Elsevier

Computers & Graphics

Volume 58, August 2016, Pages 128-138
Computers & Graphics

Special Issue on SMI 2016
Optimized subspaces for deformation-based modeling and shape interpolation

https://doi.org/10.1016/j.cag.2016.05.016Get rights and content

Highlights

  • Method to construct deformation subspaces for shape modeling and interpolation.

  • Based on automatic generation of a set of training deformations.

  • Description of our sampling strategies for the two particular methods.

Abstract

We propose a novel construction of subspaces for real-time deformation-based modeling and shape interpolation. The scheme constructs a subspace that optimally approximates the manifold of deformations relevant for a specific modeling or interpolation problem. The idea is to automatically sample the deformation manifold and construct the subspace that best-approximates these snapshots. This is realized by writing the shape modeling and interpolation problems as parametrized optimization problems with few parameters. The snapshots are generated by sampling the parameter domain and computing the corresponding minimizers. Finally, the optimized subspaces are constructed using a mass-dependent principle component analysis. The optimality provided by this scheme contrasts it from alternative approaches, which aim at constructing spaces containing low-frequency deformations. The benefit of this construction is that compared to alternative approaches a similar approximation quality is achieved with subspaces of significantly smaller dimension. This is crucial because the run-times and memory requirements of the real-time shape modeling and interpolation schemes mainly depend on the dimensions of the subspaces.

Introduction

Creating digital geometric content is an important task for applications in various areas including digital manufacturing, computer animation, and virtual reality. Acquisition technologies, like 3D-scanning, allow for creating accurate digital copies of detailed real-world objects. Therefore, methods for modeling a single shape and for synthesizing new ones from a collection of shapes are essential for customizing digital content to the demands of users and applications. Here we consider two such methods: deformation-based modeling and shape interpolation. Deformation-based modeling tools provide a user with simple and intuitive interfaces for modifying a digital shape. For example, a user can translate and rotate parts of the object, so-called handles, and the rest of the shape follows automatically. Physical models of deformable objects are used to produce deformations that match the users intuition. For shape interpolation, we consider a set of example shapes, e.g., different poses of one character. Shape interpolation allows for creating new “in-between shapes” and is a crucial module of schemes for tasks like morphing, deformation transfer, example-based shape editing, example-based materials for controlling simulations, and shape exaggeration.

A fundamental problem for both methods, modeling and interpolation, is that on the one hand, processing tools need to solve high-dimensional non-linear optimization problems to compute the deformed shapes, and, on the other hand, users expect fast or even interactive responses. Therefore, it is essential to design efficient approximation algorithms for these problems. Subspace methods proved to be very effective. The principle is to construct a low-dimensional approximation of the complex problem in a preprocess (offline phase) and to solve only the low-dimensional system in the interactive (online) phase. Different schemes for constructing subspaces for deformation-based modeling have been introduced based on space deformations, radial basis functions, bi-harmonic problems, low-frequency Laplace–Beltrami eigenfunctions or vibration modes. The common goal of these methods is to construct subspaces containing low-frequency deformations. An alternative approach is to learn the subspaces from observations. Methods following this idea, such as the method of snapshots, are prominent for the reduction of physical simulations.

In this work, we introduce constructions of subspaces that are optimized for deformation-based shape modeling and shape interpolation tasks. The constructions involve the following technical contributions. We formulate general frameworks for shape modeling and interpolation as parameterized optimization problems with low-dimensional compact parameter domains. Then, observations of the shape modeling or interpolation tasks can be obtained by samplings the solution space of the optimization problem. This in turn can be done by sampling the parameter domain of the optimization problem and computing the corresponding deformations. For shape interpolation, the parameters are the interpolation weights. Since the interpolation weights are positive and sum to one, the set of weights forms a simplex, whose dimension is one less than the number of example shapes to be interpolated. To generate the snapshots for the interpolation problem, we sample the simplex and compute the corresponding interpolating shapes. For deformation-based modeling, we consider deformation handles that can be translated and rotated in space. To obtain a compact parameter domain, we introduce a maximum translation for the handles, e.g., the length of the objects bounding box diagonal, and parametrize the rotations using Euler angles. Then the parameter domain is a rectangular box (cuboid) of dimension 6(h1), where h is the number of handles. A sample point in this box specifies locations in R3 for all handles. To generate the snapshots for deformation-based shape modeling, we sample the box and for every sample point, we compute the deformation corresponding to the handle locations. Once the snapshots for a specific modeling or interpolation task have been generated, the subspaces that optimally approximate the snapshots are constructed. A mass-orthonormal basis of a such a subspace can be obtained by a mass-weighted principle component analysis (PCA) of the snapshots.

Our approach contrast from alternative subspace construction for deform-based modeling. Whereas our approach yields low-dimensional subspaces that are optimized for containing good approximations of the deformations that are relevant for a specific setting (e.g., a set of handles defined on shape or a set of example shapes to be interpolated), alternative approaches aim at spaces containing low-frequency deformations. We analyze the quality of the resulting subspaces in experiments and comparisons to alternative approaches. In particular, we demonstrate that the proposed construction results in more efficient subspaces that achieve a comparable approximation quality with significantly smaller dimension. This is crucial because the computational cost for solving the reduced problems mainly depends on the dimension of the subspace.

