Adaptive T-spline Surface Approximation of Triangular Meshes Yimin Wang

Jianmin Zheng

School of Computer Engineering Nanyang Technological University Block N4, Nanyang Avenue, Singapore 639798 Email: [email protected]

School of Computer Engineering Nanyang Technological University Block N4, Nanyang Avenue, Singapore 639798 Email: [email protected]

Abstract—Digital 3D models represented by triangular meshes are now broadly available and they are becoming a new multimedia data type after sound, images and video. For many applications there is a need to convert such mesh models into spline surfaces. This paper describes an adaptive algorithm that automatically converts a triangular mesh into a T-spline surface based on optimization techniques. The key issues of the algorithm include the parameterization of input data points, the determination of topology of the T-spline surface, and the calculation of the least squares T-spline surface control points. Some examples are provided to demonstrate the efficiency of the algorithm.

I. I NTRODUCTION Digital geometry models are usually referred to the data that describe the freeform shape of objects and commonly occur in the style of triangular meshes. With the fast development in the data acquisition equipments such as 3D laser scanners [1], it is able to obtain high quality digital geometry models in acceptable time. Being recognized as a new type of signal, digital geometry models have now been widely involved in a variety of areas ranging from manufacturing to entertainment industry. There has been a lot of research on various issues of processing such signal, which typically include parameterization, segmentation, compression, simplification, subdivision and surface approximation. Besides triangular meshes, splines are another important representation of freeform surfaces, which serves as the major data form in CAD/CAM industry. Spline surfaces have parametric equations and thus facilitate shape analysis. Moreover, the spline surface representation is more compact and can have higher order continuity. Therefore, in many applications it is desirable to reconstruct a spline surface from the triangular mesh. There have been a lot of work done for fitting scattered data or a triangular mesh with a tensor-product B-spline surface. For the shape with uneven distribution of details, the resulting B-spline surface usually contains many control points in the control mesh, which are used just to maintain the rectangular topology in the tensor-product B-spline definition. To reduce the number of such control points, adaptive fitting is preferred. Greiner and Hormann [2] proposed to approximate triangular meshes using a collection of B-spline patches on different layers, with the base layer of patches representing the overall c 2007 IEEE 1–4244–0983–7/07/$25.00


shape and the overlay patches representing the local details. However, it is noticed that in the overlay patches, 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 keep the continuity. In this paper, we present a solution to adaptively approximating a given triangular mesh using a T-spline surface. We restrict ourself to the input models whose topology is homeomorphic to a plane disk. Models with complex topology can be handled with the help of a preprocessing step of segmentation. T-splines [3], [4] are a new surface modeling technique that generalizes tensor product non-uniform rational B-spline (NURBS) surfaces and allows local refinement of the control mesh. Our solution mainly consists of two components: a proper parameterization for the input triangular mesh and the computation of a T-spline surface which approximates the triangular mesh. They will be explained in details in the paper. Section II gives an brief overview of T-splines to make the paper self-contained. Section III describes the parameterization method that is the first component of our fitting algorithm. Then, in Section IV, we describe the steps of our adaptive T-spline surface approximation approach. Some examples are provided to demonstrate the algorithm in Section V and finally Section VI concludes the paper. II. OVERVIEW OF T- SPLINES Similar to NURBS, a T-spline surface has a control grid, which is called a T-mesh. In a T-mesh, a row or column of control points is permitted to terminate. The end point in a partial row or column is a T-junction. The T-junctions makes it possible to add a single control point to a T-spline control grid without propagating an entire row or column of control points and without altering the surface. If a T-mesh forms a rectangular grid, the T-spline surface degenerates to a NURBS surface. Each edge in a T-mesh is assigned with a knot interval, which should satisfy the following rules: Rule 1: The sum of the knot intervals on opposing edges of any face must be equal. Rule 2: If two T-junction on opposing edges of a face can be connected without violating the previous rule, that edge must be included in the T-mesh.

