Special Section on SMI 2019Real-time editing of man-made mesh models under geometric constraints
Graphical abstract
Introduction
Three-dimensional discrete meshes play an important role in shape modeling for their flexibility in representing complex shapes with arbitrary topology. This attribute makes mesh representation the most extensively used data structure for computational engineering analysis, a standard sub-routine for industrial product development. However, due to the heterogeneity between the analysis models (in discrete mesh representation) and design models (in parametric representation), design exploration and optimization would require a large amount of human intervention to convert between two types of models.
To address this issue, many shape deformation methods [1], [2], [3] have been developed and adopted to generate design variants for design exploration [4] and optimization [5]. However, since most of these methods solely rely on local geometric descriptors to produce smoothly deformed shapes, they are more suitable for editing 3D free-form models rather than mechanical parts with multiple geometric design constraints that may arise from semantic, functional or manufacturing considerations in real-world applications[6]. Although the structure of such a mechanical part is easily perceived by humans, this information is usually not explicitly defined and associated to the mesh model for computational engineering analysis. Thus, the lack of controlling explicit shape parameters and structural relationship in mesh models makes shape editing with geometric or relationship constraints one of key challenges for most of existing shape editing tools.
In recent years, there have been increasing interests in constraint-aware shape editing. To accelerate the optimization-based deformation during user interaction, Gal et al. [7] and Zheng et al. [8] use high-level structures to drive the low-level meshes and the constrained deformation system is solved using a local greedy strategy. Habbecke and Kobbelt [9] solve a system with non-linear equality constraints to derive deformation of polygonal shapes modeled with a small set of vertices. However, computing the constrained deformation of a dense mesh with a global optimization cannot be obtained efficiently till now. Moreover, deforming a 3D mesh by direct modification of shape parameters (e.g., the radius of a circle, the width of a cube, etc.) like CSG modeling is very tedious as few methods are formulated to explicitly take shape parameters into account. Thus, another key challenge we are faced with is to achieve real-time constrained deformation of mesh models of mechanical parts in an intuitive manner.
Our paper proposes a novel shape editing method that integrates the two editing approaches, i.e., defining shape parameters commonly used in CSG modeling and manipulating a set of vertices advocated by free-form modeling. We formulate this shape editing problem as a least squares problem. To solve it efficiently, we further devise an auxiliary function for exploring a compact subspace of the solution space. This will accelerate the overall deformation process to fulfil the real-time interaction goal. Our method is validated on various man-made models and the experiments are demonstrated in the results.
Our contributions can be summarized as:
- 1.
We introduce shape parameters into the constrained deformation system to explicitly and concisely represent the geometry/relationship constraints. Moreover, the shape parameters are very useful in subspace extraction and can be directly modified to obtain a desired new shape satisfying the modified shape parameters and constraints;
- 2.
The original dimension of the solution space is reduced by finding a proper subspace, which is realized by carefully designing an auxiliary function that preserves both geometric constraints and the mesh quality;
- 3.
We propose an integral end-to-end framework for mesh editing, including shape parameters analysis, constraint formulation and real-time solving.
Section snippets
Related work
Free-form deformation. Free-form deformation techniques have achieved good results to preserve low-level geometric detail and retain mesh quality. Most of these series of work preserve local discrete geometric magnitude in deformation, e.g., Laplacian coordinates [1], local rigid transformations [10], or biharmonic deformation fields [2]. However, all of those methods adopt no explicit geometric constraints. For this reason, they cannot handle man-made objects well.
High-level surface
Overview
The input data of our method is a typical 3D triangle mesh with no additional structure information. We first analyze the underlying shape parameters and relations among those shape primitives. This is done by adopting the method proposed in [19] which efficiently approximates the given model with various shape primitives, e.g., planes and cylinders. Then we use the DBSCAN method [20] to cluster the principal directions of different shapes into several groups, each of which represents a
Constraints formulation
In this section, we introduce several elementary constraints as building blocks for forming more complex shape constraints. These elementary constraints are characterized by their shape parameters and are presented as follows.
Planar constraints. A point p(x, y, z) lies in a plane, whose equation can be written as where n(nx, ny, nz) and d are the shape parameters of the plane and also variables in our constraint system. Thus, the planar constraint can be formulated as two residuals as
Optimization
In typical cases, man-made mesh models are represented using thousands of vertices. There are 3n degree of freedoms on the 3D mesh itself (with the number of vertices to be n). When considering the additional m shape parameters, it is hard to solve such a huge non-linear system in real-time. Moreover, the number of intrinsic geometry constraint residuals Eg is roughly n, which is far smaller than the dimension of the entire solution space. To accelerate the solving phase, we cast solving such
Results and discussion
Our algorithm is implemented in C++. We use the CGAL [21] library for shape primitives detection, the Eigen [22] library for basic data structure and efficient matrix products, the Sparse Eigenvalue Computation Toolkit (Spectra) [23] for finding the smallest eigenvalues and eigenvectors of a large scale real symmetric matrix (the shift-and-invert mode [24] is adopted), the levmar [25] for the LM step, and the libigl [26] for the GUI. During editing, users can set several entries of vertex
Conclusion
In this paper, we represent complex geometric constraints explicitly using multiple shape primitives and their parameters. Intuitive constrained deformation results can be obtained by solving the constraint system in a least square sense. In order to achieve real-time performance, a subspace approach is utilized to approximate the original solution space. We validate our method on various man-made models and achieve desirable edited results and real-time performance.
Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
We are grateful to the anonymous reviewers for their insightful comments and suggestions. We would like to thank Zhiming Hu for helping preparing the video. This research was supported by the National Key Technology Research and Development Program of China (No. 2017YFB1002705, 2017YFB1002601), the National Natural Science Foundation of China (NSFC) (61632003, 61661146002) and Hong Kong Innovation and Technology Fund (ITF) (ITS/457/17FP).
References (30)
- et al.
Push-pull direct modeling of solid cad models
Adv Eng Softw
(2019) - et al.
Constrained space deformation techniques for design optimization
Comput-Aided Des
(2016) - et al.
Laplacian surface editing
Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on geometry processing SGP ’04;
(2004) - et al.
Mixed finite elements for variational surface modeling
Comput Gr Forum
(2010) - et al.
Linear subspace design for real-time shape deformation
ACM Trans Graph
(2015) - et al.
Interactive design space exploration and optimization for cad models
ACM Trans Graph
(2017) - et al.
Skeleton-section template parameterization for shape optimization
J Mech Des
(2018) - et al.
Structure-aware shape processing
Proceedings of the ACM SIGGRAPH 2014 courses SIGGRAPH ’14;
(2014) - et al.
iwires: an analyze-and-edit approach to shape manipulation
ACM Trans Graph
(2009) - et al.
Component-wise controllers for structure-preserving shape manipulation
Comput Gr Forum
(2011)
Linear analysis of nonlinear constraints for interactive geometric modeling
Comput Gr Forum
As-rigid-as-possible surface modeling
Proceedings of the fifth eurographics symposium on geometry processing SGP ’07;
Pattern-aware shape deformation using sliding dockers
ACM Trans Graph
Dimensional model reduction in non-linear finite element dynamics of solids and structures
Int J Numer Methods Eng
Real-time subspace integration for St. Venant-Kirchhoff deformable models
ACM Trans Graph
Cited by (6)
Screen space shape manipulation by global structural optimization
2023, Computers and Graphics (Pergamon)Part-to-Surface Mesh Segmentation for Mechanical Models Based on Multi-Stage Clustering
2023, CAD Computer Aided DesignForeword to the Special Section on Shape Modelling International 2019
2019, Computers and Graphics (Pergamon)A Revisit of Shape Editing Techniques: From the Geometric to the Neural Viewpoint
2021, Journal of Computer Science and Technology