PCO: Precision-Controllable Offset Surfaces with Sharp Features

Abstract

Surface offsetting is a crucial operation in digital geometry processing and computer-aided design, where an offset is defined as an iso-value surface of the distance field. A challenge emerges as even smooth surfaces can exhibit sharp features in their offsets due to the non-differentiable characteristics of the underlying distance field. Prevailing approaches to the offsetting problem involve approximating the distance field and then extracting the iso-surface. However, even with dual contouring (DC), there is a risk of degrading sharp feature points/lines due to the inaccurate discretization of the distance field. This issue is exacerbated when the input is a piecewise-linear triangle mesh. This study is inspired by the observation that a triangle-based distance field, unlike the complex distance field rooted at the entire surface, remains smooth across the entire 3D space except at the triangle itself. With a polygonal surface comprising $n$ triangles, the final distance field for accommodating the offset surface is determined by minimizing these $n$ triangle-based distance fields. In implementation, our approach starts by tetrahedralizing the space around the offset surface, enabling a tetrahedron-wise linear approximation for each triangle-based distance field. The final offset surface within a tetrahedral range can be traced by slicing the tetrahedron with planes. As illustrated in the teaser figure, a key advantage of our algorithm is its ability to precisely preserve sharp features. Furthermore, this paper addresses the problem of simplifying the offset surface’s complexity while preserving sharp features, formulating it as a maximal-clique problem.

Fast Forward

What is this paper about?

The offset at a distance can be considered a specific case of the Minkowski operation, widely explored in mathematical morphology. This process computes the offset surface by allowing a sphere to traverse with its center lying on the surface:

It can be observed that even smooth surfaces may exhibit sharp features in their offsets due to the non-differential characteristics of the underlying distance field. So we introduce the PCO for generating offset surface with sharp features.

The research presented in this paper is driven by an insightful observation: A distance field derived from individual triangles, as opposed to one given by the entire surface, remains smooth throughout the 3D space, with irregularities appearing only at the triangles themselves. This inherent smoothness of a triangle-based distance field enables us to divide the space into small units and apply a linear approximation within each unit. It is important to highlight that this piecewise linear approximation method is effective provided that the units are sufficiently small. Based on the fact that the actual distance field of the entire surface, necessary for creating the offset, can be constructed by minimizing across all the triangle-based distance fields. In contrast, traditional isosurface extraction methods, which treat the boundary curve collectively, often struggle to accurately represent sharp features.

Moreover, we recommend merging the

n

triangles, the final distance field for accommodating the offset surface is determined by minimizing these

n

fields, tetrahedron by tetrahedron, to simplify the complexity of the offset surface while maintaining the integrity of sharp features. This process can be formulated as a maximal-clique problem.

Pipeline

Some results

Gallery

A gallery of our feature-preserving offsetting results from inputs of freeform and CAD models:

Datasets validation

We validate the effectiveness on extensive datasets with complex, often defect-laden meshes: Thingi10K and ABC. The average values of relative Chamfer distance and Hausdorff distance are generally below

10^{-1}

, average normal deviation values are predominantly within

[0^\circ, 5^\circ]

, and most N-Scores approach

100\%

, with all exceeding

60\%

. The runtime performance depends on the complexity of the input mesh, but the total time remains under

10^{3}

seconds even for meshes with millions of triangles.

Robustness validation

Our method efficiently recovers sharp features, as indicated by the red solid lines in~(a). Additionally, our method can handle non-manifold, self-intersecting models and ensures valid output. It can preserve fine details even in models with thin plates, as shown in highlighted windows in~(b) and~(c).

Varying offset distance

Regardless of the offset distance, our algorithm consistently produces the desired results while preserving sharp features, even when

\delta

increases significantly:

Offset comparison

To comprehensively evaluate our algorithm, we selected several state-of-the-art approaches capable of generating offset surfaces with available implementations. Here, we illustrate three visual comparison results from three kinds of models: CAD models, thin-plate models, and models reconstructed from real raw large scan data.

CAD & Thin-plate models

CAD models often exhibit prominent feature lines, thus used to test the capability of preserving sharp features. Twisted and thin-plate models often presents challenges in preserving sharp features and correctly handling intersections when perform the offsetting process on these models.

Models reconstructed from actual raw scans

These models always exhibit rich geometric details and have millions of faces, which presents challenges in preserving sharp features while maintaining high performance.

Offset simplifaction

To explore how our simplification framework affects precision control and simplification level in offset generation, we performed experiments with varying values of

\theta

, a hyper-parameter that regulates the similarity between two linear distance fields. As

\theta

increases, a greater number of linear fields can be merged, resulting in a more simplified offset mesh.

Extended Framework for Opening and Closing

Offsetting (erosion and dilation) operations are foundational building blocks in morphology. Unlike the offsetting operation, the opening and closing operations preserve the original shape while filling the overly convex or concave parts with a user-specified parameter. This is particularly useful in consolidating mechanical structures.

Acknoledgement

    The authors would like to thank the anonymous reviewers for their valuable comments and suggestions. 
This work is supported by National Key R\&D Program of China (2021YFB1715900), 
and National Natural Science Foundation of China (62272277, U23A20312, 62072284, 61772314, 61761136017, and 62072275).

BibTeX citation

    @article{wang2024pco,
    author = {Wang, Lei and Wang, Xudong and Wang, Pengfei and Chen, Shuangmin and Xin, Shiqing and Guo, Jiong and Wang, Wenping and Tu, Chenghe},
    title = {PCO: Precision-Controllable Offset Surfaces with Sharp Features},
    journal = {ACM Trans. Graph.},
    address = {New York, NY, USA},
    publisher = {ACM},
    year = {2024},
    volume = {43},
    number = {6},
    doi = {10.1145/3687920}
}