ICICS 2007

To give an analytic representation of a T-spline, a knot coordinate system is inferred by arbitrarily designating the preimage of a control point to be the knot origin. Then, every vertical or horizontal edge can be assigned with a u knot value or v knot value, respectively. The equation for a T-spline surface is thus defined as:
n wi Pi Bi (u, v) (1) S(u, v) =
i=1 n i=1 wi Bi (u, v) where the Pi are control points and the wi are control point weights. Each Bi (u, v) is a T-spline basis function for the control point Pi and is given by Bi (u, v) = N [ui ](u)N [vi ](v) where N [ui ](u), N [vi ](v) are the cubic B-spline basis functions associated with the knot vectors ui = [ui0 , ui1 , ui2 , ui3 , ui4 ] and vi = [vi0 , vi1 , vi2 , vi3 , vi4 ] respectively. The knot vectors ui and vi are extracted from the T-mesh neighborhood of Pi according to the following rule: (ui2 , vi2 ) is the knot coordinates of Pi . By casting a ray R(t) = (ui2 + t, vi2 ), t > 0, ui3 , ui4 are defined as the knot values of the first two vertical edges that intersect with R(t). Other knots in ui and vi are found in a similar manner. One important feature of T-splines is local refinement or local knot insertion, which means to insert one or more control points into a T-mesh without changing the shape of the T-spline surface. Local refinement makes T-splines suitable for adaptive fitting. It should be noted that in the T-spline local refinement algorithm, one or more extra control points may have to be added into the T-mesh when a control point is inserted into a designated position. This is to maintain the validity of the T-mesh and to ensure that the T-spline basis functions and the control points are properly associated. A T-spline surface is generally a rational surface except it is a standard T-spline or semi-standard T-spline. A standard T-spline is one for which, if all weights wi = 1, then
k i Bi (s, t) = 1. A semi-standard T-spline is one for i=1 w
k which i=1 wi Bi (s, t) = 1 and not all wi = 1. An important property of T-splines is that if we perform a local refinement on a semi-standard or standard T-spline, the result will always be a standard or semi-standard T-spline. Both standard and semi-standard T-splines define piecewise polynomial surfaces.

length preserving, or stretch minimizing, etc. It also requires that no edge in P intersects with each other and no triangle in P is degenerated (i.e. all the three vertices of a triangle lies on the same line); otherwise, undesirable effects may occur. In this section, we describe a parameterization method which is expected to provide a good parameterization for the triangular mesh D. The details are as follows. The parameter domain is first defined to be the unit square in R2 ([0, 1] × [0, 1]) which is also the domain of the Tspline surface that we are going to construct. This setting will avoid any extra trimming operation. Suppose the vertices of D are d1 , d2 , · · · , dl , dl+1 , · · · , dm , where d1+1 , · · · , dm are the vertices on the boundary of the mesh. The next step is to map the boundary vertices onto u1+1 , · · · , um on the boundary of the parameter domain using the traditional chord length parameterization. After that, the remaining task is to decide the parameters for the inner vertices. We consider the parameter ui for the inner vertex di to be a convex combination of the parameters for its neighboring vertices, which can be formulated as: m  λij uj = ui , i = 1, 2, · · · , l j=1

where λij = 0, if there is no edge between di and dj ; λij > 0,
m if di dj is an edge; j=1 λij = 1 for each i ∈ {1, 2, · · · , l}. The solution to the above linear system determines the values for u1 , u2 , · · · , ul . To choose the specific values for λij , we adopt the approach of mean value coordinates (MVC) [6] proposed by Floater, which leads to parameterization that smoothly depends on the vertices of the triangular mesh. In order to calculate all the λij for a certain i, both di and its one-ring neighborhood should be flattened into a plane, while preserving the length of edges between di and its neighboring vertices and the ratio among all the center angles αik , as illustrated in Figure 1. After that,

III. PARAMETERIZATION To perform a parametric surface approximation to a triangular mesh, a process called parameterization is usually needed, which associates each vertex in the triangular mesh with a pair of parameter values. In particular, let D denote a triangular mesh in R3 and di = (xi , yi , zi ), i = 1, 2, · · · , m denote its m vertices. A parameterization P ⊂ R2 is an embedding of the mesh D in R2 space which one-to-one maps each vertex di in D onto a pair of parameters ui (ui , vi ), i = 1, 2, · · · , m in P . The edges in D are correspondingly mapped as well, making P be a triangular mesh in R2 . Many parameterization methods have been developed [5]. Different applications may require different parameterization methods. In general, a good parameterization should satisfy some properties such as area preserving, angle preserving, edge

di

Fig. 1.

