852 QIN Kaihuai, CHANG Zhengyi et al Vol7 are generated by the recursive subdivision Each subdivision step will bring new vertices, new edges, as well

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1 Vol7 No6 J Comput Sci & Technol Nov 22 Physics-ased Loop Surface Modeling QIN Kaihuai ( ΠΞ), CHANG Zhengyi ( ffω), WANG Huawei (ΦΛΨ) and LI Denggao (± ) Department of Computer Science and Technology, Tsinghua University, eijing 84, PR China Abstract qkh-dcstsinghuaeducn Received July 5, 2; revised September 2, 2 Strongly inspired by the research on physics-based dynamic models for surfaces, we propose a new method for precisely evaluating the dynamic parameters (mass, damping and stiffness matrices, and dynamic forces) for Loop surfaces without recursive subdivision regardless of regular or irregular faces It is shown that the thin-plate-energy of Loop surfaces can be evaluated precisely and efficiently, even though there are extraordinary points in the initial meshes, unlike the previous dynamic Loop surface scheme Hence, the new method presented for Loop surfaces is much more efficient than the previous schemes Keywords subdivision, Loop surface, physics-based modeling Introduction Free-form curves and surfaces are very important and common in CAGD and CAD/CAM The Non-Uniform Rational -Spline (NURS) is a very popular tool, but there are too many parameters in NURS curves and surfaces Generally, a surface of arbitrary topology cannot be represented by a single NURS surface It is difficult for users to know how to set the knot spacings, how to manipulate the control vertices and how to modify the weights In addition, NURS surfaces are defined on a rectangular parametric domain This has hampered their widespread uses in modeling shapes On the other hand, a subdivision surface can represent any complex surface of arbitrary topology Physics-based subdivision surfaces allow users to physically modify shapes of surfaces at desired locations via forces applied Furthermore, physics-based modeling gives the users an intuitive and natural feeling for geometric modeling with clay dough Thus, in recent years, attention has been paid to physics-based dynamic subdivision surfaces [ 3] In 997, Qin, Mandal and Vemuri introduced the physical quantity" into dynamic Catmull-Clark surfaces and successfully applied it in visualization of medical data [2] ut in case of extraordinary points, the faces that contain an extraordinary point were subdivided recursively to get the mass, damping and stiffness matrices Hence not only does the number of equations of the numerical integrals increase quickly, but also the result is approximate This is because their approach used for computing the thin-plate-energy of Catmull-Clark surfaces diverges sometimes [2;4] In 2, Mandal, Qin and Vemuri [] gave an approximate solution to the physical matrices of Loop surfaces They failed to get an exactly analytic solution of the dynamic matrices like mass, damping and stiffness ones, and only got the approximations of such physical matrices by a spring-mass system Qin, Wang and Li, et al presented efficient methods for precisely computing the dynamic parameters of physics-based catmull-clark surfaces [3] In this paper, a new method for physics-based Loop surfaces is presented, the thin-plate-energy of Loop surfaces is evaluated precisely, and the physical matrices are evaluated exactly and analytically 2 Loop Subdivision Surfaces In 987, Loop generalized the box splines to irregular meshes [5] The surfaces defined by the control meshes that have extraordinary points can be processed efficiently by Loop's scheme Patches Supported by the National Natural Science Foundation of China (Grant Nos and ) and the Research Fund for the Doctoral Program of Higher Education from China Ministry of Education (Grant No2348)

