# Simple and Efficient Mesh Editing with Consistent Local Frames

## Abstract

Mesh editing methods based on differential surface representations like Poisson editing, Laplacian or rotationinvariant coordinates have recently gained much interest as they are both efficient and easy to implement. In order to reconstruct geometry from such representations, rigid local coordinate frames have to be determined for all vertices which is an inherently non-linear problem. Attempts to linearize this problem show at least one of the following problems: First, local frames degenerate for large handle transformations or second, frames can become inconsistent with the reconstructed geometry. Both results in contra-intuitive surface deformation. Existing non-linear approaches do not show these problems. However, they are comparatively slow and considerably more complex.

In this paper we present a differential representation that implicitly enforces orthogonal and geometry consistent frames while at the same time allows for an efficient reconstruction. In particular, it enforces conformal surface deformations so that local texture features are preserved. Tailored to this representation we derive a non-linear reconstruction which is fast as it avoids an iterated matrix factorization and present an efficient implementation, at which most of the calculation is done in parallel on graphics hardware.

The result is a simple and fast mesh editing framework that is robust even for large handle transformations. We demonstrate its efficiency and robustness by several examples.

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## Bibtex

@TECHREPORT{cg-2007-3, author = {Paries, Nikolas and Degener, Patrick and Klein, Reinhard}, title = {Simple and Efficient Mesh Editing with Consistent Local Frames}, number = {CG-2007-3}, year = {2007}, month = jul, institution = {Universit{\"a}t Bonn}, abstract = {Mesh editing methods based on differential surface representations like Poisson editing, Laplacian or rotationinvariant coordinates have recently gained much interest as they are both efficient and easy to implement. In order to reconstruct geometry from such representations, rigid local coordinate frames have to be determined for all vertices which is an inherently non-linear problem. Attempts to linearize this problem show at least one of the following problems: First, local frames degenerate for large handle transformations or second, frames can become inconsistent with the reconstructed geometry. Both results in contra-intuitive surface deformation. Existing non-linear approaches do not show these problems. However, they are comparatively slow and considerably more complex. In this paper we present a differential representation that implicitly enforces orthogonal and geometry consistent frames while at the same time allows for an efficient reconstruction. In particular, it enforces conformal surface deformations so that local texture features are preserved. Tailored to this representation we derive a non-linear reconstruction which is fast as it avoids an iterated matrix factorization and present an efficient implementation, at which most of the calculation is done in parallel on graphics hardware. The result is a simple and fast mesh editing framework that is robust even for large handle transformations. We demonstrate its efficiency and robustness by several examples.}, issn = {1610-8892} }