Dik Di(k 1)

dj

Mean value coordinates

the value of λij is decided by the length of di dj and the two adjoining center angles αik , α(ik+1) of di dj (refer to [6] for the formulae), as shown in Figure 1. IV. A DAPTIVE T- SPLINE SURFACE APPROXIMATION Once the parameterization is done, we are in the position to describe the algorithm for computing a T-spline surface that approximates the triangular mesh. That is, given a triangular

mesh D with vertices d1 , d2 , · · · , dm and its parameterization P with parameter uj (uj , vj ) for dj , j = 1, 2, · · · , m, we want to find a T-spline surface of (1) that approximates D within a pre-specified error tolerance ε. There has been several previous work that tries to solve the similar problem. In [7], the input data is limited to scalar value surfaces. In [8], a triangular mesh is converted into a T-NURCC surface, which however is generally incompatible with the CAD/CAM industry standard. Note that the rational representation of the surface often complicates the computations (especially those involving differentiation and integration). Therefore we try to find the best approximation within a subclass of all T-spline surfaces, which consists of standard or semi-standard T-splines. This can be accomplished by carefully setting the initial T-spline surface and applying the T-spline local refinement algorithm. Therefore now we can assume that the denominator of our T-spline surface formulation equals to one and thus our objective Tspline surface representation can be simplified as S(u, v) =

n 

wi Pi Bi (u, v).

i=1

Next in subsection IV-A, we show how to compute an geometrically optimal T-spline surface once the topology of the T-mesh is determined. In subsection IV-B, we show how an adaptive process can be carried out to refine the topology of the T-mesh. By iteratively executing these two steps (see Figure 2), the algorithm finally generates a T-spline surface that approximates the triangular mesh in a satisfactory manner.

This is the sum of squares of the parametric distances between the vertices in D and the T-spline surface. In many applications, the shape pleasingness of the surface is also a major consideration. Therefore, we modify the original objective function into the following form: F (P1 , P2 , · · · , Pn ) =

optimal T-spline surface computation

quality check / adaptive adjustment

Exit

 S(uj , vj ) − dj 2 + σJf air (S),

j=1

where σ is the fairness factor and Jf air (S) is a uniformly discretized simple thin plate energy that measures the fairness of the surface and can be expressed as Jf air (S) =

a=1 

b=1 

2 2 2 (Suu (a, b) + 2Suv (a, b) + Svv (a, b))

a=0 b=0 a+=δ b+=δ

Here Suu , Suv , Svv are the second order partial derivatives of S(u,v) respective to u, (u, v) and v, respectively. δ is a small value that controls the step of discretization. When the step is small enough, Jf air (S) would give a good approximation to its continuous counterpart. To find the P1 , P2 , · · · , Pn that minimize the objective function, we let the partial derivatives of the objective function equal to zero ∂F = 0, ∂Pg which leads to 

m


g = 0, 1, · · · m

Bi (uj , vj )Bg (uj , vj )+

 j=1  wi  Biuu (a, b)Bguu (a, b)+ 1 1

i=1  σ 2Biuv (a, b)Bguv (a, b)+ a=0 b=0 B ivv (a, b)Bgvv (a, b) m
= dj Bg (uj , vj ) n


Begin

m 

    Pi  

j=1

Combining these equations forms a linear system Fig. 2.

