Smoothing an overlay grid to minimize linear distortion in texture mapping

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Abstract
Texture is an essential component of computer generated models. For a texture mapping procedure to be effective it has to generate continuous textures and cause only small mapping distortion. The Angle Based Flattening (ABF) parameterization method is guaranteed to provide a continuous (no foldovers) mapping. It also minimizes the angular distortion of the parameterization, including locating the optimal planar domain boundary. However, since it concentrates on minimizing the angular distortion of the mapping, it can introduce relatively large linear distortion.In this paper we introduce a new procedure for reducing length distortion of an existing parameterization and apply it to ABF results. The linear distortion reduction is added as a second step in a texture mapping computation. The new method is based on computing a mapping from the plane to itself which has length distortion very similar to that of the ABF parameterization. By applying the inverse mapping to the result of the initial parameterization, we obtain a new parameterization with low length distortion. We notice that the procedure for computing the inverse mapping can be applied to any other (convenient) mapping from the three-dimensional surface to the plane in order to improve it.The mapping in the plane is computed by applying weighted Laplacian smoothing to a Cartesian grid covering the planar domain of the initial mapping. Both the mapping and its inverse are provably continuous. Since angle preserving (conformal) mappings, such as ABF, locally preserve distances as well, the planar mapping has small local deformation. As a result, the inverse mapping does not significantly increase the angular distortion.The combined texture mapping procedure provides a mapping with low distance and angular distortion, which is guaranteed to be continuous.

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