Edge based parameterization for tubular meshes Yimin Wang∗ and Jianmin Zheng† School of Computer Engineering, Nanyang Technological University

Figure 1: An example of the edge based parameterization approach. From left to right: the surface view of the bimba model (of 16k triangle faces) with two holes; the mesh view of the bimba model; the parameterization result, projected onto a cylinder surface; the front view and the back view of texture mapping with the checkboard pattern.

Abstract This paper considers the problem of parameterizing triangular meshes that have tubular shapes. Unlike an open mesh that is of plane topological type, a tubular mesh gives rise to some special issues in parameterization due to its mesh structure. In this paper, we present an edge based parameterization method, in which the edges rather than the vertices of the mesh are treated as the target for parameterization. It improves cutting-based algorithms which cut the mesh to make it a disk topologically. The problem of cutting paths is their zigzag shape that leads to suboptimal parameterizations and finding good cutting paths is usually very difficult. Our proposed method does not need cutting of the mesh. It first parameterizes the edges on the two boundaries of the tubular mesh, then parameterizes the internal edges based on the mean value coordinates, and finally computes the parameters of the mesh vertices. Some examples are provided to demonstrate the effectiveness of our method. CR Categories: I.3.7 [Three-Dimensional Graphics And Realism]: Virtual Reality; I.3.7 [Three-Dimensional Graphics And Realism]: Color, Shading, Shadowing, And Texture

1

Introduction

texture mapping, morphing, remeshing, and surface fitting [Floater 1997; Gotsman et al. 2003]. The performance of these applications tightly relies on a good parametrization, while different applications might have their own requirements for parameterization. There have been many methods for parameterization, especially when the research of digital geometry processing becomes a popular field in recent years. A survey for parametrization methods can be found in [Floater and Hormann 2005]. Since parameterization is a mapping between spaces of different dimensions, generally the distortion is inevitable by this process. Therefore most parameterization methods use some optimization processes that minimize the distortion in certain criteria which are based on various geometry properties. These geometry properties not only include vertex information, but the edge, angle or face information as well. For example, some approaches try to preserve the area of the triangle faces after they are mapped to the parameter domain [Clarenz et al. 2004], or to preserve the angles [Sheffer and de Sturler 2000] or the edge length [Maillot et al. 1993], or to minimize certain physically based energy [Greiner and Hormann 1997]. In addition, no matter which criterion is taken, one underlying requirement for almost all the parameterization methods is that in the parameter domain, no edge intersects with each other and no triangle face is degenerated (i.e., all the three vertices of a face should not lie on a same line).

This paper considers parameterization for triangular meshes. As a popular representation for free-form shapes, triangular meshes are widely used for modeling in computer graphics and other related applications, such as virtual reality. A triangular mesh, comprised of a collection of triangle faces, is able to represent complicated shapes. With the support of graphics processing units, it can be efficiently rendered with remarkably high speed. Parameterization is a geometry process that maps each vertex of the triangular mesh in R3 to a point in some parameter domain. Parameterization is an important step for various applications of triangular meshes, such as ∗ e-mail:

[email protected]

† e-mail:[email protected]

Figure 2: Some examples of tubular meshes This paper focuses on the problem of parameterizing a special type of triangular meshes: the tubular meshes, which is characterized by having two closed boundary loops and one lateral surface between