A·P=C

The iterative process of the surface approximation algorithm

which has a solution A. Optimal T-spline surface computation The construction of a T-spline surface can be divided into three parts: the T-mesh topology T (i.e. the layout of the control points in the parameter domain), the control point weights W = (w1 , w2 , · · · , wn )T and the control points P = (P1 , P2 , · · · , Pn )T . When the T-mesh topology T and the weight vector W are determined, a class of T-spline surfaces SW,T is defined. Every individual surface in this class differs to each other only in the position of the control points. Now the question is how to find a best surface out of SW,T as the approximation for the triangular mesh D. It is natural to introduce an objective function that measures the approximation error, as shown below: F (P1 , P2 , · · · , Pn ) =

m  j=1

 S(uj , vj ) − dj 2 ,

P = A−1 · C This gives the control points of the optimal T-spline surface. B. Adaptive refinement of the T-mesh So far we have described how to find a best approximate T-spline surface from a certain class by optimizing the geometry of the control points. However, if the T-mesh topology of the T-spline surface class is not appropriate, even the best surface in the class could not well represent the shape of the input triangular mesh. Therefore, in this subsection, we look into the problem of how to improve the topology of the T-mesh when the preset error tolerance is not met. Another question that also needs to be answered is how to choose an suitable initial T-mesh topology at the beginning of the algorithm.

1) Initialization: It would be better to first have a simple topology for the T-mesh and to adaptively improve the topology afterwards. Based on such consideration, we start the algorithm with a relatively simple setting: The control grid of the T-spline surface is defined to be a s × t grid with integers s and t taking values between 4 and 10. The specific values for s, t depend on the input triangular mesh. The first and the last two knot vectors along the u direction is set to be 0 − , 0, 1, 1 + , respectively, where  is a small positive value (for example, 0.001). The rest knot vectors along the u direction are evenly allocated in the [0, 1] domain. Same rules are applied to the configuration of the knot vectors along the v direction. The control points weight vector W is chosen to be (1, 1, · · · , 1)T . Under such configuration, we would have a standard T-spline surface no matter how the control points are geometrically positioned. 2) Verification and adjustment: Now we explain how to check the approximation quality and how to iteratively improve the T-mesh. After a T-spline surface is found, we need to check if the approximation meets the requirement. We compute the parametric distance of each vertex of the triangular mesh to the T-spline surface. If all the distances are below the given tolerance, the T-spline surface is accepted and the algorithm outputs the T-spline. Otherwise, the regions that contain one or more violating vertices whose distances are greater than the tolerance need to be refined. Also, we compute the maximum parametric distance among all these distances and compare it to the one obtained in the last iteration. If such a value is not decreased by a certain extent, we adjust the value of fairness factor σ by tuning down a little bit in order to increase the weight of the sum of distances in the objective function. We refine the faces of the T-mesh, which correspond to the regions containing violating vertices, by adding a new edge into that region and split the region into two halves of equal size. To add an edge, we simply insert two end points of the edge into the T-mesh, using the T-spline local refinement method. The edge could be either horizontal or vertical, depending on whether the height or the width of the region is larger. After the T-mesh is updated, we obtain a new class of Tspline surfaces. Since the new control points are added by the T-spline local refinement algorithm, the resulting class of Tspline surfaces remains to be standard or semi-standard. Therefore, we could continue with the geometrical optimization step described in IV-A. V. E XAMPLES This section provides some examples to demonstrate the effect of our surface approximation solution. In all these examples, we assume that the approximation error tolerance ε is set to be 0.6% of the scale of the triangular mesh and the initial fairness factor σ is set to be 0.01. Several steps in the algorithm involve solving linear systems, such as in the process of parameterization and in each

