In this paper, we propose a parameterization transfer algorithm for planar domains bounded by B-spline curves, where the shapes of the planar domains are similar. The domain geometries are considered to be similar if their simplified skeletons have the same structures. One domain we call source domain, and it is parameterized using multi-patch B-spline surfaces. The resulting parameterization is C1 continuous in the regular region and G1 continuous around singular points regardless of whether the parameterization of the source domain is C1/G1 continuous or not. In this algorithm, boundary control points of the source domain are extracted from its parameterization as sequential points, and we establish a correspondence between sequential boundary control points of the source domain and the target boundary through discrete sampling and fitting. Transfer of the parametrization satisfies C1/G1 continuity under discrete harmonic mapping with continuous constraints. The new algorithm has a lower calculation cost than a decomposition-based parameterization with a high-quality parameterization result. We demonstrate that the result of the parameterization transfer in this paper can be applied in isogeometric analysis. Moreover, because of the consistency of the parameterization for the two models, this method can be applied in many other geometry processing algorithms, such as morphing and deformation.

In computer-aided design (CAD), boundary representation is one of the main methods for representing three-dimensional (3D) shapes. The process of obtaining a solid spline representation of a computational domain from a given boundary is called parameterization. For planar geometries, parameterization means finding a B-spline surface representation from given boundary curves. The quality of the parameterization of the computational domain has a great impact on the computational accuracy and efficiency in isogeometric analysis (IGA), which is similar to the influence of the mesh quality in finite element analysis. Usually, the parameterization of a computational domain suitable for IGA satisfies the following three basic requirements: (1) Regularization. The mapping from the parametric domain to the physical domain should be injective. Therefore, there is no self-intersection in the iso-parametric structures. (2) Uniformity. The iso-parametric lines are as uniform as possible. (3) Orthogonality. Orthogonality of the iso-parametric structures is preferred in IGA.

In this paper, we propose a parameterization transfer algorithm from one planar domain to another domain, where the two domains have similar shapes, and the parameterization of one domain is given. The geometry with a given parameterization is called the source model, and the other is called the target model. The parameterization of the target model with the transfer algorithm is C1/G1 continuous regardless of whether the source model is C1/G1 continuous or not. The boundaries of the two planar domains are B-spline curves. Therefore, boundary control points of the two planar domains need to be matched first, and the parameterization transfer process is to map the internal control points of the source model to the interior of the target model. The parameterization transfer we proposed meets the following three requirements: (1) Injectivity. The resulting parameterization does not have self-intersection. (2) Low distortion. As the two planar domains are similar in our algorithm, if the parameterization of the source model has high quality, the resulting parameterization also has high quality if the distortion of the map is as small as possible. (3) C1/G1 continuity. Whether the parameterization of the source model satisfies C1/G1 continuity or not, we hope the resulting parameterization has this continuity without additional post-processing.

We use the method proposed by Xu et al. [

Construct boundary correspondence. First, we extract the boundary splines of the source model in the counter-clockwise direction, and then we process the boundary of the target model to obtain a boundary representation that is also in the counter-clockwise direction. Second, the number of control points on the boundaries of the target model

Construct a discrete harmonic map that preserves the continuity of the source model or adds C1/G1 continuity to the target model. In this paper, the mapping is computed on a triangle mesh of the source target model, and the position of the internal control points of the target model is obtained by linear interpolation on the triangle mesh. The C1/G1 continuity and boundary correspondence are used as constraints to obtain the parameterization transfer result.

Experiments show that if the shape of the target model approximates the shape of the source model under a rigid transformation, articulation transformation, or tiny non-rigid transformation, the transfer process can produce a high-quality parameterization for a target model, and the resulting parameterization satisfies C1/G1 continuity.

A considerable amount of work has been conducted related to the parameterization of two-dimensional domains for IGA. However, parameterizing two domains with the same spline topology has been rarely researched. Our work aims to parameterize the target model with some topology of a spline representation of the source model, which can produce high-quality parameterization of the target model, and the same topology of the two models can be used for pre-processing of other geometry processing problems. Below, we introduce the work most related to our algorithm.

Zheng et al. [

We denote the source model as

The construction of the mapping can be viewed as the following constrained optimization problem:

The procedure of our algorithm can be summarized as follows. Suppose a source model is given as parameterized spline surfaces and a target model is given as boundary spline curves. First, boundary control points of the two models will be pre-processed to be consistent. Then, the two models are triangulated based on a boundary control polygon, and a discrete map on the background triangular mesh is established with C1/G1 continuity along adjacent spline patches. The location of the interior control points of the target model can be determined with the optimized mapping.