them. The lateral surface is open along one direction (the axis direction) and closed along the other direction (the sectional direction). In mechanical engineering and medical engineering, tubular surfaces are quite common. For example, pipes and blood vessels are tubular surfaces. Therefore tubular meshes are useful to represent these shapes. Some examples of tubular meshes are shown in Figure 2. To parameterize a tubular mesh, one possible solution could be first splitting the tubular mesh along a path consisting of a set of vertices in the axis direction, making it homeomorphic to a topological disk. The cutting path then becomes the boundary of the new mesh. After that, the parameters for the new mesh could be obtained by applying one of the existing parameterization techniques for open meshes. However, such an approach would incur some inconvenience or troubles. The parameters for the vertices on the newly formed boundary should be carefully chosen in order to avoid discontinuity of the parameterization. The cutting path should also be properly selected, since some vertices might be more suitable to be boundary vertices than the others. Cursory pick of the cutting path would likely increase the distortion of the result parameterization. Furthermore, no matter which cutting path is finally selected, it is usually mapped onto a straight row or column in the parameter domain. Given the fact that the cutting path on the mesh is usually in a zigzag shape, it is likely to introduce undesirable effects when the cutting path is mapped to a straight boundary of the parameter domain. There are a few works published for tubular mesh parameterization. Z¨ockler et al. provide a parameterization method in [Z¨ockler et al. 2000], which involves first cutting the mesh and then stitching it together after a parameterization is calculated. Although the parameters are optimized again after the stitching step, the performance of the method still largely depends on finding a good cutting path. Huysmans et al. report another approch [Huysmans et al. 2005], in which no cutting of the mesh is required; however, in order to compute the parameterization, a group of progressive meshes have to be constructed, from which the parameters are gradually optimized. In this paper, we present a new approach for directly parameterizing a tubular mesh to a cylinder domain. The cylinder surface is chosen because it has the same topology as the mesh. In order to keep the parameters from being twisted too much, we propose to let the edges rather than the vertices of a mesh be the target for parameterization. Therefore we call our method edge based parameterization. The method starts by first fixing the parameters for the two boundaries of the mesh. After the edges on the two boundaries of the tubular mesh are parameterized, we build equations based on the geometry relations that the internal edges should satisfy. To generate parameterization that depends smoothly on the edges and vertices of the tubular mesh, mean value coordinates [Floater 2003] are adopted as weights for linearly combining several adjacent edges together. Finally, after the edge parameterization for the tubular mesh is calculated, the conventional vertex parametrization is derived. Figure 1 shows an example that demonstrates our edge based parameteriztion method. The rest of this paper is organized as follows. In Section 2, the mean value coordinates is breifly reviewed. In Section 3, we describe our edge based parameterization that maps a tubular mesh to a cylinder domain. Some examples that demonstrate our method are given in Section 4 and the paper is concluded in Section 5.

2

Mean value coordinates

Mean value coordinates (MVC), proposed by Floater [Floater 2003], is a set of weights that can be used to represent a vertex in a 2D triangular mesh by convexly combining its neighboring vertices.

Mean value coordinates is appreciated for depending not only continuously but also smoothly on the vertices of the mesh. It has applications in many geometry processes including parametrization. Suppose di is a vertex in the triangular mesh with one of its neighboring vertices being dj . To compute the mean value coordinates λij of vertex di with respect to vertex dj , a flat star-shaped region is constructed by flattening the 1-ring neighborhood of vertex di into a plane, as shown in Figure 3. In this flattened star-shaped region, the edges that connect di and its neighbors remain the original length and all the angles αij between two adjacent such edges are uniformly scaled so that the sum of these scaled angles equals 2π. Then we compute an initial weight wij as wij =

tan(

αij 2

αi(j−1) 2

) + tan( kdj − di k

)

(1)

After that, Pthe MVC λij can be calculated by normalizing wij using the sum j wij : wij λij = P (2) j wij P It is apparent that λij > 0 and j λij = 1. Also, it has been P proved in [Floater 2003] that j λij dj = di .

ij

di

dj

i ( j !1)

