976 Y.Guo et al.Computers Graphics 29 (2005)972-979 (a) (b) (c) Fig.4.(a)On the 3D mesh model,the green vertices belong to 1-ring of vi;the blue are 2-ring of vi.(b)The green vertices are 2-ring region of vi.(c)On the texture plane,t;is the texture point corresponding to ti.A,A2 of v;is the sum of displacement of the green points,blue points,respectively,during each adjustment of t;on the texture plane. Assume that vi,P,are a pair of constrained feature Use harmonic map (as in Section 3.2)to adjust points on the 3D mesh and the texture plane that need to those texture coordinates of the vertices included be adjusted.Normally,we adjust the positions of p on in k-ring region of ti: the texture plane to fit ti. Compute△of vi;△=△k; For description convenience,we give the following =k+1; notations for a 3D mesh model (see Fig.4): if k>K return; end topological distance between vi and vi:the number of end edges on the shortest path connecting vi with v. k-ring of vi:those vertices whose topological distance Since our approach involves solving low rank sparse to vi is k. equations,it is efficient and can be performed in real- k-ring region of v:those vertices whose topological time.If several pairs of feature points on the 3D model distance from v is not greater than k. and the texture plane need to be adjusted,the above .Ak of v:total texture coordinates'variance of the approach can be performed iteratively.Fig.5 shows an vertices within the k-ring of vi. example of real-time adjustment. After the user adjusts p,our algorithm first re- calculates the texture coordinates of vertices within the 5.Experimental results 1-ring of vi by using the harmonic model described in Section 3 and solving a set of equations whose rank is We employed several typical models and textures to equal to the vertex number of the I-ring,while the test our algorithm on an Intel Pentium IV 1.6GHz PC texture coordinates of vertices outside the 1-ring with 256 MB main memory under the Windows XP remain unchanged;if Al of t is smaller than a given operating system (In Figs.2,6-8,the red dots are the threshold,the algorithm terminates:otherwise the constraints).Table I lists the performance statistics of algorithm continues to adjust the texture coordinates our algorithm,including the number of triangles and of the 2-ring region of vi which involves solving vertices of the mesh model (#Tris/Vers),the number of equations of a higher rank.The above process is constraints specified (#Cons),the rank of Egs.(3)and iteratively conducted until the final variance is smaller (4)(#Egas),the number of iterations for solving the than the threshold or the iterative count reaches a equations (#Ites)and the computation time (Running maximum constant.This approach can be described time). briefly as follows. In Fig.6,the image of a tiger (a)and a leopard(c)are mapped onto a face model with 23 and 24 constraints, respectively.Fig.6(b)and (d)show two rendering Algorithm:Adaptive local mapping refinement results. Fig.7 shows the mapping of an image of girl shirt stepl Given a threshold and a positive constant K. onto a body model composed of 3016 triangles,it takes step2 Set k 1 and A=+o0. 0.472s to get the fine rendering result (Fig.7(c)). step3 while△>e Fig.8 shows the result of mapping a tiger image to the begin cow head model (the same one as Fig.3(c)),which hasAssume that vi, pi are a pair of constrained feature points on the 3D mesh and the texture plane that need to be adjusted. Normally, we adjust the positions of pi on the texture plane to fit vi. For description convenience, we give the following notations for a 3D mesh model (see Fig. 4):
topological distance between vi and vj: the number of edges on the shortest path connecting vi with vj.
k-ring of vi: those vertices whose topological distance to vi is k.
k-ring region of vi: those vertices whose topological distance from vi is not greater than k.
Dk of vi: total texture coordinates’ variance of the vertices within the k-ring of vi. After the user adjusts pi, our algorithm first re￾calculates the texture coordinates of vertices within the 1-ring of vi by using the harmonic model described in Section 3 and solving a set of equations whose rank is equal to the vertex number of the 1-ring, while the texture coordinates of vertices outside the 1-ring remain unchanged; if D1 of vi is smaller than a given threshold, the algorithm terminates; otherwise the algorithm continues to adjust the texture coordinates of the 2-ring region of vi which involves solving equations of a higher rank. The above process is iteratively conducted until the final variance is smaller than the threshold or the iterative count reaches a maximum constant. This approach can be described briefly as follows. Algorithm: Adaptive local mapping refinement step1 Given a threshold
and a positive constant K. step2 Set k ¼ 1 and D ¼ þ1. step3 while n4
begin Use harmonic map (as in Section 3.2) to adjust those texture coordinates of the vertices included in k-ring region of vi; Compute Dk of vi; D ¼ Dk; k ¼ k þ 1; if k4K return; end end Since our approach involves solving low rank sparse equations, it is efficient and can be performed in real￾time. If several pairs of feature points on the 3D model and the texture plane need to be adjusted, the above approach can be performed iteratively. Fig. 5 shows an example of real-time adjustment. 5. Experimental results We employed several typical models and textures to test our algorithm on an Intel Pentium IV 1.6 GHz PC with 256MB main memory under the Windows XP operating system (In Figs. 2, 6–8, the red dots are the constraints). Table 1 lists the performance statistics of our algorithm, including the number of triangles and vertices of the mesh model (#Tris/Vers), the number of constraints specified (#Cons), the rank of Eqs. (3) and (4) (#Eqas), the number of iterations for solving the equations (#Ites) and the computation time (Running time). In Fig. 6, the image of a tiger (a) and a leopard (c) are mapped onto a face model with 23 and 24 constraints, respectively. Fig. 6(b) and (d) show two rendering results. Fig. 7 shows the mapping of an image of girl shirt onto a body model composed of 3016 triangles, it takes 0.472 s to get the fine rendering result (Fig. 7(c)). Fig. 8 shows the result of mapping a tiger image to the cow head model (the same one as Fig. 3(c)), which has ARTICLE IN PRESS Fig. 4. (a) On the 3D mesh model, the green vertices belong to 1-ring of vi; the blue are 2-ring of vi. (b) The green vertices are 2-ring region of vi. (c) On the texture plane, ti is the texture point corresponding to vi. D1, D2 of vi is the sum of displacement of the green points, blue points, respectively, during each adjustment of ti on the texture plane. 976 Y. Guo et al. / Computers & Graphics 29 (2005) 972–979