2 852 QIN Kaihuai, CHANG Zhengyi et al Vol7 are generated by the recursive subdivision Each subdivision step will bring new vertices, new edges, as well as new triangles Loop's subdivision rules are listed as follows [5;6] : ffl New vertex points Suppose that for each non-boundary vertex V whose valence is N A new vertex is introduced corresponding to the old one, and the positions of the new vertex is given by (ff(n)v + V + V V N )=(ff(n)+n); ff(n) = (N( fi(n)))=fi(n); fi(n) = 5=8 (3 + 2 cos(2ß=n)) 2 =64: V is the position of vertex V, and V ; V 2 ;:::;V N are the vertex positions of N vertices connected to the vertex V ffl New edge points For each non-boundary edge E, a new edge point is introduced by (3(V + V 2 )+V F + V F2 )=8, V and V 2 are the end points of the edge E, V F and V F2 are the third points of the two triangles sharing the edge E, respectively ffl New edges New edges are formed by connecting each new vertex point to the new edge points corresponding to the old edges incident on the old vertices, and connecting each new edge point to the new edge points of other edges in two faces which share the original edge ffl New triangles New triangles are defined as faces enclosed by the new edges 3 Physics-ased Deformable Loop Surface Model In order to make the derivation below as clear and compact as possible we adopt the following notational conventions: ffl All the matrices are denoted by bold capital letters, eg, M; D; K ffl The element of the i-th row, j-th column of matrix M is denoted by (M) ij ffl The vectors are denoted by bold lowercase letters, eg, c 3 Parametric Expression of Loop Surfaces Each patch of a Loop subdivision surface [5;6] can be defined by 2 vertices Lai [7] got the basis functions of the 2 vertices by transforming the box splines into triangular ézier patches All the basis functions are shown in Appendix A In Fig, the shaded Loop patch has the formula [6] as follows: s(v; w) = C T s b(v; w); (v; w) 2 Ω; Ω = f(v; w)jv 2 [; ] and w 2 [; v]g, C T s = (c ;:::;c 2 ) is a 3 2 matrix whose elements are the coordinates of the 2 vertices, b(v; w) is a vector composed of the 2 basic functions (see Appendix A) For an irregular patch (with a vertex whose valence is not equal to 6) as shown in Fig2 Let K = N + 6 and M = K + 6, N is the valence of the extraordinary point, K is the number of vertices that define the shaded face The coordinates of the surface at any parameters (v; w) can be computed by eigenanalysis as follows [6] : s(v; w) = C T s V T Λ n (P k AV ) T b(t k;n (v; w)) () Λ is a diagonal matrix composed of the eigenvalues, V is the eigenvector matrix corresponding to the eigenvalues (see Appendix C), A is the extended subdivision matrix, P k (k = ; 2; 3) are 2 M picking" matrices, and the transformations t k;n (k = ; 2; 3) map Ω n k to the unit triangles (see Fig3): t ;n (v; w) = (2 n v ; 2 n w); t ;n 2 Ω n ; t 2;n (v; w) = ( 2 n v; 2 n w); t 2;n 2 Ω n 2 ; t 3;n (v; w) = (2 n v; 2 n w ); t 3;n 2 Ω n 3 : The definitions of A, A and P k can be found in Appendix

3 No6 Physics-ased Loop Surface Modeling 853 Fig A Loop patch Fig2 An irregular triangular patch defined by K = N +6= 3 control vertices with an extraordinary vertex of valence 7 () can be rewritten as follows: s T (v; w) = J Λ C s ; J Λ = b T (t k;n (v; w))x k Λ n V ; X k = P k AV : If the mesh has m vertices and d faces, then C T s = J i C s = J = J Λ R i C = JC ; Fig3 The parametric domain of Loop subdivision J Λ R i : (2) = (c ;:::;c m ) is a matrix whose elements are the coordinates of the m vertices R i is a k m picking" matrices (each row has only one " element, the others are ) 32 Dynamics Equation of Loop Subdivision Surfaces Consider the vertex positions C as time-varying variables, then the dynamics