iteration of optimal surface calculation. These linear systems usually have a sparse matrix. Therefore, a fast and stable linear system solver that optimizes the solving of the sparse linear systems has to be chosen. In this paper, we adopt the Preconditioned BiConjugate Gradient (PBCG) solver [9]. The results for approximating several popular triangular meshes are shown in Figure 3. Each row in the figure contains illustrations for one example, with the input triangular mesh, the mean value coordinates parameterization, the T-spline pre-image and the final result of the T-spline surface, from left to right. The T-spline pre-image is the T-mesh in the parameter domain that helps to clarify how the control points are distributed. In the first example, we further map the preimage onto the T-spline surface for better visualization. Below each group of illustrations, the basic information for the input triangular mesh, such as the number of the vertices and the number of the faces are captioned. Information about the result T-spline surface is also given which includes the number of the control points and the approximation error against the original triangular mesh. At last, it is also stated how many iterations it has undergone before the algorithm reaches the final T-spline surface. In our experiment, all the examples are well approximated under the given tolerance and it can be observed that the Tspline surfaces possess smooth appearance. Also, from the T-spline pre-images, it can be seen that some regions have higher density of control points than others and such regions are usually related to the details on the surfaces. Benefiting from the local refinement property of T-splines, the partial rows or columns in one region do not necessarily have to propagate to other regions in the control grid. Thus, the control grid can be kept in a relatively compact form. VI. C ONCLUSION In this paper, an integrated solution for adaptively approximating a triangular mesh using a T-spline surface is presented. First, a method is provided to generate parameterization for the triangular mesh using the mean value coordinates. Second, a top-down adaptive T-spline surface approximation algorithm is described with elaborate steps, all of which are carefully designed to achieve the best performance. In particular, by taking the advantage of the T-spline local refinement property, the algorithm is able to output a surface that gives a good approximation to the triangular mesh. Several examples have shown the effectiveness of the approach. ACKNOWLEDGMENTS This work is supported by A*STAR SERC TSRP GRANT (NO. 062 130 0059). R EFERENCES [1] F. Blais, “A review of 20 years of range sensor development,” Journal of Electronic Imaging, vol. 13, no. 1, pp. 231–243, January 2004. [2] G. Greiner and K. Hormann, “Interpolating and approximating scattered 3D-data with hierarchical tensor product B-splines,” in Surface Fitting and Multiresolution Methods, ser. Innovations in Applied Mathematics, 1997, pp. 163–172.

Triangular mesh

MVC Parameterization

T-spline pre-image

T-spline surface

cat model- vertices: 3533, faces: 6975, control points: 297, max error: 0.58%, average error: 0.13%, iterations: 10.

stegosaurus model- vertices: 12409, faces: 24672, control points: 984, max error: 0.47%, average error: 0.08%, iterations: 7.

face model- vertices: 24155, faces: 47976, control points: 1710, max error: 0.39%, average error: 0.02%, iterations: 12.

stamp model- vertices: 35852, faces: 71442, control points: 4389, max error: 0.26%, average error: 0.07%, iterations: 18. Fig. 3.

Examples for adaptive T-spline surface approximation

[3] T. Sederberg, J. Zheng, A. Bakenov, and A. Nasri, “T-splines and TNURCCs,” ACM Transactions on Graphics (SIGGRAPH 2003), vol. 22, no. 3, pp. 477–484, 2003. [4] T. Sederberg, D.Cardon, G.Finnigan, N.North, J. Zheng, and T. Lyche, “T-spline simplification and local refinement,” ACM Transactions on Graphics (SIGGRAPH 2004), vol. 23, no. 3, pp. 276–283, 2004. [5] M. S. Floater and K. Hormann, Surface Parameterization: a Tutorial and Survey. Springer-Verlag, Heidelberg, 2005, ch. Advances in Multiresolution for Geometric Modelling, pp. 157–186. [6] M. S. Floater, “Mean value coordinates,” Computer Aided Geometric Design, vol. 20, pp. 19–27(9), March 2003. [7] J. Zheng, Y. Wang and H. Seah, “Adaptive T-spline surface fitting to z-map models,” GRAPHITE, pp. 405–411, 2005.

[8] W. Li, N. Ray and B. Levy, “Automatic and Interactive Mesh to T-Spline Conversion,” EG/ACM Symposium on Geometry Processing, 2006. [9] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge University Press, January 1993.