this problem. It tries to preserve local rigidity as much as possible. This means, the shape is locally preserved and smooth as well.

 

Basically this is done in a two-stage approach. First, getting an initial deformation guess by propagating the handle transformation to the mesh. And second, finding the best deformation to maximize the local rigidity by minimizing an energy function. The energy function is based on the one-ring neighbourhood around vertices and measures the deviation to the rigid transformation of the cell. By summing up the energy functions for all vertices/cells of the mesh we get the global energy which is to be minimized to get the final deformation.

 

The number of iterations required to retain local properties depends mainly on the size of the mesh. Generally, just a few iterations (10-20) are needed until convergence.

The implementation is done with the HDK but instead of using Houdini’s own matrix solver I´ve used the linear algebra library Eigen. For more information about “as-rigid-as-possible” check out the following paper:

O. Sorkine, M. Alexa: As-Rigid-As-Possible Surface Modeling

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