equation can be expressed as follows: M C + D _C + KC = F p (3) M; D and K are the mass, damping and stiffness matrices of the physical model, respectively, and F p is the generalized force vector Let μ be the density of mass, fl be the coefficient of damping, then the mass and damping matrices are expressed respectively as follows: 8 >< >: M = D = μj T Jdw flj T Jdw The stiffness matrix can be obtained by computing the thin-plate-energy" of each face: K = (ff J T v J v + ff 22 J T wj w + fi J T vvj vv + fi 2 J T vwj vw + fi 22 J T wwj ww )dw ff ii and fi ij are the tension and the rigidity functions Generally, μ; fl; ff ii ;fi ij are the functions of v and w Let f(v; w; t) be the force imposed at the point (v; w), then the force vector F p can be expressed as follows: F p = J T f T (v; w; t)dw: In general, the force f(v; w; t) is equal to a constant on a small face subdivided (4)

4 854 QIN Kaihuai, CHANG Zhengyi et al Vol7 33 Evaluation of Loop Surfaces Suppose the triangular faces subdivided are small, such that μ; fl; ff ii and fi ij can be regarded as constants Combining (2) and (4) yields M = μj T Jdw = μ(j Λ R i ) T (J Λ R i )dw = μ i is the mass density of the i-th face Similarly, we can obtain the damping matrix D: D = fl i R T i J ΛT J Λ dw Ri fl i is the damping coefficient of the i-th face y evaluating μ i R T i J ΛT J Λ dw Ri J ΛT J Λ dw, one can get the matrices M and D Note that, the integral can be calculated on the sub-domains Ω i, Ω i 2 and Ω i 3 (see Fig3) Hence we have the following expressions: Ω Ω 2 J ΛT J Λ dw = J ΛT J Λ dw = Ω V T Λ X T b(2v ; 2w)b T (2v ; 2w)X Λ V dw Z = V T Λ X T = V T Λ X T 4 Ω 2 = V T Λ X T = V T Λ X T =2 Z b(2v ; 2w)b T (2v ; 2w)dw X Λ V X Λ V ; V T Λ X T b(4v ; 4w)b T (4v ; 4w)X Λ V dw Z =2 =4 4 2 Z Thus, from the above we can deduce that Ω n J ΛT J Λ dw = 4 n V T Λ n X T Similarly, we can obtain the following formulae: Ω n 2 Ω n 3 J ΛT J Λ dw = 4 V T n Λ n X T 2 J ΛT J Λ dw = 4 n V T Λ n X T 3 Z =2 v Z Z Z Hence, we can obtain the following unified expression: Ω n k J ΛT J Λ dw = 4 V T n Λ n X T k Let Z k = X T k Z Z b(4v ; 4w)b T (4v ; 4w)dw X Λ V X Λ V : X Λ n V : X 2 Λ n V ; X 3 Λ n V : Xk Λ n V ; k = ; 2; 3: Xk and Q P k = n= 4 n Λn Z k Λ n, the elements of the i-th row, j-th column of the matrices Q k and Z k are denoted by (Q k ) ij and (Z k ) ij, respectively, then we have J ΛT J Λ dw = 3X X k= n= Ω n k J ΛT J Λ dw = 3X k= V T Q k V

5 No6 Physics-ased Loop Surface Modeling 855 It is clear that (Q k ) ij = 4 X n= i j n (Zk ) ij (Z k ) ij : 4 4 i j J ΛT J Λ dw can be evaluated precisely and efficiently Since the resulting matrices are only dependent on the valence N of the extraordinary point, they can be evaluated during preprocessing Similarly, we can precisely and efficiently evaluate K and F p by computing in aance J ΛT v J Λ v dw, J ΛT w J Λ w dw, J ΛT vv J Λ vv dw, J ΛT vwj Λ vw dw, J ΛT wwj Λ ww dw and J ΛT dw, respectively Thus, we can precisely and quickly evaluate the dynamic parameters M, D, K, F p for Loop surfaces, and this is a novel contribution of this paper 34 Solving the Differential Equations We can use the difference method to solve (3) If t is small enough, then C (t + t) = C (t + t) 2C (t)+c (t t) ; _C t 2 (t + t) = C (t + t) C (t t) 2 t Then, the differential equation can be written as (2M + D t + 2 t 2 K)C (t + t) = 2 t 2 F p (t + t)+(d t 2M)C (t t)+4mc (t): The above linear system of equations can be solved quickly by the LU decomposition of the coefficient matrix Since F p is timely-varying, F p should be re-evaluated at each iteration step 35 Application Examples There are two examples dynamically generated by the presented method as follws: Fig4 A vase deformed (a) Initial shape (b) Deformed shape after seconds (c) Deformed shape after 5 seconds Fig5 Dynamically modeling a kettle (a) Initial shape (b) Deformed shape after 5 seconds (c) Deformed shape after seconds

