Real-time editing of man-made mesh models under geometric constraints

Highlights

• Geometric constraints are precisely preserved during real-time editing of man-made mesh models.

• Shape parameters are introduced to explicitly and concisely represent the geometry/relationship constraints.

• A proper subspace of the constrained deformation is explored by analyzing an auxiliary function.

Abstract

Editing man-made mesh models under multiple geometric constraints is a crucial need for product design to facilitate design exploration and iterative optimization. However, the presence of multiple geometric constraints (e.g. the radius of a cylindrical shape, distance from a point to a plane) as well as the high dimensionality of the discrete mesh representation of man-made models make it difficult to solve this constraint system in real-time. In this paper, we propose an approach based on subspace decomposition to achieve this goal. When a set of variables are edited by the user, the proposed method minimizes the residual of the constraint system in a least square sense to derive a new shape. The resulting shape shall comply with the assigned (extrinsic) constraints while maintaining the original (intrinsic) constraints analyzed from the given mesh model. In particular, we extract a meaningful subspace of the entire solution space based on the user’s edits to reduce the order of the problem, and solve the constraint system globally in real-time. Finally, we project the approximate solution back to the original solution space to obtain the editing result.

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:

• 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;

• 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;

• We propose an integral end-to-end framework for mesh editing, including shape parameters analysis, constraint formulation and real-time solving.