Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 August 15-18, 2010, Montreal, Quebec, Canada Proceedings of the ASME 2010 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2010 ´ August 15-18, 2010, Montreal, Canada

DETC2010-28301

DETC2010-28301

FLATTENABLE MESH PROCESSING BY CONTROLLABLE LAPLACIAN EVOLUTION

Hongwei Lin1,2 Yunbo Zhang2 Charlie C. L. Wang2∗ Shuming Gao1 Key Lab. of CAD&CG, Zhejiang University, Hangzhou,310058,China 2 Department of Mechanical and Automation Engineering, CUHK, Hong Kong 1 State

ABSTRACT Models represented by polygonal meshes have been more and more widely used in CAD/CAM systems. In sheet manufacturing industries, the flattenability of a model is very important. Prior methods for processing the flattenability of a mesh surface usually employ a constrained optimization framework, which takes the positions of all its non-boundary vertices as variables in computation. For a mesh surface with hundred thousands of vertices, solving such an optimization is very time-consuming, and may exceed the capacity of main memory. In this paper, we develop a controllable evolution method to process the flattenability of a given mesh patch. It decouples the global optimization problem into a sequence of local controllable evolution steps, each of which has only one variable. Therefore, mesh surfaces with large number of vertices can be processed. Keywords: Flattenable mesh processing, Laplacian evolution, geometric modeling, sheet manufacturing

ences and make revisions to generate another prototype. The prototyping and modification steps will be performed repeatedly till getting an satisfactory result. This trial-and-error procedure is very inefficient; more seriously, when designing in 2D instead of 3D space, the final product made from the 2D pattern may not give the desired shape in 3D. To improve the efficiency of product design in these industries, the 3D CAD system should be exploited. Designers can create and check a product directly in 3D environment, thus avoid the trial-and-error procedure. 3D CAD system usually represents a model using triangular mesh, since it can represent models with more complex shape and topology compared with parametric patches. Specifically, in sheet manufacturing industries, since the final products are fabricated by warping and stitching 2D patterns together, the triangular mesh S designed in 3D should be as flattenable as possible. Otherwise, the final product fabricated from 2D patterns will not give the 3D shape as designed. Following [1, 2], the mesh surface satisfying the stretch-free flattening property is defined as a flattenable mesh surface. From the knowledge of differential geometry [3], only the developable surfaces (e.g., plane, generalized cylinder surface, conical surface or tangential surface) can be flattened without stretch. Thus, there are lots of approaches modeling [4–6] or approximating [7–9] a model with developable ruled surfaces. However, the shape of products in practice could be very complicated, which can hardly be modeled by the conventional developable surfaces. Actually, in 3D CAD systems, these models are represented by free-form triangular meshes, which are rarely developable. Therefore, a flattenable mesh processing method is

1

Introduction In sheet manufacturing industries, the products are fabricated from planner materials (e.g., metal in ship industry, fabric in apparel industry and toy industry, and leather in shoe industry and furniture industry). However, the traditional product design procedure in these industries is conducted in a trial-anderror manner. A designer first drafts 2D pieces on a paper and then makes a prototype by the 2D patterns to check whether resultant shape is satisfactory. If the result is not desirable, the designer needs to modify the patterns according to his experi-