6 856 QIN Kaihuai, CHANG Zhengyi et al Vol7 Example Fig4 shows a vase The initial control vertices, the Loop surface and 6 initial forces on the boundary of the surface are shown in Fig4(a); the deformed shapes are shown in Fig4(b) and (c), respectively Example 2 Fig5 shows a kettle" generated dynamically Fig5(a) shows the initial shape, control vertices and force on the extraordinary point whose valence is 5; Fig5(b) and (c) show the deformed surfaces after 5 and seconds, respectively 4 Computational Complexity First, let us recall the previous physics-based Loop surfaces [] proposed by Mandal, Qin and Vemuri They used 4 j triangular faces obtained after j subdivision steps to approximate the smooth patch in the limit surface corresponding to a face in the initial mesh In their method, the mass matrix M is calculated using the following formula (the same way for calculating D, K and f p ): M abc = kx μ(v j i )fj abc (vj i )gt f j abc (vj i )g; k is the number of vertices of the 4 j faces, v j i is the i-th vertex of the 4 j faces in the j-th level mesh, j is the collection of abc basic functions at the vertices v, which is of dimension 3r and is abc the concatenation of the (x; y; z) positions for all r vertices that are within the 2-neighourhood of the vertices a, b, c in the initial mesh as show in Fig6 Note that the cost for calculating M is dependent on the subdivision level j, all the matrices (M; D; K and f p ) must be calculated Fig6 Points a; b; c and their 2-neighbourhoods again in the next subdivision step, and the results are approximative At level j of the subdivision, the number of vertices of the 4 j faces k = 3j+ + 2 Suppose that 2 there are m triangular faces in the initial mesh, then the computational complexity of M is O(m3 j ), increasing exponentially However, the complexity of the method proposed in the paper for calculating M is O(m), depending only on the number of faces in the initial mesh In addition, all the matrices (M; D; K and f p ) can be calculated exactly and saved in aance So the computational complexity of the new method is a constant, much better than the previous method 5 Conclusions and Future Work In this paper, based on the dynamic model, we can obtain the precise solution of physics-based Loop surfaces We have proposed a unified method to evaluate the physical parameters of the dynamic model regardless of a regular or an irregular face The matrices M; D; K and F p can be obtained in much less time than the previous schemes, because they can be calculated using the analytic formulae in our method without recursive subdivision and numerical integrals [] So our method is much more efficient The thin-plate-energy of Loop surfaces can be evaluated precisely and analytically, even though there are extraordinary points in the control meshes The method proposed can be used easily for modeling any complex shapes of surfaces of arbitrary topology, and the shapes can be modified conveniently It is easy to be implemented It can also be used in medical visualization, which is our future work References [] Mandal C, Qin H, Vemuri C A novel FEM-based dynamic framework for subdivision surfaces Computer-Aided Design, 2, 32: [2] Qin H, Mandal C, Vemuri C Dynamic Catmull-Clark subdivision surfaces IEEE Transactions on Visualization and Computer Graphics, 998, 4(3): [3] Qin K, Wang H, Li D, Kikinis R, Halle M Physics-based subdivision surface modeling for medical imaging and simulation In Proc of MIAR'2, IEEE Computer Society Press, Hong Kong, 2, pp7 24

