Curvature-guided adaptive -spline surface fitting☆
Introduction
Triangular meshes and splines are two popular representations for free-form surfaces. Triangular meshes are convenient for visualization and description of complicated shapes of arbitrary topology. With the development of laser scanners or other 3D data acquisition equipments [1], triangular mesh models are easily obtained. Spline surfaces are widely used in the computer-aided design and manufacturing (CAD/CAM) industry and engineering computation as well. They are represented concisely by parametric equations and are good for the description of smooth shapes and shape analysis. In particular, non-uniform rational -splines (NURBS) have been an industry standard in CAD/CAM [2]. For some applications, there is a need to convert a model in triangular mesh representation into splines. The conversion can be achieved by surface fitting or reconstruction. This paper investigates effective techniques for spline surface fitting.
Extensive research has been done on surface fitting. Some interesting methods have been developed for reconstructing -spline surfaces to fit a triangular mesh [3], [4], [5]. These methods either initiate the fitting process with a control mesh that is sufficiently dense or refine the control mesh globally when the approximation error is large. Both of these strategies often result in an over-refined control mesh. In fact, the refinement of the control mesh is desired only in the areas where the fitting result is poor, but the -spline surfaces do not have local refinement. An adaptive approach was proposed in [6] that uses a collection of -spline patches on different layers, with the base layer of patches representing the overall shape and the overlay patches representing the local details. In the overlay patches, however, several rings of control points near the margin of the control mesh must be fixed with the corresponding control points in the lower level in order to maintain continuity. In the work of [7], an approach based on patchwork surface construction was presented. The adaptivity of this method lies in that, when the resulting surface is not accurate enough, the selected faces are chosen to be split and re-approximated. One problem of the method in [7] is that the total number of patches in the result surface could be very large and many unnecessary extraordinary points might be introduced even though the topology of the object is simple.
Besides, geometrical features [8], [9], [10] in free-form surfaces seem to be overlooked in many previous surface fitting approaches. In fact, surface features are important for shape description in that they reflect some identities of a model. Therefore, features should be appropriately reconstructed during surface fitting in order to achieve satisfactory result. A feature sensitive parameterization technique was introduced in [11] to embed the feature information into the parameterization of the triangular mesh and distort the parameter domain so that larger parameter intervals are allocated to feature areas. While this technique could indirectly facilitate non-adaptive -spline surface fitting, its use becomes less apparent in adaptive fitting.
The research effort reported in this paper initially was on fitting spline surfaces to triangular meshes of arbitrary topology as in [4], [7], [12]. However, we soon realized that even the surface fitting of a mesh with simple topology has many open questions, such as how to efficiently achieve adaptive fitting, how to generate good parameterization for fitting purpose, how to generate good initial knot placement for spline surface fitting, and how to automatically allocate more control points in those highly curved feature regions to better represent the features. The solutions to these problems also surely facilitate the surface fitting of arbitrarily topological models. Thus this paper focuses on fitting a spline surface to a triangular mesh that is topologically equivalent to a disc and presents several plug-and-play technical components and strategies, which can be easily integrated into many existing surface fitting algorithms for enhancement to produce visually smooth spline surfaces with concise control meshes and geometrical features well respected. The main contributions of the paper include:
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We propose to use -spline surfaces for adaptive surface fitting. Adaptive surface fitting refines fitting only in the required areas where the fitting results are not satisfactory. Thus adaptive fitting is attractive in handling complicated models with unevenly distributed details. -splines are a new free-form surface technology that solve most of the limitations inherent in the NURBS representation [13], [14]. In particular, -splines allow local refinement, making them a good candidate for adaptive fitting [15], [12]. While -splines are more powerful than -splines, they also bring some technical challenges. For example, -splines in a general setting are likely rational, which makes the computation (especially differentiation) complicated. While previous -spline fitting methods [12], [16] directly use -splines and thus output rational surfaces, we deliberately design our algorithm so that all the -splines constructed in our algorithm are standard or semi-standard [14], which guarantees that they are piecewise polynomial, thus simplifying the fitting process.
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We propose to use curvature to guide the fitting process so that the feature regions receive special attention. Features are characterized as the regions where there is a large variation in surface normal. While features were not paid enough attention to in many previous fitting algorithms, which often results in loss of features in the fitting results, we present a simple strategy that will automatically allocate more control points in the feature regions so that the features can be well represented in the fitting surface.
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We introduce a process called re-parameterization. This is inspired by Hoschek’s work for approximation [17]. With re-parameterization using the current fitting result, the parameterization of the triangular mesh is improved towards an intrinsic one that is able to help optimize the fitting result as shown in [17]. We present a method that guarantees the re-parameterization to be faithful.
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We present a process for re-placing the initial control grid structure of the spline surface. Since -spline surfaces have flexible knot structure, it is very difficult to provide a good initial -spline knot (or structure) placement which is needed in the surface fitting procedure. Our proposed method heuristically constructs knot re-placement by taking the current fitting result and the characteristics of the input model into consideration and ensures that the -splines defined over the generated knot structure are standard or semi-standard. As a result, we can obtain a reasonable -spline structure, which helps generate a concise fitting result.
The rest of the paper is organized as follows. First, we present an overview of our adaptive fitting algorithm in Section 2. Then, several technical components are explained in subsequent sections. In particular, Section 3 describes parameterization and re-parameterization; Section 4 discusses initial -spline structure placement and re-placement; and Section 5 gives adaptive least-squares -spline fitting with curvature guidance. Next, Section 6 presents experimental results to demonstrate the effectiveness of our algorithm. Finally, the paper is concluded in Section 7.
Section snippets
Overview of the proposed algorithm
A -spline surface is defined by [13], [14] where and are the control points, weights, and -spline blending functions, respectively. As a major difference from the control points of -spline surfaces, in can be a -junction, which means the terminal point of a partial row or column. The existence of -junction points makes the -spline structure more flexible and allows local refinement.
Our surface fitting problem can roughly
Parameterization and re-parameterization
This section describes two processes related to parameter calculation: parameterization and re-parameterization.
Initial -spline structure placement and re-placement
This section describes two processes related to the initial -spline structure: placement and re-placement.
Curvature-guided least-squares -spline fitting
Our -spline fitting process itself is an iterative least-squares procedure consisting of three sub-components, shown in the flowchart in Fig. 9. Given a -spline knot structure, the optimal -spline surface is computed using least-squares optimization. Then the quality of the obtained surface is checked, taking surface features into account. If the surface passes the quality checking, the obtained -spline surface is considered a qualified candidate. Otherwise, the current -spline structure
Experiments
This section presents our experimental results to demonstrate the proposed algorithm and various components. In our implementation, we adopt the preconditioned complex bi-conjugate gradient (PCBCG) solver [25] to solve linear systems due to its good performance and stability. Example 1 This example demonstrates how the curvature-guided -spline fitting, one of the major components of our algorithm, is carried out, given a parameterization and an initial -spline structure placement. As depicted by Fig. 9
Conclusion
We have described several plug-and-play components and strategies to enhance the spline surface fitting. The use of these components and strategies promises an adaptive -spline fitting algorithm that well captures the surface features and generates a compact -spline representation. The experiments have successfully demonstrated the effectiveness of these components and strategies. Adapting them to more general surface fitting problems such as fitting an arbitrarily topological mesh is a
Acknowledgment
This work is supported by ARC 9/09 Grant (MOE2008-T2-1-075) from Singapore.
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This paper has been recommended for acceptance by Tae-wan Kim.