Section snippets

Related work

Deformation-based shape modeling and interpolation have received much attention in recent years. We can distinguish between linear and non-linear frameworks. For an in-depth discussion of linear schemes, which are not in the focus of this paper, we refer to the survey [1]. Examples of non-linear frameworks are co-rotated iterative Laplacian editing schemes[2], [3], PriMo [4] and As-Rigid-As-Possible [5], [6].

Dimensional reduction proved to be a powerful concept for designing computational

Parametrized optimization problems in shape deformation

Computing optimal deformations of shapes is important for many applications in geometry processing. Here, we consider optimization problems that are controlled by few parameters. This means, we are looking at problems of the formargminxRnF(x,ω),where xRn is the search space, ωΩRm is a vector listing the parameters, and we assume that n is large and m is small. We specifically address two applications, namely deformation-based shape modeling and interpolation. Before discussing these two

Sampling the space of solutions

Before we consider the construction of optimized subspaces in the next section, we look at strategies for sampling the solution space of the optimization problems. A reasonable approach is to sample the parameter domain following a particular distribution and compute minimizers of the samples. In this section, we first discuss the sampling of the parameter domains for modeling and interpolation, then the efficient computation of minimizers for lifting the sampling from the parameter domain to

Subspace construction

In this section, we discuss the dimensional reduction of the optimization problem (1). The motivation is that the optimization problem is controlled by only a few parameters (rigid motions of handles for deformation and weights for the interpolation), which means that in the solution space the n degrees of freedom of the shape (e.g., the vertex positions) are correlated. In other words, the solution space of the optimization problem is a low-dimensional object in Rn. The idea is to construct a

The reduced optimization problem

After the subspace has been constructed, we restrict the optimization problem to this space. This means, we consider the reduced objective function F^(q,ω)=F(Uq+τ,ω)and instead of (1) and solve the reduced problem argminqRdF^(q,ω).Our goal is to construct the reduced optimization problem such that the cost for solving it depends only on the subspace dimension and the number of parameters of the optimization problem. In particular, it should be independent of the resolution of the shapes to be

Results and discussion

The supplementary video shows examples of real-time shape modeling and interpolation performed with our implementation of the proposed framework. In all the shown examples, we use very low-dimensional subspaces (6–60 dim.) and demonstrate that large deformations including twists can be represented. One example, is the interpolation between a cylindrical shape and a twisted helix, shown in Fig. 4. The example nicely illustrates the high-quality of the shape interpolation resulting from the

Conclusion

We present a novel subspace construction for deformation-based shape modeling and interpolation. In contrast to previous approaches, our construction produces subspaces that are explicitly optimized for approximation of the deformations reachable with a specific user interface. With this method very low-dimensional subspace for modeling and interpolation can be produced. Comparisons to alternative approaches illustrate the benefits of subspace optimization. These spaces are well-suited for

Acknowledgments

We would like to thank Christopher Brandt, Christian Schulz and Christoph von Tycowicz for inspiring discussions and for sharing code and data, Yu Wang, Alec Jacobson, Jernej Barbič and Ladislav Kavan for making their biharmonic coordinates code available as part of libigl, and the anonymous reviewers for helpful comments and suggestions.

References (60)

  • G. Rong et al.

    Spectral mesh deformation

    Vis Comput

    (2008)
  • Rustamov RM. On mesh editing, manifold learning, and diffusion wavelets. In: IMA international conference on...
  • T. Dey et al.

    Eigen deformation of 3d models

    Vis Comput

    (2012)
  • Y. Wu et al.

    HIRM: a handle-independent reduced model for incremental mesh editing

    Comput Aided Geom Des

    (2015)
  • J. Barbič et al.

    Real-time subspace integration for St. Venant–Kirchhoff deformable models

    ACM Trans Graph

    (2005)
  • K. Hildebrandt et al.

    Interactive surface modeling using modal analysis

    ACM Trans Graph

    (2011)
  • C. von Tycowicz et al.

    An efficient construction of reduced deformable objects

    ACM Trans Graph

    (2013)
  • R.W. Sumner et al.

    Embedded deformation for shape manipulation

    ACM Trans Graph

    (2007)
  • B. Adams et al.

    Meshless shape and motion design for multiple deformable objects

    Comput Graph Forum

    (2010)
  • X. Wu et al.

    Real-time symmetry-preserving deformation

    Comput Graph Forum

    (2014)
  • A. Jacobson et al.

    Bounded biharmonic weights for real-time deformation

    ACM Trans Graph

    (2011)
  • Y. Wang et al.

    Linear subspace design for real-time shape deformation

    ACM Trans Graph

    (2015)
  • R.Y. Wang et al.

    Real-time enveloping with rotational regression

    ACM Trans Graph

    (2007)
  • O. Weber et al.

    Context-aware skeletal shape deformation

    Comput Graphics Forum

    (2007)
  • M. Botsch et al.

    Adaptive space deformations based on rigid cells

    Comput Graph Forum

    (2007)
  • Y. Lipman et al.

    Green coordinates

    ACM Trans Graph

    (2008)
  • M. Ben-Chen et al.

    Variational harmonic maps for space deformation

    ACM Trans Graph

    (2009)
  • F.G. García et al.

    *cages: : A multilevel, multi-cage-based system for mesh deformation

    ACM Trans Graph

    (2013)
  • J. Zhang et al.

    Local barycentric coordinates

    ACM Trans Graph

    (2014)
  • E. Landreneau et al.

    Poisson-based weight reduction of animated meshes

    Comput Graph Forum

    (2010)
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