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needed to generate more flattenable models. Wang and Tang [10] adopt the discrete definition of Gaussian curvature in [11] to process the given mesh surface through a constrained optimization to make it more flattenable. The method is improved to the flattenable Laplacian mesh processing method in [1], which is more stable. Recently, Wang [2] presents a least-norm solution based flattenable mesh processing method to speed up the computation. All of the methods take the positions of the non-boundary vertices of a mesh as the optimization variables. Therefore, when the number of vertices on a given mesh surface is large, the computation of the optimization could be very slow. Sometimes the procedure will run out the memory. In fact, the processing of mesh’s flattenability can be considered as a special mesh fairing process: the mesh is processed to be more and more flattenable. In general, there are two kinds of mesh fairing methods [12]: global optimization based and local evolution based. The global optimization approaches construct an optimization framework which minimizes certain energy functions [13, 14], such as thin plate energy, membrane energy [15], or total curvature energy [16, 17]. The flattenable mesh processing method developed in [1] is global optimization based. Another kind of fairing methods, local evolution, is developed from the diffusion equation [18, 19]. Different from the global optimization methods, which take the positions of all the non-boundary vertices of a mesh surface as variables, the local evolution methods process a mesh in a more efficient manner by adjusting the positions of vertices one by one [20]. Therefore, large meshes can be processed. Although it has been shown that the local evolution methods are equivalent to minimizing some energy functions in theory [19], it cannot be guaranteed that after each vertex adjustment, the energy functions will decrease. In this paper, we develop a controllable local evolution method to process a given mesh to make it more flattenable. The method is controllable so that after each vertex adjustment, the energy function measuring the flattenability is ensured to decrease, and then the small region incident to this vertex is more flattenable. The paper is organized as follows. In section 2, we formulate the non-linear optimization problem to process the flattenability of a given mesh. Section 3 develops the controllable evolution scheme. In section 4, we demonstrate the experimental results. Finally, section 5 concludes the paper.

In [11], the discrete Gaussian curvature at a mesh vertex v is proposed as K=

2π − ∑i θvi , 1 3A

(1)

where θvi is the vertex angle of triangle i at v, A is the summed area of the triangles adjacent to v. In the remaining part of this paper, θv = ∑i θvi is named as the summed inner angle of vertex v. Therefore, a triangular mesh S can be flattenable, if and only if 2π − θv = 2π − ∑ θvi = 0,

(2)

i

at each vertex. Specifically, it has been proved in [1] that, a triangular mesh patch S with only one boundary loop can be flattened onto 2D plane without stretch, if Eq. (2) is satisfied at every vertex not on the boundary – called inner vertex. To represent the flattenability at each vertex v of a given mesh surface S, we define dv = 2π − θv = 2π − ∑ θvi

(3)

i

as the flattenable degree at a vertex v. We further define an energy function

∑

v∈Vint

dv2

(4)

to measure the flattenability of whole surface S, where Vint is the collection of all inner vertices of S. Thus, processing the mesh is equivalent to the following optimization problem, minv∈Vint

∑

v∈Vint

dv2 .

(5)

When the number of vertices in Vint is very large, to solve such an optimization problem becomes very time-consuming, and may run out of memory. To overcome these difficulties, a controllable evolution method is developed in the following section to efficiently solve the optimization problem in Eq. (5).

2

Problem Formulation A flattenable mesh surface is defined as a triangular mesh surface which can be flattened into a two-dimensional region without any stretch [1, 2]. As stated above, in differential geometry, the parametric surface with such property is called a developable surface. A parametric surface is developable if and only if the Gaussian curvature at every point on it is zero [3].

3 Controllable Laplacian Evolution In this section, a controllable Laplacian evolution method is developed to move inner vertices on by one. We call this method 2

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controllable, because after each vertex adjustment, the energy function defined in Eq. (4) is guaranteed to decrease. The evolution methods are designed to fair a mesh surface S by reducing surface energies, such as membrane energy Emembrane (S) =

1 2

Z Ω

Su2 + Sv2 dudv,

(6)

which approximates the surface area [21]. The variational derivative of Emembrane is a Laplacian operator, L(S) = Suu + Svv .

(7)

Figure 1.

Therefore, the geometric meaning of the evolution process ∂S = λL(S) ∂t

degree dv at each inner vertex v, all the inner vertices can be classified into three types (as illustrated in Fig. 1): planner vertex with dv = 0, convex or concave vertex with dv > 0, and saddle vertex with dv < 0. For a planner vertex, as the flattenable degree on such a vertex is zero, there is no need to process it. Therefore, we only need to process the other two types of vertices.