7 No6 Physics-ased Loop Surface Modeling 857 [4] Halstead M, Kass M, DeRose T Efficient, fair interpolation using Catmull-Clark surfaces Computer Graphics (Proceedings of SIGGRAPH'93), 993, pp35 44 [5] Loop C Smooth subdivision surfaces based on triangles [Thesis] Department of Mathematics, University of Utah, 987 [6] Stam J Evaluation of Loop subdivision surface In Proceedings CD of SIGGRAPH'98, 998 [7] Lai M J Fortrain subroutines for -nets of box splines on three- and four-directional meshes Numerical Algorithms, 992, 2: QIN Kaihuai is a professor in Department of Computer Science and Technology, Tsinghua University His research interests include graphics, computer data visualization, image processing, computer aided geometric design, physics-based geometric modeling, virtual reality and CAD/CAM CHANG Zhengyi is a software engineer He got his bachelor's degree in Hunan University in 988 He is now a graduate student in the Department of Computer Science and Technology, Tsinghua University His research interests include computer graphics and computer aided geometric modeling WANG Huawei is a PhD candidate in the Department of Computer Science and Technology, Tsinghua University His research interests include computer graphics, computer aided geometric design and physicsbased geometric modeling LI Denggao is a PhD candidate in the Department of Computer Science and Technology, Tsinghua University His research interests include computer graphics, computer aided geometric design and physicsbased computer aid geometric modeling Appendix A The vector b(v; w) contains 2 bi-quadratic ezier basis functions as follows [7] : b T (v; w) = 2 (u4 +2u 3 v; u 4 +2u 3 w; u 4 +2u 3 w +6u 3 v +6u 2 vw +2u 2 v 2 +6uv 2 w +6uv 3 +2v 3 w + v 4 ; 6u 4 u = v w +24u 3 w +24u 2 w 2 +8uw 3 + w 4 +24u 3 v +6u 2 vw +36uvw 2 +6vw 3 +24u 2 v 2 +36uv 2 w +2v 2 w 2 +8uv 3 +6v 3 w + v 4 ;u 4 +6u 3 w +2u 2 w 2 +6uw 3 + w 4 +2u 3 v +6u 2 vw +6uvw 2 +2vw 3 ; 2uv 3 + v 4 ;u 4 +6u 3 w +2u 2 w 2 +6uw 3 + w 4 +8u 3 v +36u 2 vw +36uvw 2 +8vw 3 +24u 2 v 2 +6uv 2 w +24v 2 w 2 +24uv 3 +24v 3 w +6v 4 ;u 4 +8u 3 w +24u 2 w 2 +24uw 3 +6w 4 +6u 3 v +36u 2 vw +6uvw 2 +24vw 3 +2u 2 v 2 +36uv 2 w +24v 2 w 2 +6uv 3 +8v 3 w + v 4 ; 2uw 3 + w 4 ; 2v 3 w + v 4 ; 2uw 3 + w 4 +6uvw 2 +6vw 3 +6uv 2 w +2v 2 w 2 +2uv 3 +6v 3 w + v 4 ;w 4 +2vw 3 ); Appendix The K K subdivision matrix A and the M K extended subdivision matrix A respectively have the following block forms: ψ S S A = ; A = S S 2 the matrix S is an (N +) (N +)matrix as follows [6] : S =! S S 2 S 2 S 22 a N b N b N b N b N b N b N b N b N b N c c d d c d c d c d c d c d c d c d c d c d d c C A ;

8 858 QIN Kaihuai, CHANG Zhengyi et al Vol7 a N = ff N ; b N = ff N =N; c = 3=8; d = =8; ff N = 5 (3 + 2 cos(2ß=n))2 ; S 2 = 6 2 C A ; S = C A ; S 2 = C A ; S22 = C A : 3 3 P k (k = ; 2; 3) is a 2 M picking" matrix, there is only one " in each line of P k, and other elements equal P k (k = ; 2; 3) are defined by introducing the following 3 vectors: q = (3; ;N +4; 2;N +;N +9;N +3;N +2;N +5;N +8;N +7;N + ); q 2 = (N +;N +7;N +5;N +2;N +3;N +6;N +; 2;N +4;N;; 3); q 3 = (;N;2;N +;N +6;N +3;N +2;N +5;N +2;N +7;N +;N + ): If the i-th element of q k equals j, then (P k ) ij = and other elements of the i-th row of P k all equal Appendix C The eigenstructure of matrix A is denoted by (V ; Λ), Λ is the diagonal matrix containing the eigenvalues of A, and V is an invertible matrix whose columns are the corresponding eigenvectors, that is AV = V Λ, [6] Λ = ±; ;U ; W can be evaluated as follows: ± ; V = U : U W ± = ρ diag(;μ2;μ 3;μ 3;:::;μ (N )=2;μ (N )=2); diag(;μ 2;μ 3;μ 3;:::;μ N=2 ;μ N=2 ; =8); if N is odd; if N is even μ = ;μ 2 = 5 ffn ; μ3 = f();:::;μn+ = f(n ); 8 and f(k) = ßk cos( 8 N ); = diag 8 ; 8 ; 8 ; 6 ; ; 6 W = C A ; U = C(k) = cos(2ßk=n); S(k) = sin(2ßk=n); 8 ffn 3 C() S() C(2) C(2) S(2) C(4) C(N ) S(N ) C(2(N )) C A ; U is computed by solving the linear system of equation S U + S 2U = U ±: