Special Issue on SMI 2016Optimized subspaces for deformation-based modeling and shape interpolation
Graphical abstract
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 , where h is the number of handles. A sample point in this box specifies locations in 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 formwhere is the search space, 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 . 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 and instead of (1) and solve the reduced problem 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.
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