(8)

is to decrease the surface area. Here, λ ∈ [0, 1] is a constant to control the time step of evolution. To perform the evolution process in Eq. (8) on the triangular mesh surface S, Taubin [20] developed the discrete Laplacian operator at each vertex vi as L(vi ) =

∑

ω j v j − vi ,

Three classes of vertices

(9)

j∈N1 (i)

where v j are the 1-ring neighbors of vi , ω j > 0, and ∑ j ω j = 1. In this paper, we choose the edge length based coefficients Moving the vertex v along the Laplacian operator will decrease the energy function in Eq. (5). (a.) Convex vertex; (b.) concave vertex;

Figure 2.

ωj =

kv j − vi k . ∑ j kv j − vi k

(c.) the apex angle of the triangle Tλ is larger than that of the triangle T .

For a convex vertex (dv > 0), it is important to note that the Laplacian operator points inwards (Fig. 2(a)); for a concave vertex (dv > 0), the Laplacian operator points outwards (Fig. 2(b)). If the vertex v moves along the Laplacian operator to the vertex vλ , we get two pyramids: T adjacent to v and Tλ adjacent to vλ , which share the same base (Fig. 2(c)). Since the height of the pyramid Tλ adjacent to vλ is shorter than the height of T adjacent to v, the apex angle of Tλ is larger than that of T . Therefore, the flattenable degree dv = 2π − θvλ decreases. For a saddle vertex v where dv < 0 (see Figs. 1 and 3), if the vertex v moves to any point on the line CD (e.g., v1 ), or any point on the line AB (e.g., v2 ), its flattenable degree dv will become zero. This is because that the summed inner angle at any point on line AB or CD is always 2π. Since lines CD and AB are on the convex hull of the triangles in-

In the classical Laplacian evolution vk+1 = vki + λL(vki ), k = 0, 1, 2, · · · i

(10)

for every vertex, λ ∈ [0, 1] is selected manually by users. Since the classical Laplacian evolution processes mesh vertices one by one, very large meshes can be computed. However, as illustrated in Figs. 5 and 6, the classical Laplacian evolution does not guarantee the decreasing of energy. Hence, it must be improved to be controllable, so that after each vertex adjustment, the energy function of Eq. (4) will decrease, and then the mesh surface S is more flattenable. As stated above, only the inner vertices need to be adjusted when processing a mesh. According to the sign of the flattenable 3

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cident to the saddle vertex v, by adjusting v along the Laplacian operator with suitable value of λ, v will approach v1 or v2 . In other words, the flattenable degree dv2 will become smaller. The analysis above has shown that the flattenable degree at a convex, concave, and saddle vertex will decrease if we move the vertex along the direction of Laplacian operator. Moreover, to obtain the optimal step that makes the flattenable degree smallest, we need to find an optimal value of λ. Note that, the adjustment of vertex v influences not Figure 3. A saddle vertex. only the flattenable degree of v but also the flattenable degrees of its 1-ring neighbors v0 , v1 , · · · , vn−1 (Fig. 4). Hence, the objective function of the local optimization problem should be

In this way, the global optimization problem Eq. (11) is converted into a sequence of controllable Laplacian evolution procedures defined in Eq. (12) and Eq. (13) at each inner vertex. Since the conFigure 4. Vertex v and its 1trollable local evolution has ring neighbors. only one variable, it can be computed efficiently. As a result, the controllable evolution is able to process mesh surfaces with large number of vertices, as illustrated in section 4. 3.1 Derivative of the Energy Function In our implementation, we adopt the subspace trust region method based on the interior-reflective Newton algorithm [22,23] to solve Eq. (13). The exact derivative formula of bv (λ) makes the subspace trust region method reliable to find the optimum of Eq. (13). That is, b0v (λ) = 2dvλ dv0 λ +

n−1

bv (λ) = dv2λ + ∑ dv2j .

(11)

j=0

s.t. λl < λ < λu .

(12)

2dv j θ0v j . (15)

(16) where α j = ∠v j vλ v j+1 , j = 0, 1, · · · , n − 1, and θ0v j (λ) = Ã

(13)

j (v − v j ) · L(v j ) − (vλ − v j−1 ) · L(v j ) ° °° ° − csc β1 λ °vλ − v j ° °v j − v j−1 °

In our implementation, we choose λl = 0 and λu = 1. As aforementioned, the flattenable degree at vertex vλ is dvλ = 2π − θvλ , and the summed inner angle of vertex vλ can be computed by (see Fig. 4)

j

+ cot β1 Ã

j

kvλ − v j k2 + kvλ − v j+1 k2 − kv j − v j+1 k2 , 2kvλ − v j k · kvλ − v j+1 k j=0 (14) where the index j is cycled. Other flattenable degrees dv j at v0 s 1-ring neighbors, can be calculated similarly.

j

+ cot β2

∑ arccos

j

!

(vλ − v j ) · L(v j ) ° ° °vλ − v j °2

+ − csc β2

n−1

θvλ (λ) =

∑

v j ∈N(vλ )

n−1 µ (v − v j ) · L(v) + (vλ − v j+1 ) · L(v) θ0vλ (λ) = ∑ − csc α j λ kvλ − v j k · kvλ − v j+1 k j=0 !! Ã (vλ − v j ) · L(v) (vλ − v j+1 ) · L(v) , + cot α j ° ° + ° ° °vλ − v j °2 °vλ − v j+1 °2

for each inner vertex v, where λ is to be determined by solving the following non-linear least square problem which has only one variable, minλ bv (λ),

2dv j dv0 j = −2dvλ θ0vλ −

We further derive the formula of θ0vλ (λ) as

Then, the optimization problem Eq. (5) can be converted into the controllable Laplacian evolution vλ = v + λL(v)

∑

v j ∈N(vλ )

(vλ − v j ) · L(v j ) − (vλ − v j+1 ) · L(v j ) ° °° ° °vλ − v j ° °v j − v j+1 °

!

(17)

(vλ − v j ) · L(v j ) , ° ° °vλ − v j °2 j

where β1 = ∠v j−1 v j vλ , β2 = ∠vλ v j v j+1 , and j = 0, 1, · · · , n − 1(Fig. 4). Again, the index j here is cycled. By substituting 4

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Eq. (16), Eq.(17), as well as the flattenable degrees dvλ , dv j into Eq. (15), the derivatives of bλ (v) is obtained.

Figure 5. Compare the convergence of the controllable Laplacian evolution (blue) and the classical Laplacian evolution (magenta).

Figure 7. Some patches of a wetsuit (a,c) and their processed counterparts (b,d).

and Eavg =

(19)

where n is the number of inner vertices on a mesh surface. We also display a discrete Guassian curvature map in all the figures of results to show the variation of flattenability on the meshes, where red color indicates high Guassian curvature (therefore non-flattenable), and blue represents low Guassian curvature (thus flattenable). First of all, to verify the convergency of the controllable Laplacian evolution, two data curves are shown in Fig. 5, where the x-axis is the time, and the y-axis is the maximum flattenable degree Emax . The magenta curve records Emax in each step when processing a given mesh surface shown in Fig. 6 by using the classical Laplacian evolution with a fixed λ = 0.5. The blue curve records Emax in each iteration on the same surface using controllable Laplacian. Also, referring to Fig. 6, the maximum flattenable degree of the original patch is Emax = 0.1271 before processing (Fig. 6(a)). After 50 steps of controllable Laplacian evolution, Emax is reduced to 0.0383 (Figs. 5 and 6(b)). However, after 50 times Laplacian evolution with λ = 0.5, Emax does even raise to 0.2357 (Figs. 5 and 6(c)). From Figs. 5 and 6, we can see that, different from the classical Laplacian evolution,

Figure 6. process the original patch by the controllable Laplacian evolution and the classical Laplacian evolution, respectively. (a.) Original patch; (b.) result by controllable Laplacian evolution; (c.) result by the classical Laplacian evolution with λ = 0.5.

4

Results and Discussion The controllable Laplacian evolution method has been implemented using VC++ 6.0. All results presented in this paper are generated on a PC with Intel 2.4GHz Quad-Core CPU and 2GB RAM. To measure the effectiveness of our method, we define two error terms. One is the maximum flattenable degree Emax , and other is the average flattenable degree Eavg . Emax = maxv {|dv |}

∑v |dv | , n

(18) 5

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Figure 9.

A more dense mesh with 195093 vertices shown in (a) and its processed counterpart in (b).

Figure 8. Two dense meshes shown in (a) and (c) and their processed counterparts (b) and (d).

Eavg = 0.00004. After ten steps processing, Emax is reduced to 0.0228 and Eavg is reduced to 0.00003. The computational statistics of all examples are listed in Table 1. From the statistics, we can conclude that the controllable Laplacian evolution method can reduce the flattenable degrees within a limited iteration steps (less than 10) for varieties of mesh surfaces. In general, the flattenable degrees of processed mesh surfaces can be reduced by around five times, and some of them can even be reduced by more than 30 times (Figs. 7(d) and 9(b)).

which cannot be guaranteed to converge, the controllable Laplacian evolution can ensure that the mesh surface is more flattenable after each evolution. Fig. 7 illustrates some patches from a wetsuit and the processed results. The patch in Fig. 7(a) has 4234 vertices with Emax = 0.3212, Eavg = 0.0055. After ten steps of evolution, the processed result is with Emax = 0.0999 and Eavg = 0.0020 (Fig. 7(b)). In Fig. 7(c), the original patch has 5088 vertices with Emax = 0.6319 and Eavg = 0.0018. The processed patch with Emax = 0.0231 and Eavg = 0.0010 is generated after ten steps of evolution (shown in Fig. 7(d)). As mentioned above, the controllable Laplacian evolution method can process dense mesh. Fig. 8 shows some processed patches with dense meshes. The patch in Fig. 8(a) has 12859 vertices with Emax = 0.0241 and Eavg = 0.00019. After four steps processing, Emax is reduced to 0.0036 and Eavg is reduced to 0.00016 (see Fig. 8(b)). Another patch in Fig. 8(c) has 59513 vertices, with Emax = 0.3490 and Eavg = 0.00020. After ten steps processing, Emax is reduced to 0.0756 and Eavg is reduced to 0.00017 (shown in Fig. 8(d)). Our method can even process more dense meshes. In Fig. 9, the mesh patch has 195093 vertices with Emax = 0.6988 and

5 Conclusion and Limitation In this paper, the controllable Laplacian evolution method is developed to reduce the flattenable degree of a mesh patch, especially those meshes with a large number of vertices. Different from the classical Laplacian evolution method, where the adjustment step λ is specified manually, the step λ in the controllable Laplacian evolution method is computed by solving a non-linear least square problem locally. It ensures that after adjusting each vertex, the energy function measuring the flattenable degree of a mesh will decrease. Since each non-linear least square problem has only one variables, it can be solved efficiently. As a result, the controllable Laplacian evolution method is able to process dense mesh efficiently. 6

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directions, which has more degree of freedom. Compared with this flexible vertex movement, moving vertices along Laplacian directions cannot be guaranteed to minimize flattenable degrees and then prevents the minimization. To verify the analysis above, we did some tests on a saddle shape mesh surface S shown in Fig. 11(a). Following Eq.(12) in [2], we compute the gradient ∂θ(v) ∂v of the summed inner angle θ(v) at each inner vertex v on S. In each iteration, if v moves along the the direction of - ∂θ(v) ∂v , eventually the flattenable degree of v will converge to zero. In contrast, if v moves along the direction which is orthogonal to ∂θ(v) ∂v , the minimization will be stuck. According to this, we evaluate the dot product of directions of Laplacian operator and ∂θ(v) ∂v on each v, and highlight those vertices on which the two directions are always nearly orthogonal (see Fig. 11(c)). As a result, moving vertices along such directions can never converge to the optimum of flattenable degrees. To overcome this problem, we need to find a more flexible way to move vertices. This problem will be addressed in our future work.

