Elsevier

Computers & Graphics

Volume 30, Issue 6, December 2006, Pages 936-946
Computers & Graphics

Interactive mesh deformation using equality-constrained least squares

https://doi.org/10.1016/j.cag.2006.08.012Get rights and content

Abstract

Mesh deformation techniques that preserve the differential properties have been intensively studied. In this paper, we propose an equality-constrained least squares approach for stably deforming mesh models while approximately preserving mean curvature normals and strictly satisfying other constraints such as positional constraints. We solve the combination of hard and soft constraints by constructing a typical least squares system using QR decomposition. A well-known problem of hard constraints is over-constraints. We show that the equality-constrained least squares approach is useful for resolving such over-constrained situations. In our framework, the rotations of mean curvature normals are treated using the logarithms of unit quaternions in R3. During deformation, mean curvature normals can be rotated while preserving their magnitudes. In addition, we introduce a new modeling constraints called rigidity constraints and show that rigidity constraints can effectively preserve the shapes of feature regions during deformation. Our framework achieves good performance for interactive deformation of mesh models.

Introduction

Mesh deformation is useful in a variety of applications in computer graphics and computer-aided design. In recent years many discrete deformation techniques based on differential properties have been published [1], [2], [3], [4], [5]. They represent the differential properties of a given surface as a linear system and deform the surface so that the differential properties are preserved. In typical interactive mesh editing, the user first selects the region to be fixed, and then the vertices to be moved as the manipulation handle. Such user-specified conditions on vertices are called positional constraints. When the user drags the positions of handle vertices on the screen, the surface is deformed according to the handle manipulation.

Meyer et al. [6] approximated the mean curvature normals using cotangent weights. This method can produce good results even when triangles are not uniformly constructed. When a linear system is solved using the least squares method, positional constraints are satisfied by placing large weights in the least squares system. However, it is not easy to predict weight values so that positional constraints are satisfied within the allowable margin of error, because the magnitude of a mean curvature normal may be largely different at each vertex. As the values of weights on positional constraints become increasingly large, the solver satisfies positional constraints more strictly. However, very large weights make the solver numerically unstable when the mean curvature normals are represented using cotangent weights.

We propose a novel equality-constrained least squares approach in this paper. In our method, positional constraints are described as hard constraints. Then the combination of hard and soft constraints is converted to a typical least squares system using QR decomposition. This method stably and efficiently computes vertex positions that satisfy positional constraints precisely and preserve mean curvature normals in the least squares sense.

A well-known problem of hard constraints is over-constraints. If over-constraints are involved, the solver may halt the computation. Our framework is useful for resolving such over-constrained situations, which include redundant and conflicting constraints.

When handle vertices are rotated, mean curvature normals need to be rotated automatically [1]. We introduce a new rotation-propagation method that interpolates the logarithms of unit quaternions. Our method realizes the spherical linear interpolation of rotations from fixed vertices.

In addition, we propose a new modeling constraints called rigidity constraints, which preserve the shapes of feature regions during deformation. Rigidity constraints are very useful not to deform important features, such as eyes.

Our main contribution in this paper is as follows:

  • A new deformation framework by means of the equality-constrained least squares method, which

    • incorporates hard constraints as well as soft constraints,

    • robustly calculates cotangent weights for Laplacian discretization, and

    • can detects conflicting and redundant constraints in hard constraints.

  • A new rotation-propagation method by means of the interpolation of quaternion logarithms.

  • Introduction of rigidity constraints that preserve the shapes of feature regions during deformation.

In the following section, we review the related work on 3D shape deformation. In Section 3, we describe our mesh deformation framework using the equality-constrained least squares method. In Section 4, we introduce rigidity constraints for feature regions. In Section 5, we will show how to manage over-constraints in hard constraints. In Section 6, we evaluate our framework and show experimental results. We conclude the paper in Section 7.

Section snippets

Related work

Interactive mesh editing techniques have been intensively studied [7]. Such research aims to develop modeling tools for intuitively modifying free-form surfaces while preserving the geometric details.

There are several types of approach for mesh editing: free-from deformation (FFD), multiresolution mesh editing, and partial differential equation (PDE)-based mesh editing.

FFD methods are very popular approaches. They modify shapes implicitly by deforming 3D space in which objects are located [8],

Preliminaries

Let M=(V,E,F) be a given triangular mesh with n vertices. V, E and F are the set of vertices, edges and faces, respectively. Vertex iV has a three-dimensional coordinate pi. The original position of pi is referred to as pi0.

A discrete Laplacian operator L(pi) and the original Laplacian vector δi are defined asL(pi)=jN(i)wij(pi-pj),δi=jN(i)wij(pi0-pj0),where N(i)={j|(i,j)E} is the set of immediate neighbors of vertex i. Then the detail shape of a mesh model is encoded as the following

Rigidity constraints for feature regions

In addition to positional constraints, we introduce rigidity constraints that constrain the relative positions of vertices aspi-pj=uij,where uijR3 is a certain constant vector. These constraints are added to a linear system defined in Eq. (7).

When rigidity constraints are specified to edges in a feature region, the shape of the feature region is preserved as a rigid body while deformation. The user can specify such a feature region by drawing a closed curve on a mesh model. Then the spanning

Resolution of over-constraints

In our framework, positional constraints are strictly satisfied as hard constraints. Therefore, if conflicting or redundant constraints are involved, they lead to the rank deficiency in matrix B, and the solver may halt the computation.

We can resolve such over-constraint problems by detecting and removing the deficiency of the rank during the processes of QR decomposition.

QR decomposition BT=QR can be computed using Householder factorization [30]. Each column in BT corresponds to a hard

Experimental results

In this section, we show some results of deformation based on the equality-constrained least squares method.

Fig. 9 shows examples of deformed shapes. This mesh has badly shaped triangles, as shown in Fig. 3. Our method produced good results.

Fig. 10 shows sample models that have been commonly used for evaluation in computer graphics. We solved the linear systems using TAUCS [27], which is a well-known solver of Cholesky factorization.

Table 1 shows the numerical stability. When all constraints

Conclusion

In this paper, we proposed an equality-constrained least squares approach for stably computing mesh deformation that strictly satisfies positional constraints and approximately preserves mean curvature normals. Our experimental results for mesh models showed that our method is sufficiently stable compared to existing methods based on the least squares method. The performance of our method was as good as that of the method that manages only soft constraints. We also showed that our framework is

Acknowledgments

The Armadillo, Bunny and Dragon models are courtesy of Stanford University, and the Mannequin model is courtesy of the Graphics and Imaging Laboratory, University of Washington.

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