Volume and Shape Preservation via Moving Frame Manipulation

Yaron Lipman Daniel Cohen-Or Ran Gal David Levin

ACM Transactions on Graphics, 2007

Abstract:

This paper introduces a method for mesh editing, aimed at preserving shape and volume. We present two new developments: the first is a minimization of a functional expressing a geometric distance measure between two isometric surfaces. The second is a local volume analysis linking the volume of an object to its surface curvature. Our method is based upon the moving frames representation of meshes. Applying a rotation field to the moving frames defines an isometry. Given rotational constraints, the mesh is deformed by an optimal isometry defined by minimizing the distance measure between the original and the deformed meshes. The resulting isometry nicely preserves the surface details, but, when large rotations are applied, the volumetric behavior of the model may be unsatisfactory. Using the local volume analysis, we define a scalar field by which we scale the moving frames. The scaled and rotated moving frames restore the volumetric properties of the original mesh, while properly maintaining the surface details. Our results show that even extreme deformations can be applied to meshes, with only minimal distortion of surface details and object volume.

Technical Report:

Acrobat, ~27 MB

BibTeX entry:

@article{LIPMAN2006,
author = {Yaron Lipman and Daniel Cohen-Or and Ran Gal and David Levin},
title = {Volume and Shape Preservation via Moving Frame Manipulation},
journal = {Technical report},
volume = {},
number = {},
year = {2006},
}

Keywords:

surface deformations, moving frames, surface reconstruction, volume preservation, shape

More examples:

A comparison with the method of Lipman et al.[2005]. In (a) the bar is rotated by just less than pi radians. On the left (a-1) a bar with bunnies is deformed by the technique of Lipman et al.[2005] and by our technique (a-2). Note how the error is evenly distributed by our technique. The colored close-up views (a-3),(a-4) of the bunny head, show the differences in the mean curvature with respect to the original shape. Another example is shown in (b) with a bumpy plane model.

The elephant model is deformed in one step with a 2pi rotation applied to the tip of his trunk.	A general deformation is applied to a bumpy sphere. In (a), the handle set is drawn (yellow) over the original model. (b) is the result of applying the deformation, and (c) is the result of applying two such deformations.

A bar with bunnies (a) (110K polygons) is deformed in (b) by two rotations of 3pi/2 each. In (c) a single large rotation of 3pi is applied.

The body of the Armadillo model is bent by pi radians and the hands are further bent to create a bridge-like pose. Note the preservation of the details and volume under the deformation.	The Armadillo model is twisted by 8pi/9 radians. Shape preserving deformations, without volume correction (a) and with volume correction in (b).

The volume correction algorithm is applied to a twisted bar. A rotation of approximately 4pi in one step is obtained by applying our technique, without volume correction in (a) and with volume correction in (b). Applying a rotation of 4p and then another rotation of 3pi/4 around another rotation axis yields (c). (d) shows the result of applying one big rotation of 3pi, together with position constraints to form a helical shape. Rotation of over 2pi is applied to a bar. In (a) the technique of Lipman et al.[2005] is applied three times with 2pi/3 in each step. In (b) the shape preserving isometric deformation is applied in one single step. (c) a volume correction is applied to (b).

The tip of the tentacle of the octopus is rotated by 3pi radians. Note preservation of details despite the highly irregular triangulation of the mesh.