Figure 10. Comparison with method in [2]. Original patches are shown in (a,d), and results processed by our method is shown in (b,e) vs results shown in (c,f) processed by method in [2].

Acknowledgement The research conducted in this paper is supported by Shun Hing Institute of Advanced Engineering (SHIAE) Research Grant (CUHK/8115022). The first author of the paper is also supported by 973 program of China (No. 2004CB719400) and NSF of China (No. 60736019). ∂θ(v)

Figure 11. Analysis on directions of Laplacian operator and ∂v . (a) Original mesh; (b) processed mesh; (c)highlighted vertices at which the two directions are always near orthogonal.

REFERENCES [1] Wang C.C.L. Towards flattenable mesh surfaces. Computer-Aided Design, 40(1), 109-122, 2008. [2] Wang C.C.L. A least-norm approach to flattenable mesh surface processing. IEEE International Conference on Shape Modeling and Applications, 131-138, New York, 2008. [3] do Carmo M. Differential Geometry of Curves and Surfaces. Prentice-Hall, Englewood Cliffs, NJ, 1976. [4] Chu C. and Sequin ´ C.. Developable Bezier ´ patches: properties and design. Computer-Aided Design, 34(7), 511-527, 2002. [5] Leopoldseder S. and Pottmann H. Approximation of developable surfaces with cone spline surfaces. Computer-Aided Design, 30(7), 571-582, 1998. [6] Pottmann H. and Wallner J. Approximation algorithms for developable surfaces. Computer Aided Geometric Design, 16(6), 539-556, 1999. [7] Chen H., Lee I., Leopoldseder S., Pottmann H., Randrup T., and Wallner J. On surface approximation using developable surfaces. Graphical Models and Image Processing, 61(2), 110-124, 1999.

The major limitation of this work is that the optimization of flattenable degrees may be stuck at a certain level. For all the models we tested, the flattenable degrees will decrease significantly in the first few iteration steps, and then converge but never reach the optimum. To evaluate the convergency of flattenable degrees, we take the approach introduced in [2] as our bench mark. From the color maps shown in the last column of Fig. 10, the approach in [2] gives much better results than ours. Both the Emax and Eavg of the processed results by the approach of [2] are less than 10−4 , which are quite near the optimum. According to Table 1, none of the results can converge to this level by our approach. The reason why the convergence of our approach is not good as [2] can be analyzed from the perspective of vertices moving direction during computation. We minimize the energy function in Eq. (13) by moving every inner vertex along the direction of its Laplacian operator. However, in [2], the vertices can move in all 7

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Table 1.

Experimental Data of the Controllable Laplacian Evolution Algorithm

Models

#Vertices number

Fig.7(a),7(b)

Before processing

After processing

#Iterations

Time(in seconds)

0.0020

10

28

0.0231

0.0010

10

44

0.00019

0.0036

0.00016

4

4

0.3490

0.00020

0.0756

0.00017

10

97

0.6988

0.00004

0.0228

0.00003

10

159

Emax

Eavg

Emax

Eavg

4234

0.3212

0.0055

0.0999

Fig.7(c),7(d)

5088

0.6319

0.0018

Figs.8(a),8(b)

12859

0.0241

Figs.8(c),8(d)

59513

Figs.9(a),9(b)

195093

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[20] Taubin, G. A signal processing approach to fair surface design. In SIGGRAPH’95 Proceedings, 351-358, 1995. [21] Wang X., Cheng F., and Barsky B. Energy and B-spline interproximation. Computer-Aided Design, 29(7), 485-496, 1997. [22] Coleman T.F. and Li Y. On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds. Mathematical Programming, 67(2), 189-224, 1994. [23] Coleman, T.F., Li Y. An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds. SIAM Journal on Optimization, 6, 418-445, 1996.

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