Figure 3: A star-shaped region around di

3 Edge based parameterization Denote a tubular triangular mesh by T(D, E) where D = {d1 , d2 , · · · , dm } is a vertex list and E = {e1 , e2 , · · · , en } is a directed edge list. Given a tubular triangular mesh T with its vertex list D ⊂ R3 and a parameter domain P corresponding to the lateral surface of a unit cylinder, our goal is to find a parameterization ψv which establishes a mapping between vertex di (xi , yi , zi ) and a parameter pair (ui , vi ) that forms a point ti = (ui , vi ) in P, i = 1, 2, · · · , m, and thus a mapping between vertex di (xi , yi , zi ) and a point on the cylinder surface. The edges of T are correspondingly mapped as well. The mapping of the edges onto the cylinder surface forms a curved triangular mesh on the surface. Here, we define the unit cylinder to be a cylinder whose the height and the perimeter of the cross sectional circle are both 1. The u direction is closed and the v direction is open, as illustrated on the left of Figure 4. To better reflect the geometric characteristic of the cylinder, we let domain P consist of [0+k, 1+k)×[0, 1], k = 0, ±1, ±2, · · · with each [0 + k, 1 + k) × [0, 1] mapping to the cylinder surface, and thus each vertex di is actually associated with a series of parameters (ui + k, vi ), k = 0, ±1, ±2, · · ·. Figure 4 (right) shows a part of domain P obtained by outspreading the cylinder to a plane. For the parameter pair (ui , vi ), it can be understood that ui corresponds the arc length along the rotational direction of the cylinder and vi corresponds to the height value along the axial direction of the cylinder. Having two individual closed boundaries is one of the most salient characteristics that differentiate a cylinder domain from an open

vi

ti

ψe is constructed, the conventional vertex parameterization ψv for the vertices in D of the tubular mesh can be inferred by specifying an arbitrary boundary vertex as the origin of the (u, v) coordinate system.

ti

vi

V V

U

U

ui

boundary edge mapping

ui

Figure 4: The parameter domain for tubular meshes

disk domain. It is essential that our parameterization method should map the vertices on the bottom boundary of the tubular mesh to the boundary v = 0 in P and the vertices on the top boundary to the boundary v = 1 in P. The rest vertices of the mesh should be mapped to the remainder part of P, namely [0 + k, 1 + k) × (0, 1), k = 0, ±1, ±2, · · ·. However, finding such parameterization incurs some difficulties that the conventional open mesh parameterization does not have. In order to generate a parameterization with low distortion, the vertices on either boundary should be assigned with reasonable parameters. There has to be certain correspondence between the parameters for the two boundaries. Both boundaries should have a consistent direction for the parameters, either clockwise or counterclockwise. Moreover, two vertices on the opposite boundaries should be assigned with similar u parameters if they are almost along the same axis on the tubular mesh. Otherwise, the result triangulation on the parameter domain would be twisted, as in the example given in Figure 5. However, finding correct correspondence between the two boundaries is not straightforward and the situation could become quite complicated.

tj

dj

di

ti

Figure 5: A tubular triangular mesh (left) and a twisted parameterization (right) Another difficulty for tubular mesh parameterization is that each vertex di is actually associated with a series of parameters (ui + k, vi ), k = 0, ±1, ±2, · · ·. When we use the mean value coordinates to construct equations to find parameters for the vertices, we have to choose a proper value of k for them and it is not easy to determine such a value. To overcome these difficulties, we propose our edge based parameterization method in this section. The main idea of our method is that the edges, rather than the vertices of a mesh are treated as the target for parameterization. Since each edge is a vector that represents the location offset between two vertices, the offset values are unique no matter where the knot origin is. Therefore the messy problems such as finding correspondence of vertices on two boundaries and deciding a suitable value k for the parameters could be avoided. An edge base parameterization ψe maps each directed edge ei in E to a curve segment on the cylinder and thus a 2D edge vector ui (ui , vi ) in the domain P, i = 1, 2, · · · , n. Note that here ui and vi are the u and v components of vector ui . Once

internal edge mapping

vertex parameter generation

Figure 6: A flowchart of our parameterization approach A flowchart for our parameterization approach is given in Figure 6. The whole parameterization process is divided into three steps. The first step is to assign parameters to the boundary edges. The second step is to calculate parameters of the internal edges. The third step is to convert edge parameterization to vertex parameterization. The details of these steps are described in the following subsections.

3.1

Parameterizing boundary edges

Our method first assigns parameters to the edges on two boundaries of the tubular mesh, i.e. the boundary edges. The edges that are not on any boundary or only have one vertex on the boundary are defined as internal edges. For clarification, in the directed edge list E, we denote e1 , e2 , · · · , ed to be internal edges and ed+1 , ed+2 , · · · , en to be boundary edges without losing any generality. For each directed edge in the mesh, we have two choices on its specific direction. An edge ei that connects di1 and di2 can originate from either di1 or di2 . For the internal edges, we let their direction be arbitrarily chosen. For the boundary edges, we make some constraints such that edges belong to the same boundary have same direction, and edges belong to different boundaries have opposite directions. These constraints would simplify the description of our method. If certain mesh data structure such as half-edge [Weiler 1985] is adopted for representation, we can simply achieve the above arrangements by letting the direction of the boundary edges conform with that of the corresponding boundary half edge. For the boundary edges, the v components would always be zero. We determine the u components of the boundary edges using the chordal length approach. Because the representations of vectors are invariant under translation, edge parameterization can be carried out without specifying a knot origin for the coordinates system. Therefore, depending on which boundary it belongs to, the boundary edge ei is parameterized as either (li /L1 , 0) or (−li /L2 , 0), where li is the length of the edge. L1 and L2 are the sum of lengths of the edges on the two respective boundaries and are used to normalize the parameters for boundary edges. This makes the lengths of parameter range for both boundaries the same and equal 1. Note that at the end of this stage, the parameters for either boundary have been fixed but the relative position of the two boundaries is still undecided yet.

3.2

Computing internal edge parameters

We now describe how to compute the parameters for the internal edges e1 , e2 , · · · , ed . We set up a system of linear equations that the edge parameters should satisfy. The edge parameters are then obtained by solving the equations. First, each triangle face in the tubular mesh corresponds to a triangle in the parameter domain. Suppose ui , uj , uk are the parameters for three directed edges that form a triangle in the tubular mesh. Then ui , uj , uk form a triangle in the parameter domain, as shown on the left part of Figure 7. Thus ui , uj , uk should satisfy the following

equation, which we call the face related equation: ui + [uj ] + [uk ] = 0

(3)

Here, the operator [ ] is introduced, which changes the direction of a vector to its opposite when necessary, to make sure that after the adjustment, the edges corresponding to the three vectors have different starting points.

u i,1

uk

ui

u i,2

ui,ki

di

uj

Figure 7: Build equations from face relation and vertex relation Second, for each internal vertex, we look at its 1-ring neighborhood. To generate a good parameterization, we use mean values coordinates [Floater 2003]. That is, we let the parameter of the internal vertex be represented as a linear combination of the parameters of its surrounding vertices in the 1-ring neighborhood. After a simple arrangement, this turns out a constraint on the edge, which we call the vertex related equation: ki X

λij [ui,j ] = 0

(4)

j=1

where ki is the valence of the interval vertex di and λij is the mean value coordinates of di with respect to its j-th neighboring vertex dj , computed from the tubular mesh using the approach described in the preceding section. Again, here ui,j is first passed into operator [ ] in order to guarantee that the edge corresponding to [ui,j ] emits from di . So far we have established n1 face related equations and n2 vertex related equations, where n1 is the number of faces and n2 is the number of internal vertices. These equations contain n3 unknowns, where n3 is the number of internal edges. A careful analysis using Euler formula shows that n1 + n2 = n3 , which means the number of the equations equals the number of unknowns. However, it can be found that the face related equations are linearly dependent. This is because the sums of the edge vectors of two boundaries have the same length but opposite direction. To fix this problem, we select one internal edge, assign appropriate parameters to it, and remove one equation derived from a face that contains the edge. To choose the edge, we compute the angle θ between all the boundary edges and the edges adjacent to them and pick the edge e∗ whose θ is closest to 21 π. The parameters for e∗ is set to be (cos θ, sin θ). In this way, we will have n3 − 1 equations with n3 − 1 unknowns. Solving the equations gives the parameters for all the edges.

3.3

Inferring vertex parameterization

Once we have obtained the edge parameterization ψe for a triangular mesh of tubular topological type, we can convert the edge parameterization ψe to an equivalent vertex parameterization ψv , where each vertex di is associated with a pair of parameter values. First we arbitrarily pick a vertex on either boundary of the mesh and assign to it parameters (0, 0). After that, parameters for the remaining vertices are obtained by traversing the triangular mesh

under the breadth-first search algorithm ([Cormen et al. 2001]). We set the previously picked vertex as the root vertex and put it into an empty queue structure. We then keep retrieving a vertex from the top of the queue and attempt to compute parameters for all of its unattended neighboring vertices. From a vertex di , the parameter for its neighboring vertex dj is computed by either adding or subtracting to the parameter of di the parameter of the edge that connects di and dj , depending on the direction of that edge. We require that the u direction parameter for any vertex be in the [0, 1) domain. If ui is outside [0, 1) after being computed from its neighbor, we try to find an integer k such that ui + k ∈ [0, 1). In that case, we use ui + k as the u direction parameter for di instead. We put all the newly parameterized vertices into the bottom of the queue and repeatedly deal with the top vertex in the queue. A vertex goes out of the queue when attempts have been made to parameterize all of its neighboring vertices. We continues this process until the queue finally becomes empty, which means that all the vertices of the triangular mesh have been given proper parameters. Finally, after the above procedure stops, we normalize the v coordinates of all the vertices by a translation or/and a scaling to make the range be [0,1]. This yields a vertex parameterization in [0, 1) × [0, 1] for the tubular mesh.

4 Examples We have tested several models of different types to demonstrate the performance of the edge based parameterization. The geometry information of the models is given in Table 1. The experimental results are displayed in Figure 8. The first column of Figure 8 shows the surface view of the models. The second column shows the mesh

Model mannequin screw driver knot Venus bumpy sphere

Table 1: Information of the models Vertex number Face number Edge number 680 1328 2008 514 999 1513 234 456 690 710 1389 2099 5610 11181 16791

view of the models. All the triangular meshes used here have the same topological type as a cylinder and there are two closed boundaries for each of these meshes. We apply the edge based parameterization method to these models, and the results of parameterization are shown in the third column of Figure. 8. We project the parameterization onto the cylinder 2πu sin 2πu through a function: (u, v) 7→ ( cos2π , 2π , v) for a clearer visualization. It can be seen that although we do not make any effort on aligning the parameters of the two boundaries, correct correspondence can be automatically obtained. Another way of visualizing the result is provided in the fourth column of Figure. 8. Here each parameterization is cut open along a path of vertices from one boundary to another and then displayed in a plane. The edges shown in red lines connect a right-end vertex to a left-end one. The edges shown in dash lines form the cutting path. Finally, in the fifth column of Figure. 8, we demonstrate the effect of mapping a checkboard texture to the tubular meshes, using the result parameterization. It can be seen that the distortion of the parameterizations is very low.

5

Conclusion

In this paper we have presented an edge based parameterization method that is especially suitable for parameterizing tubular triangular meshes. Instead of parameterizing vertices directly, we first compute the parameters of the edges. The mean value coordinates are used to parameterize the edges. After that, the parameterization for the vertices is derived. Since our method is based on intrinsic geometry of the mesh, there is no need to perform cutting for parameterization and a low distortion can be achieved. We demonstrated our approach by several examples of different types. The effectiveness of the method suggests that the idea of treating edges as the main target may also be applicable in other mesh related problems in geometry processing. From the parameterization, further applications such as texture mapping and surface fitting of the tubular meshes could be developed.

Acknowledgments This work is supported by the A*STAR SERC Grant (No 0621010034) of Singapore.

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Figure 8: Examples of edge based parameterization