In this paper, we suppose the two models are similar, which means their simplified skeletons have the same structure. The target boundary shape approximates the source shape through rigid transformations (such as scaling, translation, and rotation) or small non-rigid transformations (such as articulation and local transformation). However, the boundary representation is not necessarily the same. To establish a one-to-one correspondence between the boundary control points of the two models, we need to sample control points on their boundaries. If the density of the target boundary is less than that of the source boundary, the target model will be refined through knot insertion. The target boundary is approximated based on the sampled boundary control points so that the target boundary curve representation is the same as that of the source boundary. The process can be roughly divided into the following two steps:

Let

For each curve

For the B-spline curve fitting, given

Inserting

The control points

Finally, we can establish

Once the boundary representation of the target model is consistent with the source model, there is a one-to-one correspondence between the boundaries of the two models. The goal of parametrization transfer is to determine the location of the internal control points under mapping

We construct a Delaunay triangulation based on the Triangle library with control points of the source model and boundary control polygons as constraints to obtain the source mesh

Sets

The locations of the control points inside the target geometry are determined by the mapping

where

To obtain the C1/G1 continuity between the patches of transferred parameterization, we can define the constraints in a linear form

These are then inserted into the following equation to obtain the G1 continuity constraint that should be satisfied at the singular points

After obtaining the background mesh of the source model, the vertices satisfying the constraint

We use block coordinates to solve the optimization problem similar to the reversible harmonic map, and the result obtained is shown in

When the difference between the target and source shape is large, the initial mapping result by

The singular value decomposition (SVD) decomposition of

If the smallest singular value

Determination of

Coordinates

The unknown quantities in the above equation are the bounded conformal mapping

Another important evaluation indicator is the condition number of the Jacobian matrix

For a planar parameterization, the closer the scaled Jacobian matrix is to 1.0, the better the quality of the parameterization result will be. Similarly, the closer the condition number of the parametric Jacobian matrix is to 2.0, the better the parametrization will be. In the following example, we use the colormap of the scaled Jacobian matrix on the iso-parametric edges in the source and target models whose parameterizations were obtained by transfer, which is shown in

Model | Scaled Jacobians | Condition number | ||||
---|---|---|---|---|---|---|

Minimum | Average | Maximum | Minimum | Average | Maximum | |

Rabbit_s | 0.0013 | 0.9308 | 1 | 2 | 2.4689 | 2011.8 |

Boarder_s | 0.0128 | 0.8905 | 1 | 2 | 2.5718 | 156.667 |

Two_hole_s | 2.92 × 10^{−5} |
0.9543 | 1 | 2 | 4.3951 | 73852 |

Human_s | 0.0013 | 0.8957 | 1 | 2 | 2.5958 | 1598.8 |

Rabbit | 0.0048 | 0.9244 | 1 | 2 | 2.4940 | 439.59 |

Boarder | 0.0017 | 0.8770 | 1 | 2 | 2.8581 | 1751.4 |

Two_hole | 0.00487 | 0.9590 | 1 | 2 | 2.2722 | 4113.7 |

Human | 0.000189 | 0.9083 | 1 | 2 | 2.6942 | 1030.1 |

The right terminal term in the Poisson equation is

Based on the definition of the spline surface, the approximating numerical solution in each subdomain can be formulated as follows:

We proposed a novel method to transfer the parameterization results of a model to another model with a similar shape, and the C1/G1 continuity was satisfied between adjacent patches of the target model. The parameterization results can be used in IGA and other geometry processing problems. Experiments showed that if the shape of the target model is similar to the original model, e.g., if the two models can be transformed by rotation, translation, scaling, and other rigid or articulation transformations, with a small degree of nonlinear deformation, the algorithm can obtain a more ideal parametrization transfer result. However, determining how to measure the similarity of the two shapes and the transfer result theoretically has not been defined clearly, which is a problem to be solved in the future.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 62072148 and U22A2033), the National Key R&D Program of China (Grant Nos. 2022YFB3303000 and 2020YFB1709402), the Zhejiang Provincial Science and Technology Program in China (Grant No. 2021C01108), the NSFC-Zhejiang Joint Fund for the Integration of Industrialization and Informatization (Grant No. U1909210), and the Fundamental Research Funds for the Provincial Universities of Zhejiang (Grant No. 490 GK219909299001-028).

The authors declare that they have no conflicts of interest to report regarding the present study.