Multiresolution Geometric Modeling

Authors: Stefanie Hahmann, Georges-Pierre Bonneau, Basile Sauvage

[1] B. Sauvage, S.Hahmann, GP. Bonneau
Volume preservation of multiresolution meshes
Computer Graphics Forum (Proceedings of Eurographics 2007), Volume 26, Number 3 (2007)   

[2] B. Sauvage, S. Hahmann, G.-P. Bonneau, G. Elber
Detail Preserving Deformation of B-spline Surfaces with Volume Constraint
Computer Aided Geometric Design, Volume 25, Number 8, p. 678-696 (2008)   

[3] Stefanie Hahmann, Basile Sauvage, Georges-Pierre Bonneau,
Area preserving deformation of multiresolution curves ,
Computer Aided Geometric Design 22 (4), pp. 349--367 (2005)   

[4] Basile Sauvage, Stefanie Hahmann, Georges-Pierre Bonneau,
Length preserving multiresolution editing of curves,
Computing 72 (1-2), pp. 161-170 (2004).   

Theme 1: Volume preservation of multiresolution meshes

Geometric constraints have proved to be efficient for improving the realism of shape animation. The present paper addresses the computation and the preservation of the volume enclosed by multiresolution meshes. A wavelet based representation allows the mesh to be handled at any level of resolution. The key contribution is the calculation of the volume as a trilinear form with respect to the multiresolution coefficients. Efficiency is reached thanks to the pre-processing of a sparse 3D data structure involving the transposition of the filters while represented as a lifting scheme. A versatile and interactive method for preserving the volume during a deformation process is then proposed. It is based on a quadratic minimization subject to a linearization of the volume constraint.


Figure: The horse is animated by skinning, which is combined with the MR mesh and our volume preserving method. The MR mesh is deformed through its control mesh (wireframe upper left) according to a skeleton. Upper row: initial mesh (left) and various positions. The lower row show several frames extracted from a walking animation

Figure: Filter Bank Algorithm: From left to right: analysis process (filters A and B). From right to left: synthesis process (filters S and C).

Figure: Lifting scheme based on the Loop subdivision scheme. Left: analysis represents the filters A and B . Right: synthesis represents the filters S and C.

Theme 2: Area preserving multiresolution editing of curves

We describe a method for multiresolution deformation of closed planar curves that keeps the enclosed area constant. We use a wavelet based multiresolution representation of the curves which are represented by a finite number of control points at each level of resolution. A deformation can then be applied to the curve by modifying one or more control points at any level of resolution. This process is generally known as multiresolution editing to which we add the constraint of constant area. A multiresolution representation for the area moment is also developed. We make sure that all computations are fast and that the deformations can be performed interactively. Diverse types of deformation are discussed.


Area preserving editing

Without / with area preservation

Theme 3: Detail Preserving Deformation of B-spline Surfaces with Volume Constraint

Geometric constraints have proved to be helpful for shape modeling. Moreover, they are efficient aids in controlling deformations and enhancing animation realism. The present paper adresses the deformation of B-spline surfaces while constraining the volume enclosed by the surface. Both uniform and non-uniform frameworks are considered. The use of level-of-detail (LoD) editing allows the preservation of tne details during coarse deformations of the shape. The key contribution of this paper is the computation of the volume with respect to the appropriate basis for LoD editing: the volume is expressed through all levels of resolution as a trilinear form and recursive formulas are developped to make the computation e±cient. The volume constrained is maintained through a minimization process for which we develop closed solutions. Real-time deformations are reached thanks to sparse data structures and e±cient algorithms.


left: initial model,
middle: deformation without volume preseravtion,
right: deformation with volume deformation

Theme 4: Wrinkle generation via length preserving MR editing

In this paper a method for multiresolution deformation of planar piecewise linear curves that preserves the curve length is presented. In a wavelet based multiresolution editing framework, the curve can be deformed at any level of resolution through its control points. Enforcing the length constraint is carried out in two steps. In a first step the multiresolution decomposition of the curve is used in order to approximate the initial curve length. In a second step the length constraint is satisfied exactly by iteratively smoothing the deformed curve. Wrinkle generation is an application the paper particularly focuses on. It is shown how the multiresolution definition of the curve allows to explicitly and intuitively control the scale of the generated wrinkles.


MR editing: the initial curve (a) is at first edited at the coarsest level (b) and then at an intermediate level (c) and (d).

The Figure shows cross deformations on a regular quadrangular mesh of size 129x129. Our method is applied on each line. The initial surface is square and quasi planar. It is first pinched at the coarsest level, creating large wrinkles. Then a part of the mesh is compressed in the transverse direction: small wrinkles are superimposed on the largest ones.

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